Journal of Applied Nonlinear Dynamics

Volume 7 Issue 2 June 2018

ISSN 2164‐6457 (print) ISSN 2164‐6473 (online)

Journal of Applied Nonlinear Dynamics

Journal of Applied Nonlinear Dynamics Editors J. A. Tenreiro Machado ISEP-Institute of Engineering of Porto Dept. of Electrical Engineering Rua Dr. Antonio Bernardino de Almeida 431, 4200-072 Porto, Portugal Fax:+ 351 22 8321159 Email: [email protected]

Albert C.J. Luo Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, IL 62026-1805 USA Fax: +1 618 650 2555 Email: [email protected]

Associate Editors J. Awrejcewicz Department of Automatics and Biomechanics K-16, The Technical University of Lodz, 1/15 Stefanowski St., 90-924 Lodz, Poland Fax: +48 42 631 2225, Email: [email protected]

Shaofan Li Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, CA 94720-1710, USA Fax : +1 510 643 8928 Email: [email protected]

C. Steve Suh Department of Mechanical Engineering Texas A&M University College Station, Texas 77843-3123 USA Fax:+1 979 845 3081 Email: [email protected]

Dumitru Baleanu Department of Mathematics and Computer Sciences Cankaya University Balgat, 06530, Ankara, Turkey Fax: +90 312 2868962 Email: [email protected]

Antonio M Lopes

Vasily E. Tarasov Skobeltsyn Institute of Nuclear Physics Moscow State University 119991 Moscow, Russia Fax: +7 495 939 0397 Email: [email protected]

Nikolay V. Kuznetsov Mathematics and Mechanics Faculty Saint-Petersburg State University Saint-Petersburg, 198504, Russia Fax:+ 7 812 4286998 Email: [email protected]

Miguel A. F. Sanjuan Miguel A. F. Sanjuan Department of Physics Universidad Rey Juan Carlos 28933 Mostoles, Madrid, Spain Email: [email protected]

UISPA-LAETA/INEGI Faculty of Engineering University of Porto Rua Doutor Roberto Frias, s/n, 4200-465 Porto, Portugal Email:[email protected]

Editorial Board Ahmed Al-Jumaily Institute of Biomedical Technologies Auckland University of Technology Private Bag 92006 Wellesley Campus WD301B Auckland, New Zealand Fax: +64 9 921 9973 Email:[email protected]

Giuseppe Catania Department of Mechanics University of Bologna viale Risorgimento, 2, I-40136 Bologna, Italy Tel: +39 051 2093447 Email: [email protected]

Liming Dai Industrial Systems Engineering University of Regina Regina, Saskatchewan Canada, S4S 0A2 Fax: +1 306 585 4855 Email: [email protected]

Alexey V. Borisov Department of Computational Mechanics Udmurt State University, 1 Universitetskaya str., Izhevsk 426034 Russia Fax: +7 3412 500 295 Email: [email protected]

Riccardo Caponetto DIEEI, (Electric, Electronics and Computer Engineering Department), Engineering Faculty University of Catania Viale A. Doria 6, 95125 Catania, Italy Email: [email protected]

Mark Edelman Yeshiva University 245 Lexington Avenue New York, NY 10016, USA Fax: +1 212 340 7788 Email: [email protected]

Continued on back materials

Journal of Applied Nonlinear Dynamics Volume 7, Issue 2, June 2018

Editors J. A. Tenreiro Machado Albert Chao-Jun Luo

L&H Scientific Publishing, LLC, USA

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Printed in USA on acid-free paper.

Journal of Applied Nonlinear Dynamics 7(2) (2018) 111-122

Journal of Applied Nonlinear Dynamics https://lhscientificpublishing.com/Journals/JAND-Default.aspx

Trajectory Controllability of Fractional-order α ∈ (1, 2] Systems with Delay V. Srinivasan†, N. Sukavanam Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand-247667, India Submission Info Communicated by J.A.T. Machado Received 24 January 2017 Accepted 15 May 2017 Available online 1 July 2018 Keywords Fractional derivatives and integrals Control systems Trajectory controllability Delay systems

Abstract This paper is concerned with trajectory controllability of a class of fractional-order systems of order α ∈ (1, 2] with delay in state variable and with a nonlinear control term. Firstly, the existence and uniqueness of the system is proved under suitable conditions on the nonlinear term involving state variable. Then the trajectory controllability of this class of systems is studied using Mittag-Leffler functions and Gronwall-Bellman inequality. Finally, examples are given to illustrate the proposed theory.

©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction Fractional-order systems are generalization of the classical integer-order systems. In recent years, many researchers believe that fractional-order systems are more appropriate to model the real world problems. In the following, two examples are shown to depict the importance of fractional-order systems. As a first example, let us consider the following simple population dynamics C

Dtα x(t) = ax(t), x(t0 ) = x0 ,

0 < α ≤ 1,

(1)

where x(t) represents the population at time t, x0 is the initial population at t = t0 , a = birth rate − death rate, and C Dtα is the Caputo fractional derivative of order 0 < α ≤ 1 (see, Definition 3). The solution of the system (1) can be written as follows: x(t) = Eα ,1 [a(t − t0 )α ]x0 ,

0 < α ≤ 1,

(2)

where Eα ,1 [a(t − t0 )α ] is the Mittag-Leffler function, which will reduce to exponential function ea(t−t0 ) when the fractional-order α = 1 (see, Definition 4). Suppose that z1 is the actual population for the year t1 . Then the estimated population for the same year t1 using (2) with α = 1 (integer-order) may not close to represents the actual population z1 . However, by taking the fractional-order 0 < α < 1 † Corresponding

author. Email address: [email protected] ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2018.06.001

112

V. Srinivasan, N. Sukavanam / Journal of Applied Nonlinear Dynamics 7(2) (2018) 111–122

in the equation (2), one can get a suitable α such that x(t1 ) is close to z1 , using (2). This illustrates that models based on fractional-order systems will yield better results than the integer-order systems. Consider the following system as a second example C

Dtα x(t) = δ t δ −1 , x(0) = 1, 0 < δ < 1,

0 < α ≤ 1,

(3)

The solution of (3) for α = 1 is x(t) = 1 + t δ . As t → ∞, x(t) approaches ∞ when 0 < δ < 1. The solution of (3) for 0 < α < 1 can be given as x(t) = 1 + Γ(δ + 1)/Γ(δ + α )t δ +α −1 . As t → ∞, x(t) approaches 1 when 0 < δ ≤ 1 − α . These imply that the fractional-order (0 < α < 1) system (3) is stable; whereas, the corresponding integer-order (α = 1) system (3) becomes unstable. The above observations clearly signify the importance of consideration of the fractional-order systems in place of integer-order systems. The applications of fractional-order systems are in many fields like signal processing, economics, population dynamics, viscoelastic materials, astrophysics and control theory (see, Bagley and Torvik [1], Kilbas et al [2] and Rivero et al [3]). The existence and uniqueness of solutions and numerical schemes of fractional-order dynamical systems can be found in Pitcher and Sewell [4], Diethelm [5], Miller and Ross [6] and Podlubny [7]. In nature a nonzero time delay occurs always between the instants at which a cause and its effects take place. Therefore the occurrence of delay in fractional dynamical systems seems natural to model real world problems. Fractional-order systems with delay have been studied by many authors. In 2008, Benchohra et al [8], in 2008, Lakshmikantham [9], and in 2008, Maraaba et al [10] studied the existence and uniqueness theorem for fractional-order differential equations with delay. In 2011, Bhalekar and Gejji [11] provided a numerical scheme for fractional-order differential equations with delay. A system is said to be controllable if and only if it is possible, by means of an input, to transfer the system from any initial state to any other state in a finite time. The controllability of fractional order systems with or without delay have been studied by many authors in 2008, Adams and Hartley [12], in 2010, Monje et al [13], in 2012, Surendra Kumar and Sukavanam [14] and in 2012, Wei [15]. A system is said to be trajectory controllable if and only if it is possible, by means of an input, to transfer the system from any initial state to any other desired state along a prescribed trajectory. Therefore trajectory controllability is stronger notion of controllability. In recent years many authors studied trajectory controllability of various kind of dynamical systems. In 2010, Chalishajar et al [16] studied trajectory controllability of abstract nonlinear integro-differential system in finite and infinite dimensional space settings. In 2013, Bin and Liu [17] studied trajectory controllability of semilinear evolutions equations with impulses and delay. In 2015, Klamka et al in [18] investigated the trajectory controllability of finite-dimensional semilinear systems with point delay in control and in nonlinear term. In 2016, Govindaraj et al [19] discussed the trajectory controllability of fractional-order α ∈ (0, 1] systems. In this paper we prove the trajectory controllability of fractional-order α ∈ (1, 2] system with delay. For that, we consider the following nonlinear fractional-order system C Dα x(t) = Ax(t) + B(t, u(t)) + f (t, x(t − τ ), x(t)), 0 0. When α = β = 1, one has E1,1 (z) = ez . Definition 5 (Joshi and Bose [20]). Let X be a reflexive real Banach space. A mapping F : X → X ∗ is said to be of type (M) if the following conditions hold: (a) If a sequence {xn } ∈ X converges weakly to x ∈ X and {Fxn } converges weakly to y ∈ X ∗ and lim sup(Fxn , xn ) ≤ (y, x), n

then Fx = y. (b) F is continuous from finite dimensional subspaces of X into X ∗ endowed with weak* topology. Lemma 1 (Joshi and Bose [20]). Let X be a real Banach space and F : X → X ∗ be a mapping of type (M). If F is coercive then the range of F is all of X ∗ .

114


3 Existence and uniqueness of solution In this section we prove the existence and uniqueness of solution of (4) by using method of steps. In what follows we assume the following conditions on the nonlinear functions B and f . (A1) B(t, u) is measurable with respect to t for all u ∈ L2 ([0, T ], Rn ) := U and continuous with respect to u for almost all t ∈ [0, T ] and it satisfies the growth condition B(t, u)Rn ≤ b0 (t) + b1 uU ,

∀ u ∈ U, t ∈ [0, T ].

(A2) f (t, r, s) is measurable with respect to t for all r, s ∈ Rn and continuous with respect to r and s respectively for almost all t ∈ [0, T ] and it satisfies the growth condition f (t, r, s)Rn ≤ f0 (t) +C1 rRn +C2 sRn ,

∀ r, s ∈ Rn , t ∈ [0, T ],

where C1 > 0,C2 > 0 and f0 (t) is continuous in the interval [0, T ]. (A3) f : [0, T ] × Rn × Rn → Rn is continuous in the first variable and Lipschitz continuous in the second and third variables. That is, f (t, r1 , s1 ) − f (t, r2 , s2 ) ≤ L1 r1 − r2 + L2 s1 − s2 , where L1 > 0 and L2 > 0 are Lipschitz constants. First consider the interval 0 ≤ t ≤ τ . Here, since y(t − τ ) = φ (t − τ ), (4) becomes C

Dtα x(t) = Ax(t) + B(t, u(t)) + gτ (t, x(t)), 0 < t ≤ τ ,

where gτ (t, x(t)) = f (t, φ (t − τ ), x(t)). Assumption (A3) implies that gτ is Lipschitz continuous in x and continuous function of t. Hence for each control function u(t) there exists a unique solution for (4) in the interval [0, τ ]. Its solution in [0, τ ] is of the form (see, Kilbas et al [2]), xτ (t) = Eα ,1 [At α ]φ (0) + tEα ,2 [At α ]x (0) ˆ t + (t − s)α −1 Eα ,α [A(t − s)α ][B(s, u(s)) + f (s, φ (s − τ ), x(s))]ds.

(5)

0

Hence in the interval [0, τ ] the solution of (4) exists and is unique. Now, in the interval [0, 2τ ], the system (4) may be written as C

where g2τ (t, x(t)) =

Dtα x(t) = Ax(t) + B(t, u(t)) + g2τ (t, x(t)), 0 < t ≤ τ ,

f (t, φ (t − τ ), x(t)), 0 < t ≤ τ . Then the solution of (4) in the interval [0, 2τ ] for f (t, xτ (t − τ ), x(t)), τ < t ≤ 2τ

each u is given by x(t) =

xτ (t), 0 ≤ t ≤ τ x2τ (t), τ ≤ t ≤ 2τ

(6)

´t where x2τ (t) = Eα ,1 [A(t − τ )α ]x(τ ) + (t − τ )Eα ,2 [A(t − τ )α ]x (τ ) + τ (t − s)α −1 Eα ,α [A(t − s)α ][B(s, u(s)) + f (s, xτ (s − τ ), xτ (s))]ds, τ ≤ t ≤ 2τ and xτ (t) is given by (5). Proceeding in a similar way we can easily prove the following theorem:


115

Theorem 2. Let x0τ (t) = φ (t), t ∈ [−τ , 0], x0τ (0) = x0 and let k be the greatest positive integer such that kτ ≤ T and let ⎧ 0 0, the angle has the property that it diverges in the limit of vanishingly action and is added, by a finite function dependent on a free parameter γ , when the action is larger than zero. The case γ = −1 reproduces the expression of the angle for the traditional standard mapping. The phase space is mixed and shows, for certain ranges of control parameters, a set of periodic islands, chaotic seas and invariant spanning curves. Statistical properties for an ensemble of noninteracting particles starting in the chaotic sea with very low action is considered and we show: (i) the saturation of chaotic orbits grows with ε α ; (ii) the regime of growth scales with nβ ; and (iii) the regime that marks the changeover from the diffusive dynamics to the stationary state scales with ε z . The exponents α and z depend on γ and are independent of the nonlinear function f while β is universal. To illustrate the theory here proposed, we obtain an estimation for the critical parameter Kc for a generalized standard mapping considering three different periodic functions. We also find α , β and z for different nonlinear functions. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

† Corresponding


124

Diogo Ricardo da Costa et al. / Journal of Applied Nonlinear Dynamics 7(2) (2018) 123–129

After the seminal paper from Moser [1] discussing the conditions for the existence of invariant curves for area preserving mappings of an annulus, the subject and interest for the topic increased significantly. The invariant curve has the property of separate different portions of the phase space. Therefore it blocks the passage of particles through it. Because of its crucial importance in transport properties and on the scaling in chaotic seas, among other applications, many different recent investigations were carried out. As for example, the Slater criteria [2], which precedes to Moser results, was used to estimate the breakup of invariant curves in dynamical systems [3] and to prove the existence of non-twist phenomenon in reversible non Hamiltonian systems [4]. Moreover, elementary proofs of the existence and of some properties of non autonomous analogues of rotational tori are discussed in [5] while the construction of a tori in the phase space and the inverse problem of Poincaré was made in [6]. Analytical results were made also to understand the topic, particularly the works of Wang [7], who has made investigation of the destruction of invariant circles for Gevrey area preserving twist map, Gentile [8] studying invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers and Kaloshin and Zhang [9] who gave a proof of a strong form of Arnold diffusion for smooth convex Hamiltonian systems. In this letter we describe how to construct a family of generic mapping by using a Hamiltonian formalism similar as the one used in [10]. We consider a function describing the angle with the following properties: (i) is a continuous function; (ii) is controlled by a parameter γ . For γ = −1 the expression of the angle for the standard map [11] is recovered and; (iii) for γ > 0 the angle diverges in the limit of vanishingly action. The variable action is dependent on a parameter ε controlling the intensity of a generic nonlinear function f which is smooth, continuous, infinitely many differentiable and periodic. Our main goal is to understand, obtain and describe how the position of the lowest action invariant spanning curve influences the scaling features of the chaotic sea in the low energy regime particularly 2 focusing on the description of the observable Jrms = J at the short, intermediate and long time dynamics. As we will see, three control parameters are important in order to understand the behaviour of Jrms , named α , β and z. We will show how to find them for any nonlinear function chosen. In order to to this, it is necessary to obtain an estimation for the critical parameter Kc for a generalized standard mapping. To start we consider the following Hamiltonian H(J1 , J2 , θ1 , θ2 ) = H0 (J0 , J1 ) + ε H1(J0 , J1 , θ0 , θ1 ), where the variables Jl and θl with l = 1, 2 correspond respectively to the action and angle [10]. The term H0 corresponds to the integrable part of the Hamiltonian while H1 is a non integrable contribution. The parameter ε controls a transition from integrability, from ε = 0, to non integrability with ε = 0. Because the Hamiltonian is time independent, one of the four variables can be eliminated, say J2 , and only three are left. A Poincaré surface of section defined by the plane J1 × θ1 at θ2 constant (mod 2π ) is considered. As a result, the perturbed twist mapping is given by Jn+1 = Jn − ε f (Jn+1 , θn ) and θn+1 = θn + κ (Jn+1 ) + ε g(Jn+1 , θn ), where f and g are periodic functions in θ . The function F2 = Jn+1 θn + A(Jn+1 ) + ε f˜(Jn+1 , θn ) generates the previous mapping, with κ = dA/dJn+1 , f = ∂ f˜/∂ θn and f = 0 needs to be satisfied. For many g = ∂ f˜/∂ Jn+1 . For area preservation, the expression ∂∂θgn − ∂ J∂n+1 mappings g(Jn+1 , θn ) = 0 ( f˜ is independent of Jn+1 ), therefore the following mapping is obtained [10] Jn+1 = Jn − ε f (θn ), (1) T: θn+1 = θn + κ (Jn+1 ) mod(2π ). When g = 0 and f˜(θn ) = cos(θn ), many applications can be found in the literature [10]. To illustrate few of them, the standard mapping is reproduced considering κ (Jn+1 ) = Jn+1 [11]. For κ = 2/Jn+1 the Fermi-Ulam model is reproduced [12, 13]. κ = ζ Jn+1 , where ζ is a constant, describes the bouncer 2 , the logistic twist mapping is obtained [15]. An application model [14]. For κ (Jn+1 ) = Jn+1 + ζ Jn+1 merging properties of the Fermi-Ulam [12] and bouncer [14] leading to a the hybrid Fermi-Ulam bouncer model can be seen in [16].


-2

Jrms

10

Jsat

β ε=5ε−5 ε=1ε−4 ε=5ε−4 ε=1ε−3

-4

α

10 (a)

125

2

10

nx

4

6

10

10

n

Jrms/ε

-1

10

-2

10

(b) 10-4

-2

10

10

0

z

n/ε

10

2

10

4

Fig. 1 Plot of: (a) Jrms vs. n for different values of ε and considering f˜ = cos(θ ) in mapping (1). (b) After a properly rescale in the axis, all curves overlapped onto each other in a single and hence universal curve. The parameters used were γ = 1 and M = 1000 different initial conditions chosen as J0 = 10−3 ε and θ0 ∈ [0, 2π ).

As one of the properties of f , we assume it is a smooth and periodic function such that f (θ ) = f (θ + 2π ). For κ (Jn+1 ) = Jn+1 , and choosing f˜ = cos(θ ) the standard map is recovered [10]. For ε < Kc ∼ = 0.9716 . . ., the standard map has invariant spanning curves which prevent the chaos to diffusing unlimitedly. For ε > Kc a chaotic sea can widespread unlimitedly from ±∞ in action axis. In our approach we consider that κ (Jn+1 ) = −|Jn+1 |−γ in the mapping (1), where γ is a control parameter. γ = −1 yields the standard mapping [11] while γ = 1 gives the Fermi-Ulam [12, 13]. For γ = 3/2 the Keppler map is recovered [17]. The phase space of the mapping is mixed and contains periodic islands, invariant spanning curves and chaotic seas. We are interested in the scaling properties of the chaotic sea considering γ > 0. Then, we describe the dynamics under three regimes: (i) short time; (ii) long time and; (iii) intermediate time. We begin with case (i). For short time and starting the dynamics using an ensemble of initial conditions in the low action regime along the chaotic sea, the ensemble of particle evolves using mapping 2 = Jn2 − 2Jn ε f (θn ) + ε 2 [ f (θn )]2 , where f (θn ) (1). The first equation of mapping (1) is written as Jn+1 attends to the conditions above. Moreover, it has a well defined average so that averaging the equation 2 2 = Jn2 + ε2π c( f ), where c( f ) is a constant that depends over an ensemble of θ ∈ [0, 2π ], we obtain Jn+1 ´ 2π on the f chosen and is written as c( f ) = 0 [ f (θn )]2 d θn . An important observation that needs to ´ 2π be taken into account is that 0 [ f (θn )] d θn = 0 for any function f . For sufficiently small ε , we use 2 − J2 = the following approximation Jn+1 n ε2 2 Jrms (n) = J0 + 2π c( f )n.

2 −J 2 Jn+1 n (n+1)−n

∼ =

dJ 2 dn

=

ε2 2π c( f ).

After a direct integration we have

Figure 1(a) shows a plot of Jrms vs. n. We see that for short n, Jrms can be described as Jrms ∝ nβ . A power law fit for short n gives β ≈ 0.5. For large enough regime of growth is changed to a n the

1 n 2 saturation. In our simulations Jrms was obtained as Jrms = J 2 = M1 ∑M i=1 [ n ∑ j=1 Ji, j ], where M is the number of different initial conditions. Analyzing Jsat , where the subindex sat indicates saturation, as

126


fi(θ)

1

f0(θ) f1(θ) f2(θ)

0

-1 0

2

θ

4

6

Fig. 2 Plot of the functions fi (θ ) against θ for θ ∈ (0, 2π ] with i = 0, 1, 2.

function of the parameter ε , a law Jsat ∝ ε α is obtained. Therefore, the saturation of J has a link to the position of the first invariant spanning curve, indeed a fraction of it, which depends on the parameter γ and not on the function f˜. Our simulations for γ = 1 lead to an exponent α ∼ = 0.5. Let us now discuss the saturation given by case (ii), i.e., large enough time. The exponent α indirectly controls the position of the lowest invariant spanning curve denoted as J ∗ . Near the spanning curve, the dynamics of the mapping can be made locally via a connection with the generalized standard mapping. We assume then Jn+1 = J ∗ + ΔJn+1 where ΔJn+1 is a small perturbation of the curve. We stress J ∗ > 0 and ε > 0 so that the absolute value of |Jn+1 | is simply written as Jn+1 . Considering the second equation of Map (1) and Taylor expanding under the limit ΔJn+1 /J ∗ → 0 we end up with θn+1 = θn + In+1 , where In+1 = γ ΔJn+1 (J ∗ )−(1+γ ) − (J ∗ )−γ . Rewriting the first equation of mapping (1), multiplying both sides by γ J ∗ −(1+γ ), adding −J ∗ −(1+γ ) in both sides of the equation and rearranging the terms, the following expression is obtained In+1 = In − γε J ∗ 1+γ f (θn ). These changes lead to the following map ⎧ ⎨In+1 = In − γε f (θn ), J ∗ (1+γ ) (2) ⎩ θn+1 = (θn + In+1 ) mod 2π . This mapping is structurally the same of the generalized standard mapping and the term γε /J ∗ (1+γ ) corresponds to a local parameter Kc ( f ) in the generalized standard mapping which marks the transition from local to global chaos Kc = γε J ∗ −(1+γ ). For each given function f a different Kc is obtained. Therefore, the position of the first invariant 1 1 spanning curve is J ∗ = [ Kcγε( f ) ] (1+γ ) . In a more simplified way, it is written as J ∗ ∝ ε 1+γ . This expression allows an immediate comparison with the exponent α , so that α = (1 + γ )−1 , resulting in J ∗ ∝ ε α , as mentioned in [18]. When the regime of growth meets with the saturation, case (iii) applies. Matching the equation of growth with the one marking the saturation we end up with

nx = c(2πf ) [γ 2

−2γ

Kc ( f )] (1+γ ) ε 1+γ

. As it is known in the

γ literature [18], nx ∝ ε z , hence z = −2 1+γ which also proves that z is independent on the function f . After obtaining the values of α and z, all curves in Fig. 1(a) overlap onto each other in a single and universal curve with the following scaling transformations n → n/ε z and Jrms → Jrms /ε α as shown in Fig. 1(b). Let us now discuss the localization of the invariant spanning curve for a more general function f , hence considering a generalized standard mapping with γ = −1. We consider the following cases f (θ ) = fi (θ ), with i = 0, 1, 2, hence f0 (θ ) = cos(θ ), f1 (θ ) = 12 [cos(θ ) + cos(2θ )] or f2 (θ ) =


127

6

J

4 2 0 (a) 0

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

6

θ

J

4 2 0 (b) 0 6

θ

J

4 2 0 (c) 0

θ

Fig. 3 Plot of the phase space for: (a) f0 (θ ) and ε = 1.2; (b) f1 (θ ) and ε = 0.5; (c) f2 (θ ) and ε = 0.24. The parameter γ was fixed at γ = 1. The mixed structure of the phase space is evident in both figures.

[cos(θ ) + cos(2θ ) + cos(3θ )]. A pure cosine function (i = 0) is our reference, as shown in Fig. 2. Figure 3(a) shows a phase space for f = f0 and considering ε = 1.2. The phase space is clearly mixed-type, containing periodic islands and a large chaotic sea. This parameter produces a global chaos because unlimited diffusion in the action may occurs, while only limited diffusion is observed for ε < Kc . We then search for values of Kc in the different functions fi chosen. A phase space for f = f1 and ε = 0.5 is shown in Fig. 3(b), and the red crosses mark a period 2 fixed point. The period of the lowest periodic fixed point depends on which i is used. For example, if f = f0 the period is one (see Fig. 3(a)), but for f = f2 and ε = 0.24 the period is three (see the red crosses in the Fig. 3(c)). We used the following procedure to estimate numerically Kc for the different function fi . The method consists of iterate the initial conditions up to nmax = 1011 times for different values of the parameter ε . The algorithm stores the minimum value of J of the trajectory associated to the initial condition (θ0 , J0 ) = (10−6 , 10−6 ) and, similarly, the maximum value of J of the orbit associated to (θ0 , J0 ) = (10−6 , 2π − 10−6 ). If the minimum and maximum reach the value J = π , then we conclude spanning 1 3

128


3.4 3.2 3

5

J

4

0.97

0.96

Kc

0.98

3 2

0.8

0.9

1

1.1

ε

1.2

Fig. 4 Plot of minimum value of J found for an orbit starting from J0 = 2π − 10−6 and θ0 = 10−6 until it reaches J = π . The red squares show the maximum value of J for an orbit starting from J0 = 10−6 and θ0 = 10−6. The orbits were iterated up to 1011 times and the point where these two orbits touch J = π is a good estimation for Kc . -1

J*

10

f0 f1 f2 Power law fit

0(3)

0.51 α

(a)

8(1)

1)

0.50

05( =0.5

-2

10

-4

-5

-3

10

10

10

γεJ*

-(1+γ)

0

10

-1

10 (b) 10-5

-4

10

ε

-3

10

Fig. 5 Plot of: (a) Approximated position for the minimum of the lowest invariant spanning curve (J ∗ ) as function of the control parameter ε . The orbits were iterated up to 1011 , and we have considered J0 = θ0 = 10−6 and γ = 1. (b) Rescale in the vertical axis of item (a) J ∗ → γε J ∗−(1+γ ) , where the dashed lines are the values of Kc ( fi ) found numerically for the generalized standard mapping.

curves do not exist in phase space. Therefore ε ≥ Kc and the iteration process of the corresponding trajectory is interrupted. Otherwise, both trajectories are iterated up to nmax . When parameter ε is chosen such that a crossing from J = π happens, another ε must be taken (500 different values of ε were selected in our simulations). In Fig. 4 we present the results for orbits iterated up to nmax times for f = f0 . The orbits touch J = π simultaneously at ε ∼ = 0.9728. This value is approximately the same found using Greene’s residue criterion [19], which is equal to Kc = 0.9716 . . .. The same procedure was used for the functions f1 and f2 leading to the values Kc = 0.42 . . . and Kc = 0.219 . . ., respectively. Now we estimate the position of the lowest action invariant spanning curve. For this, we divide θ ∈ (0, 2π ] in a stripe of 103 equally spaced cells. For each cell, the highest value of J in the chaotic sea is collected after a long run of nmax iterations of the mapping. With this, an inferior limit for


129

the invariant spanning curve is obtained numerically. Figure 5(a) shows J ∗ as a function of ε for the initial conditions J0 = θ0 = 10−6 . The curves are described by a power law of ε with slope α ∼ = 1/2 (the values found were 0.505(1), 0.508(1) and 0.510(3) respectively for f0 , f1 and f2 . Other values of γ where tested too and result is in good agreement with the theory. Figure 5(b) shows the rescaled axis J ∗ → γε J ∗ −(1+γ ) as a function of ε for the different functions fi considered. The dashed lines are the critical values Kc found for the generalized standard mapping considering each function fi . As seen the numerical results are in agreement with the analytical ones (dashed lines). The larger ε the worse is the approximation for Kc . As a short summary, we obtained the localization of the last invariant spanning curve in a family of generalized standard mappings. An estimation for the critical parameter Kc considering three different periodic functions was found. Theses values were used to compare with the result obtained from theoretical prediction. We also obtain the exponents α , β and z describing the curves of Jrms for different values of fi . We demonstrated that the critical exponents α and z depend on γ and are independent of the nonlinear function f while β is universal. Acknowledgments DRC acknowledges Brazilian agencies PNPD/Capes and FAPESP (2013/22764-2). EDL thanks to CNPq (303707/2015-1) and FAPESP (2012/23688-5), Brazilian agencies. This research was supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

Moser, J.K. (1962), Nachr. Akad. Wiss. G¨ ottingen Math.-Phys. Kl II, 1962, 1. Slater, N.B. (1950), Mathematical Proceedings of the Cambridge Philosophical Society, 46, 525. Abud, C.V., Caldas, I.L., (2015), Physica D, 308, 34. Altmann, E.G., Cristadoro, G. and Paz´ o, D. (2006), Phys. Rev. E, 73, 056201. Canadell, M. and de la Llave, R. (2015), Physica D, 310, 104. Laakso, T., Mikko, K. (2016), Physica D, 315, 72. Wang, L. and Dyn. J. (2015), Diff. Equat., 27, 283. Gentili, G. (2015), Nonlinearity, 28, 2555. Kaloshin, V. and Zhang, K. (2015), Nonlinearity, 28, 2699. Lichtenberg, A.J. and Lieberman, M.A. (1992), Regular and Chaotic Dynamics, Appl. Math. Sci., 38, Springer, New York. Chirikov, B.V. (1979), Phys. Rep., 52, 263. Lieberman, M.A. and Lichtenberg, A.J. (1971), Phys. Rev. A, 5, 1852. Karlis, A.K., et. al (2006), Phys. Rev. Lett., 97, 194102. Pustylnikov, L.D. (1978), Trans. Mosc. Math. Soc., 2, 1. Howard, J.E. and Humpherys, J. (1995), Physica D, 80, 256. Leonel, E.D. and McClintock, P.V.E. (2005), J. Phys. A, 38, 823. Petrosky, T. Y. (1986), Phys. Lett. A, 117, 328. Leonel, E.D., de Oliveira, J.A., and Saif, F. (2011), Phys. A: Math. Theor., 44, 302001. MacKay, R.S. (1992), Nonlinearity, 5, 161.



Runge-Kutta Method of Order Four for Solving Fuzzy Delay Differential Equations under Generalized Differentiability S. Indrakumar†, K. Kanagarajan Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore 641 020, India Assistant professor in Kongu Engineering College, Erode, Tamilnadu 638 052, India Submission Info

Abstract

Keywords

This paper portrays and interpret the fuzzy delay differential equations using the generalized differentiability concept by applying the Generalized Characterization Theorem. Subsequently we also investigate the problem of finding a numerical approximation of solutions. Moreover, the Runge-Kutta approximation methods is implemented and its error analysis are also discussed. The applicability of the theoretical results are illustrated with some examples.

Fuzzy delay differential equations Generalized differentiability Runge-Kutta method


Communicated by A.C.J. Luo Received 25 February 2017 Accepted 27 May 2017 Available online 1 July 2018

1 Introduction The concept of fuzzy set was first introduced by Zadeh [1]. Since then, the theory has been developed and it is now emerged as an independent branch of Applied Mathematics. Initially, the elementary fuzzy calculus based on the extension principle was studied by Dubois and Prade [2]. When a dynamical system is modeled by deterministic Ordinary Differential Equations(ODE) we cannot usually be sure that the model is perfect because, in general of dynamical system is often incomplete or vague. If the underlying structure of the model depends upon subjective choices, one way to incorporate these into the model is to utilize the aspects of fuzziness, which normally leads to the consideration of Fuzzy Differential Equations(FDE) and were regularly treated by Seikkala [3] and Kaleva [4]. A more realistic model must include some of the past history of the system. Whereas, models incorporating past history generally include Delay Differential Equations (DDE) or functional differential equations. By combining fuzzy mathematics and functional differential equations we get fuzzy functional differential equations. The numerical solution of FDE was studied by many researchers [5–10]. Generalized differentiability concept was first introduced by Bede [11] and was used to solve FDE [12–16]. Khastan et.al., [17] proved † Corresponding

author. Email address: [email protected]

ISSN 2164-6457, eISSN 2164-6473/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JAND.2018.06.003

132

S. Indrakumar, K. Kanagarajan / Journal of Applied Nonlinear Dynamics 7(2) (2018) 131–146

the existence and uniqueness of solution for Fuzzy Delay Differential Equations (FDDE) by using the concept of generalized differentiability. In this paper, we find the numerical solution of FDDE by using Runge-Kutta method of order four(RK4) under generalized differentiability concept. The structure of this paper is organized as follows. In section 2, we collect some basic concepts and preliminary results. In section 3, we give the Generalized Characterization Theorem for FDDE under generalized differentiability which have been discussed by Bede [18]. In section 4, we present Runge-Kutta method for finding the numerical solution of FDDE by giving the convergence results. In section 5, the proposed algorithm is illustrated by solving some examples of Malthusian model with delay and an Ehrlich ascites tumor model, finally the conclusion is given in section 6. 2 Preliminaries In this section, we consider some basic definitions and notations which will be applied throughout in this paper, we refer [19]. Definition 1. Let X be a nonempty set. A fuzzy set u in X is characterized by its membership function u : X → [0, 1]. For each x ∈ X , u(x) is interpreted as the degree of membership of an element x in the fuzzy set u, for each x ∈ X . We denote by RF the class of fuzzy subsets of the real axis, u : R → RF = [0, 1], satisfying the following properties: (i) u is normal, i.e., there exist an s0 ∈ R such that u(s0 ) = 1, (ii) u is fuzzy convex, i.e., u(ts + (1 − t)r) ≥ min {u(s), u(r)} , ∀ t ∈ [0, 1], s, r ∈ R, (iii) u is upper semicontinuous on R, (iv) cl{s ∈ R|u(s) > 0} is a compact set, where cl denotes the closure of a subset. Then RF is called the space of fuzzy numbers. Obviously, R ⊂ RF . For 0 < α ≤ 1, we above [u]α = {s ∈ R|u(s) ≥ α } and [u]0 = cl{s ∈ R|u(s) > 0}. From the conditions (i)-(iv), it follows that the α -level set [u]α is a nonempty compact interval, for all 0 ≤ α ≤ 1 and any u ∈ RF . The notation [u]α = [uα , uα ], denotes explicitly α -level set of u, for α ∈ [0, 1]. We refer to u and u as the lower and upper branches of u respectively. For u, v ∈ RF and λ ∈ R, the sum u + v, the scalar product λ u and multiplication uv are defined as follows: [u + v]α = [u]α + [v]α , [λ u]α = λ [u]α , [uv]α = [min{uα vα , uα vα , uα vα , uα vα }, max{uα vα , uα vα , uα vα , uα vα }] , ∀α ∈ [0, 1]. The metric structure D : RF × RF → R+ ∪ {0}, is based on the Hausdorff distance and is given by D(u, v) = sup max{|uα − vα |, |uα − vα |}. α ∈[0,1]

For the metric D defined on RF , we know that D(u + w, v + w) = D(u, v), ∀ u, v, w ∈ RF , D(ku, kv) = |k|D(u, v), ∀ k ∈ R, u, v ∈ RF , D(u + v, w + e) ≤ D(u, w) + D(v, e), ∀ u, v, w, e ∈ RF , and (RF , D) is a complete metric space.


133

Definition 2. [12] Let u, v ∈ RF . If there exist w ∈ RF such that u = v + w, then w is called the H-difference of u, v and it is denoted by u ⊖ v. We fix I = (a, b) for a, b ∈ R. The concept of generalized differentiability is as follows Definition 3. [12] Let F : I → RF and fix t0 ∈ I. We say that F is differentiable at t0 if there exist an element F ′ (t0 ) ∈ RF such that either (1) for all h > 0 sufficiently close to 0, F(t0 + h) ⊖ F(t0 ), F(t0 ) ⊖ F(t0 − h) and the limits lim

h→0+

F(t0 ) ⊖ F(t0 − h) F(t0 + h) ⊖ F(t0 ) = lim+ = F ′ (t0 ) h h h→0

exist; or (2) for all h > 0 sufficiently close to 0, F(t0 ) ⊖ F(t0 + h), F(t0 − h) ⊖ F(t0 ) and the limits lim+

h→0

F(t0 ) ⊖ F(t0 + h) F(t0 − h) ⊖ F(t0 ) = lim+ = F ′ (t0 ) −h −h h→0

exist. In Definition 3, the existence of the limits is considered in the metric D. Bede and Gal [12] indicated that a fuzzy function for F and t0 ∈ I, if F is differentiable in the sense (1) and (2) simultaneously, then, for h > 0 sufficiently small, it follows F(t0 +h) = F(t0 )+u1 , F(t0 ) = F(t0 −h)+u2 , F(t0 −h) = F(t0 )+v1 and F(t0 ) = F(t0 + h) + v2 , with u1 , u2 , v1 , v2 ∈ RF . In consequence, F(t0 ) = F(t0 ) + (u2 + v1 ), i.e., u2 + v1 = χ{0} , leading to two possibilities: u2 = v1 = χ{0} if F ′ (t0 ) = χ{0} ; or u2 = χ{a} = −v1 , with a ∈ R, if F ′ (t0 ) ∈ R. Finally, it justifies that, if there exists F ′ (t0 ) in the first form (second form, respectively) with F ′ (t0 ) ∈ / R, then F ′ (t0 ) does not exist in the second form (first form, respectively). Remark 1. In the previous definition, case (1) corresponds to the H-derivative introduced in [20], so this concept of differentiability is a generalization of the H-derivative. Remark 2. In [12], the authors consider four cases in the definition of derivative. Here, we consider only the two first cases of Definition 5 in [12]. In the other cases, the derivative reduces to a crisp element(more precisely, F ′ (t0 ) ∈ R); for details, see Theorem 7 in [12]. Definition 4. Let F : I → RF . We say that F is (1)-differentiable on I if F is differentiable in the sense (1) of Definition 2.3 on I, in this case its derivative is denoted by D1 F. Similarly, for (2)-differentiability, we write the derivative as D2 F. Next, we select some properties from [14] in relation with the concept of (2)-differentiability. Theorem 1. [14] Let F : I → RF and write [F(t)]α = [ fα (t), gα (t)], for each α ∈ [0, 1], t ∈ I. (i) If F is (1)-differentiable, then fα and gα are differentiable functions and we have [D1 F(t)]α = [ fα′ (t), gα′ (t)]. (ii)If F is (2)-differentiable, then fα and gα are differentiable functions and we have [D2 F(t)]α = [gα′ (t), fα′ (t)]. Proof. See [14].

134


Theorem 2. [19]. Let I be a real interval and F : I → RF . If, for arbitrary fixed t0 ∈ I and ε > 0, there exist δ > 0 (depending on t0 and ε ) such that t ∈ I, |t − t0 | < δ ⇒ D(F(t), F(t0 )) < ε , then F is said to be continuous on I. If J = [a, b] is a compact interval in R, then C(J, RF ) represents the set of all continuous fuzzy functions from J into RF . In the space C(J, RF ), we consider the following metric: D(u, v) = sup D[u(t), v(t)]. t∈J

Following the notation in [21], for a positive number σ , we denote by Cσ the space C([−σ , 0], RF ) equipped with the metric defined by Dσ (u, v) = sup D[u(t), v(t)]. t∈[−σ ,0]

Remaining faithful to the classical notation used in the field of functional differential equations [22], for a given u ∈ C([−σ , ∞), RF ), ut denotes, for each t ∈ [0, ∞), the element in Cσ is defined by ut (s) = u(t + s), s ∈ [−σ , 0]. Lemma 9. If F : [0, ∞) × RF ×Cσ → RF is a jointly continuous function and u : [−σ , ∞) × RF → RF is a continuous function, then the function is as follows t 7→ F(t, u(t), u(t − σ )) : [0, ∞) × RF ×Cσ → RF and also it is continuous. Remark 3. [21]. If F : [0, ∞) × RF × Cσ → RF is jointly continuous and u : [−σ , ∞) × RF → RF is continuous, then the function t 7→ F(t, u(t), u(t − σ )) : [0, ∞)×RF ×Cσ → RF is integrable on each compact interval [0, T ]. Theorem 3. Let F be a fuzzy function continuous on I and define ˆ t u(t) = β ⊖ −F(τ )d τ , t ∈ I, 0

where β ∈ RF is such that the previous H-difference exists, for t ∈ I. Then u is (2)-differentiable and u′ (t) = F(t),

t ∈ I.

Proof. See [23]. 3 Generalized characterization theorem for FDDE under generalized differentiability Let us consider the FDDE u′ (t) = f (t, u(t), u(t − σ )), for t0 ≤ t ≤ T,

(1)

where σ > 0 is a constant delay, u ∈ RF is a n-vector-valued fuzzy function and f is a continuous fuzzy function defined on a mapping f : [t0 , T ] × RF ×Cσ → RF with the initial conditions u(t0 ) = u0 , u(t) = ϕ (t), for t0 − σ ≤ t < t0 ,

(2)


135

where u0 ∈ RF , ϕ : [t0 − σ ,t0 ) → RF is a given continuous fuzzy function. The parametric forms are [u′ (t)]α = [u′ (t; α ), u′ (t; α )], [u(t)]α = [u(t; α ), u(t; α )], α

[u(t − σ )] = [u(t − σ ; α ), u(t − σ ; α )],

(3) for t ∈ [t0 , T ], α ∈ [0, 1].

Theorem 4. Let f : [0, ∞) × RF ×Cσ → RF be a continuous fuzzy function such that there exists L > 0 such that D( f (t, ϕ ), f (t, ψ )) ≤ LDσ (ϕ , ψ ), ∀ t ∈ [0, ∞), ϕ , ψ ∈ RF . Then problem (1) has two solutions (one (1)-differentiable and the other one (2)-differentiable) on [0, ∞). Theorem 5. Let u : [0, ∞) × RF × Cσ → RF be a fuzzy function such that D1 u or D2 u exists. If u and D1 u satisfy the problem (1), we say u is a (1)-solution of problem (1). Similarly, if u and D2 u satisfy problem (1), we say u is a (2)-solution of problem (1). Then Theorem 2.7 shows us a way to translate the FDDE (1) into a system of Ordinary Delay Differential Equations(ODDE). Let [u(t)]α = [u(t; α ), u(t; α )]. If u(t) is (1)-differentiable then [D1 u(t)]α = [u′ (t; α ), u′ (t; α )] and (1) translates into the following system of ODDE: u′ (t) = f α (t, u(t; α ), u(t; α ), u(t − σ ; α ), u(t − σ ; α )) = F(t, u(t), u(t), u(t − σ ), u(t − σ )), t0 ≤ t ≤ T ′ u (t) = f α (t, u(t; α ), u(t; α ), u(t − σ ; α ), u(t − σ ; α )) = G(t, u(t), u(t), u(t − σ ), u(t − σ )), t0 ≤ t ≤ T u(t0 ) = u0 , u(t0 ) = u0 u(t) = ϕ (t) u(t) = ϕ (t); t0 − σ ≤ t ≤ t0 . Also, if u(t) is (2)-differentiable then [D2 u(t)]α = [u′ (t; α ), u′ (t; α )] and (1) translates into the following system of ODDE: u′ (t) = f α (t, u(t; α ), u(t; α ), u(t − σ ; α ), u(t − σ ; α )) = G(t, u(t), u(t), u(t − σ ), u(t − σ )), t0 ≤ t ≤ T u′ (t) = f α (t, u(t; α ), u(t; α ), u(t − σ ; α ), u(t − σ ; α )) = F(t, u(t), u(t), u(t − σ ), u(t − σ )), t0 ≤ t ≤ T u(t0 ) = u0 , u(t0 ) = u0 u(t) = ϕ (t) u(t) = ϕ (t); t0 − σ ≤ t ≤ t0 . Then, the authors of [14] state that if we ensure that the solution [u(t; α ), u(t; α )] of the system (4) and (4) are valid level sets of a fuzzy number valued function and if [u′ (t; α ), u′ (t; α )] are valid level sets of a fuzzy valued function, then by the Stacking Theorem [4], it is possible to construct the (1)-solution of FDDE (1). Also, for the (2)-solution, we can also proceed in a same way. 4 Numerical solution of FDDE by Generalized characterization theorem In this section we present numerical methods for solving FDDE (1) by the Generalized Characterization Theorem. Here we consider the FDDE (1) under the following assumptions (i) There exist L > 0 such that D[ f (t, ϕ ), f (t, ψ )] ≤ LDσ (ϕ , ψ ) for all ϕ , ψ ∈ Cσ and t ≥ 0. (ii) f : [0, ∞) × RF ×Cσ → RF is jointly continuous. ˆ ≤ Mebt for all t ≥ 0, 0ˆ = χ{0} , 0ˆ ∈ [0, ∞). (iii) There exist M > 0 and b > 0 such that D[F(t, 0), 0]

136


Lemma 6. [12]. If u ∈ C([0, ∞), RF ) is such that u0 = ϕ and the H-difference in FDDE (1) is equivalent to one of the following integral equations:  for t0 − σ ≤ t ≤ t0 ,  ϕ (t), ˆ t u(t) :=  ϕ (0) + for t0 ≤ t ≤ T, f (s, u(s), u(s − σ ))ds, 0

or   ϕ (t), u(t) :=

 ϕ (0) − (−1) ⊙

ˆ

t

for t0 − σ ≤ t ≤ t0 , f (s, u(s), u(s − σ ))ds, for t0 ≤ t ≤ T,

0

depending on the strong differentiability considered, (1)-differentiability or (2)-differentiability, respectively. Based on Generalized Characterization Theorem, we replace the FDDE by two ODE systems. Eqns (4) and (4) represent two ordinary Cauchy problems for which any converging classical numerical procedure can be applied. In the following, we generalize the Runge-Kutta method of order four and give its error analysis. We consider the FDDE ((1) and (2)) adapt the Runge-Kutta method so is to express the difference between the value of u at tn+1 and tn as ν

ki (tn , u(tn ), ut ) = hn+1 f (tn + ci hn+1 , u(tn ) + ∑ ai j k j (tn , u(tn ), ut ), ut ), j=1

ν

(4)

u(tn+1 ) = u(tn ) + ∑ bi ki (tn , u(tn ), ut ), i=1

where u(tn + ci hn+1 − σ ) is a delay term and the non-zero constants are c2 = c3 = 21 , c4 = 1, a21 = a32 = 1 1 1 1 1 2 , a43 = 1, b1 = 6 , b2 = 3 , b3 = 3 , b4 = 6 . Let ∆ = {t0 ,t1 , . . . ,tN = T } be given grid points and let hn+1 = tn+1 − tn , n = 0, 1, 2, . . . , N − 1 denote the corresponding stepsizes, at which the exact solution [U1 ]α = [U 1 (t; α ),U 1 (t; α )] and [U2 ]α = [U 2 (t; α ),U 2 (t; α )] are approximated by some [u1 ]α = [u1 (t; α ), u1 (t; α )] and [u2 ]α = [u2 (t; α ), u2 (t; α )] respectively. The exact and approximate solutions at tn , 0 ≤ n ≤ N are denoted by U1 (tn ; α ),U2 (tn ; α ), u1 (tn ; α ) and u2 (tn ; α ) respectively. The generalized Runge-Kutta method of order four based on the first-order ′ ′ approximation of U ′1 (t; α ),U 1 (t; α ), U ′2 (t; α ),U 2 (t; α ) and Eqns (4) and (4) is obtained as solved below:  ν   ai j k1, j (tn , u, ut ; α ), ut ) (t , u, u ; α ) = h f (t + c h , u + k  n t n+1 n i n+1 1,i ∑    j=1      u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn + ci hn+1 − σ ; α ), u1 (tn + ci hn+1 − σ ; α )],     ν    ai j k1, j (tn , u, ut ; α ), ut ) k (t , u, u ; α ) = h f (t + c h , u +  1,i n t n+1 n i n+1 ∑  j=1

 u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn + ci hn+1 − σ ; α ), u1 (tn + ci hn+1 − σ ; α )],     ν     u bi k1,i (tn+1 , u, ut ; α ), (t ; α ) = u (t ; α ) +  1 n+1 1 n ∑   i=1     ν    u1 (tn+1 ; α ) = u1 (tn ; α ) + ∑ bi k1,i (tn+1 , u, ut ; α ), i=1

(5)


137

 ν   k ai j k2, j (tn , u, ut ; α ), ut ) (t , u, u ; α ) = h f (t + c h , u +  2,i n t n+1 n i n+1 ∑    j=1      u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn + ci hn+1 − σ ; α ), u2 (tn + ci hn+1 − σ ; α )],     ν    ai j k2, j (tn , u, ut ; α ), ut ) k (t , u, u ; α ) = h f (t + c h , u +  2,i n t n+1 n i n+1 ∑  j=1

  u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn + ci hn+1 − σ ; α ), u2 (tn + ci hn+1 − σ ; α )],     ν    u bi k2,i (tn+1 , u, ut ; α ), (t ; α ) = u (t ; α ) +  2 n+1 2 n ∑    i=1    ν    u2 (tn+1 ; α ) = u2 (tn ; α ) + ∑ bi k2,i (tn+1 , u, ut ; α )

(6)

i=1

and k1,1 (tn , u, ut ; α ) = min{hn+1 f (tn , u, ut )| u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn − σ ; α ), u1 (tn − σ ; α )]}, 1 1 k1,2 (tn , u, ut ; α ) = min{hn+1 f (tn + hn+1 , u + k1,1 (tn , u, ut ; α ), ut )| 2 2 1 1 u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn + hn+1 − σ ; α ), u1 (tn + hn+1 − σ ; α )]}, 2 2 1 1 k1,3 (tn , u, ut ; α ) = min{hn+1 f (tn + hn+1 , u + k1,2 (tn , u, ut ; α ), ut )| 2 2 1 1 u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn + hn+1 − σ ; α ), u1 (tn + hn+1 − σ ; α )]}, 2 2 k1,4 (tn , u, ut ; α ) = min{hn+1 f (tn + hn+1 , u + k1,3 (tn , u, ut ; α ), ut )| u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn + hn+1 − σ ; α ), u1 (tn + hn+1 − σ ; α )]}, k1,1 (tn , u, ut ; α ) = max{hn+1 f (tn , u, ut )| u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn − σ ; α ), u1 (tn − σ ; α )]}, 1 1 k1,2 (tn , u, ut ; α ) = max{hn+1 f (tn + hn+1 , u + k1,1 (tn , u, ut ; α ), ut )| 2 2 1 1 u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn + hn+1 − σ ; α ), u1 (tn + hn+1 − σ ; α )]}, 2 2 1 1 k1,3 (tn , u, ut ; α ) = max{hn+1 f (tn + hn+1 , u + k1,2 (tn , u, ut ; α ), ut )| 2 2 1 1 u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn + hn+1 − σ ; α ), u1 (tn + hn+1 − σ ; α )]}, 2 2 k1,4 (tn , u, ut ; α ) = max{hn+1 f (tn + hn+1 , u + k1,3 (tn , u, ut ; α ), ut )| u ∈ [u1 (tn ; α ), u1 (tn ; α )], ut ∈ [u1 (tn + hn+1 − σ ; α ), u1 (tn + hn+1 − σ ; α )]}, F1 [tn , u1 (tn ; α ), u1 (tn ; α ), u1 (tn − σ ; α ), u1 (tn − σ ; α )] = k1,1 (tn , u, ut ; α ) + 2k1,2 (tn , u, ut ; α ) + 2k1,3 (tn , u, ut ; α ) + k1,4 (tn , u, ut ; α ), G1 [tn , u1 (tn ; α ), u1 (tn ; α ), u1 (tn − σ ; α ), u1 (tn − σ ; α )] = k2,1 (tn , u, ut ; α ) + 2k2,2 (tn , u, ut ; α ) + 2k2,3 (tn , u, ut ; α ) + k2,4 (tn , u, ut ; α ),

(7)

138


 1  u1 (tn+1 ; α ) = u1 (tn ; α ) + F1 [tn , u1 (tn ; α ), u1 (tn ; α ), u1 (tn − σ ; α ), u1 (tn − σ ; α )],    6    1 u1 (tn+1 ; α ) = u1 (tn ; α ) + G1 [tn , u1 (tn ; α ), u1 (tn ; α ), u1 (tn − σ ; α ), u1 (tn − σ ; α )], 6     u1 (t0 ; α ) = u0 (α ), u1 (t0 ; α ) = u0 (α ),    u1 (t, α ) = ϕ (t; α ), u1 (t, α ) = ϕ (t; α ), t0 − σ ≤ t ≤ t0

t ≥ 0,

(8)

and for (2)-differentiability k2,1 (tn , u, ut ; α ) = min{hn+1 f (tn , u, ut )|u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn − σ ; α ), u2 (tn − σ ; α )]}, 1 1 k2,2 (tn , u, ut ; α ) = min{hn+1 f (tn + hn+1 , u + k2,1 (tn , u, ut ; α ), ut )| 2 2 1 1 u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn + hn+1 − σ ; α ), u2 (tn + hn+1 − σ ; α )]}, 2 2 1 1 k2,3 (tn , u, ut ; α ) = min{hn+1 f (tn + hn+1 , u + k2,2 (tn , u, ut ; α ), ut )| 2 2 1 1 u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn + hn+1 − σ ; α ), u2 (tn + hn+1 − σ ; α )]}, 2 2 k2,4 (tn , u, ut ; α ) = min{hn+1 f (tn + hn+1 , u + k2,3 (tn , u, ut ; α ), ut )| u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn + hn+1 − σ ; α ), u2 (tn + hn+1 − σ ; α )]}, k2,1 (tn , u, ut ; α ) = max{hn+1 f (tn , u, ut )|u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn − σ ), u2 (tn − σ )]}, 1 1 k2,2 (tn , u, ut ; α ) = max{hn+1 f (tn + hn+1 , u + k2,1 (tn , u, ut ; α ), ut )| 2 2 1 1 u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn + hn+1 − σ ; α ), u2 (tn + hn+1 − σ ; α )]}, 2 2 1 1 k2,3 (tn , u, ut ; α ) = max{hn+1 f (tn + hn+1 , u + k2,2 (tn , u, ut ; α ), ut )| 2 2 1 1 u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn + hn+1 − σ ; α ), u2 (tn + hn+1 − σ ; α )]}, 2 2 k2,4 (tn , u, ut ; α ) = max{hn+1 f (tn + hn+1 , u + k2,3 (tn , u, ut ; α ), ut )| u ∈ [u2 (tn ; α ), u2 (tn ; α )], ut ∈ [u2 (tn + hn+1 − σ ; α ), u2 (tn + hn+1 − σ ; α )]}, F2 [tn , u2 (tn ; α ), u2 (tn ; α ), u2 (tn − σ ; α ), u2 (tn − σ ; α )] = k2,1 (tn , u, ut ; α ) + 2k2,2 (tn , u, ut ; α ) + 2k2,3 (tn , u, ut ; α ) + k2,4 (tn , u, ut ; α ), G2 [tn , u2 (tn ; α ), u2 (tn ; α ), u2 (tn − σ ; α ), u2 (tn − σ ; α )]

(9)

= k2,1 (tn , u, ut ; α ) + 2k2,2 (tn , u, ut ; α ) + 2k2,3 (tn , u, ut ; α ) + k2,4 (tn , u, ut ; α ),  1  u2 (tn+1 ; α ) = u2 (tn ; α ) + G2 [tn , u2 (tn ; α ), u2 (tn ; α ), u2 (tn − σ ; α ), u2 (tn − σ ; α )],    6    1 u2 (tn+1 ; α ) = u2 (tn ; α ) + F2 [tn , u2 (tn ; α ), u2 (tn ; α ), u2 (tn − σ ; α ), u2 (tn − σ ; α )], t ≥ 0, 6     u2 (t0 ; α ) = u0 (α ), u2 (t0 ; α ) = u0 (α ),    u2 (t, α ) = ϕ (t; α ), u2 (t, α ) = ϕ (t; α ), t0 − σ ≤ t ≤ t0 .

(10)

Our next result determines the pointwise convergence of the generalized Runge-Kutta’s approximations to the exact solutions. Let F(t, a, b, c, d) and G(t, a, b, c, d) be the functions F and G of Eq ((4)


139

and (4)), where a, b, c and d are constants and a ≤ c, b ≤ d. The domain where F and G are defined is therefore K = {(t, a, b, c, d)| t0 ≤ t ≤ T, −∞ < c < ∞, −∞ < a ≤ c} . Theorem 4.3. Let F(t, x, xt , y, yt ) and G(t, x, xt , y, yt ) belong to C1 (K) and let the partial derivatives of F, G be bounded over K. Then, for arbitrary fixed α : α ∈ [0, 1], the generalized Runge-Kutta’s approximate of Eq ((8) and (10)) converge to the exact solutions U1 (t; α ), U2 (t; α ) uniformly in t. 5 Numerical examples We consider the FDDE:

u′ (t) = f (t, u(t − σ )); u(t) = ϕ (t),

t ≥ 0, −σ ≤ t ≤ 0.

(11)

Using Theorem 2.7, we get [u(t)]α = [u(t; α ), u(t; α )], t ≥ −σ , and

[ϕ (t)]α = [ϕ (t; α ), ϕ (t; α )], t ∈ [−σ , 0]

[ f (t, u(t − σ ))]α = [ f (t, u(t − σ ; α ), u(t − σ ; α ); α ), f (t, u(t − σ ; α ), u(t − σ ; α ); α )], t ≥ 0.

By applying the generalized differentiability concept and Zadeh’s extension principle, we have the following alternatives for solving problem (11) Case (i) : If we consider the derivative u(t) by using (1)-differentiability, then from Theorem 2.7, we have [u′ (t)]α = [u′ (t; α ), u ′ (t; α )], for t ≥ 0 and α ∈ [0, 1]. Now, we proceed as follows: (i) Solve the parameterized delay differential system  ′  u (t; α ) = f (t, u(t − σ ; α ), u(t − σ ; α ); α ), t ≥ 0, (12) u ′ (t; α ) = f (t, u(t − σ ; α ), u(t − σ ; α ); α ), t ≥ 0,   −σ ≤ t ≤ 0, 0 ≤ α ≤ 1, u(t; α ) = ϕ (t; α ), u(t; α ) = ϕ (t; α ), for α ∈ [0, 1], to find u and u. (ii) Ensure that [u(t; α ), u(t; α )], [u′ (t; α ), u ′ (t; α )] are valid level sets. (iii) Using the Stacking Theorem [4], construct a fuzzy solution u(t) such that [u(t)]α = [u(t; α ), u(t; α )], for α ∈ [0, 1] and t ≥ 0. Case (ii) : Similarly to [14], if we consider the derivative of u(t) by using (2)-differentiability, then from Theorem 2.7, we have [u′ (t)]α = [u′ (t; α ), u ′ (t; α )], for t ≥ 0 and α ∈ [0, 1]. These considerations allow to proceed as follows: (i) Solve the parameterized delay differential system  ′  u (t; α ) = f (t, u(t − σ ; α ), u(t − σ ; α ); α ), t ≥ 0, u ′ (t; α ) = f (t, u(t − σ ; α ), u(t − σ ; α ); α ), t ≥ 0, (13)   u(t; α ) = ϕ (t; α ), u(t; α ) = ϕ (t; α ), −σ ≤ t ≤ 0, 0 ≤ α ≤ 1. for α ∈ [0, 1], to get u and u. (ii) Ensure that [u(t; α ), u(t; α )], [u′ (t; α ), u ′ (t; α )] are valid level sets. (iii) Using the Stacking Theorem [4], construct a fuzzy solution u(t) such that [u(t)]α = [u(t; α ), u(t; α )], for α ∈ [0, 1] and t ≥ 0.

140

S. Indrakumar, K. Kanagarajan / Journal of Applied Nonlinear Dynamics 7(2) (2018) 131–146 2

1 Ŧ Exact o Euler . RKŦ4 0.8

Ŧ

Exact

o

Euler

.

RKŦ4

1

Alpha

u(t)

0.6

0

0.4

Ŧ1 0.2

Ŧ2

0

0.5

1 t

1.5

0 Ŧ2

2

Ŧ1

0 u(t)

1

2

Fig. 1 Comparison between the exact, Euler and RK4 solutions for (1)-differentiability at t = 2.

Alpha

1

0.5

0 2 2

1 1.5

0 1

Ŧ1

0.5 Ŧ2

u(t)

0 t

Fig. 2 The approximate solution by RK4 for (1)-differentiability at t = 2.

5.1

Example

We consider a fuzzy time-delay Malthusian model [24] ′ t ≥ 0, r > 0, u (t) = ru(t − σ ); u(t) = u0 , −1 ≤ t ≤ 0,

(14)

where, u(t) refers the population at time t. Suppose the initial value is [u0 ]α = [α − 1, 1 − α ] = (1 − 1 , σ = 1. Here, f (t, φ ) = rφ (−1) and α )[−1, 1], α ∈ [0, 1] and r > 0. In this example, we set r = 10 α α [ϕ (t)] = [u0 ] = [α − 1, 1 − α ]. If we consider [u′1 (t)]α = [u1 ′ (t; α ), u1 ′ (t; α )] in the notion of (1)-differentiability, by using (12) we get the exact solution


141

Alpha

1

0.5

0 1 2

0.5 1.5

0 1

Ŧ0.5

0.5 Ŧ1

0

u(t)

t

Fig. 3 The approximate solution by RK4 for (2)-differentiability at t = 2.

[u(t)]α = [(α − 1)(1 + rt), (1 − α )(1 + rt)], for t ∈ [0, 1] and 1 1 [u(t)]α = [(α − 1)(1 + rt + r2 (t − 1)2 ), (1 − α )(1 + rt + r2 (t − 1)2 )], 2 2 for t ∈ (1, 2], α ∈ [0, 1]. On the other hand, if [u′ (t)]α = [u′ (t; α ), u ′ (t; α )] is (2)-differentiability, by using (13) we get the exact solution [u(t)]α = [(1 − α )(rt − 1), (1 − α )(1 − rt)], for t ∈ [0, 1] and 1 1 [u(t)]α = [(α − 1)(1 − rt + r2 (t − 1)2 ), (1 − α )(1 − rt + r2 (t − 1)2 )], for t ∈ (1, 2], α ∈ [0, 1]. 2 2 We compare the results obtained by RK4 with the exact solution of (1)-differentiability of the problem (14) and the errors are shown in Table 1 and the Figures 1 and 2 respectively. The comparison of exact and the approximate solutions of (2)-differentiability of the problem (14) at t = 2 and the errors are shown in the following Table 2, Figures 3 and 4 respectively. 5.2

Example

We consider the fuzzy version of logistic equation for the Ehrlich ascites tumor model [25] ′ u (t) = ru(t − σ )(1 − u(t − σ )), t ≥ 0, u(t) = u0 , −1 ≤ t ≤ 0,

(15)

where [u0 ]α = 14 [α , 2 − α ], α ∈ [0, 1], r = 41 , σ = 1. The approximate solutions of both the cases are shown below

142


Table 1. Comparison between the exact and the approximate solutions for (1)-differentiability at t = 2. α

Euler Apprx

RK-4

Exact

Error Euler

Error RK-4

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

0.0

-1.2036e-00

1.2036e-00

-1.2046e+00

1.2046e+00

-1.2050e+00

1.2050e+00

-1.4000e-03

1.4000e-03

-3.6045e-04

3.6045e-04

0.1

-1.0832e-00

1.0832e-00

-1.0842e+00

1.0842e+00

-1.0845e+00

1.0845e+00

-1.2600e-03

1.2600e-03

-3.2440e-04

3.2440e-04

0.2

-9.6288e-01

9.6288e-01

-9.6371e-01

9.6371e-01

-9.6400e-01

9.6400e-01

-1.1200e-03

1.1200e-03

-2.8836e-04

2.8836e-04

0.3

-8.4252e-01

8.4252e-01

-8.4325e-01

8.4325e-01

-8.4350e-01

8.4350e-01

-9.8000e-04

9.8000e-04

-2.5231e-04

2.5231e-04

0.4

-7.2216e-01

7.2216e-01

-7.2278e-01

7.2278e-01

-7.2300e-01

7.2300e-01

-8.8400e-04

8.8400e-04

-2.1627e-04

2.1627e-04

0.5

-6.0180e-01

6.0180e-01

-6.0232e-01

6.0232e-01

-6.0250e-01

6.0250e-01

-7.0000e-04

7.0000e-04

-1.8022e-04

1.8022e-04

0.6

-4.8144e-01

4.8144e-01

-4.8186e-01

4.8186e-01

-4.8200e-01

4.8200e-01

-5.6000e-04

5.6000e-04

-1.4418e-04

1.4418e-04

0.7

-3.6108e-01

3.6108e-01

-3.6139e-01

3.6139e-01

-3.6150e-01

3.6150e-01

-4.2000e-04

4.2000e-04

-1.0813e-04

1.0813e-04

0.8

-2.4072e-01

2.4072e-01

-2.4093e-01

2.4093e-01

-2.4100e-01

2.4100e-01

-2.8000e-04

2.8000e-04

-7.2089e-05

7.2089e-05

0.9

-1.2036e-01

1.2036e-01

-1.2046e-01

1.2046e-01

-1.2050e-01

1.2050e-01

-1.4000e-04

1.4000e-04

-3.6045e-05

3.6045e-05

1.0

0.0000e+00

0.0000e+00

0.0000e+00

0.0000e+00

0.0000e+00

0.0000e+00

0.0000e+00

0.000e+00

0.0000e+00

0.000e+00

Table 2. Comparison between the exact and the approximate solutions for (2)-differentiability at t = 2. α

Euler Apprx

RK-4

Exact

Error Euler

Error RK-4

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

0.0

-8.0360e-01

8.0360e-01

-8.0354e-01

8.0354e-01

-8.0500e-01

8.0500e-01

-1.4000e-03

1.4000e-03

-1.4581e-03

1.4581e-03

0.1

-7.2324e-01

7.2324e-01

-7.2319e-01

7.2319e-01

-7.2450e-01

7.2450e-01

-1.3000e-03

1.3000e-03

-1.3123e-03

1.3123e-03

0.2

-6.4288e-01

6.4288e-01

-6.4283e-01

6.4283e-01

-6.4400e-01

6.4400e-01

-1.1000e-03

1.1000e-03

-1.1665e-03

1.1665e-03

0.3

-5.6252e-01

5.6252e-01

-5.6248e-01

5.6248e-01

-5.6350e-01

5.6350e-01

-9.8000e-04

9.8000e-04

-1.0207e-03

1.0207e-03

0.4

-4.8216e-01

4.8216e-01

-4.8213e-01

4.8213e-01

-4.8300e-01

4.8300e-01

-8.4000e-04

8.4000e-04

-8.7485e-04

8.7485e-04

0.5

-4.0180e-01

4.0180e-01

-4.0177e-01

4.0177e-01

-4.0250e-01

4.0250e-01

-7.0000e-04

7.0000e-04

-7.2904e-04

7.2904e-04

0.6

-3.2144e-01

3.2144e-01

-3.2142e-01

3.2142e-01

-3.2200e-01

3.2200e-01

-5.6000e-04

5.6000e-04

-5.8323e-04

5.8323e-04

0.7

-2.4108e-01

2.4108e-01

-2.4106e-01

2.4106e-01

-2.4150e-01

2.4150e-01

-4.1999e-04

4.1999e-04

-4.3742e-04

4.3742e-04

0.8

-1.6072e-01

1.6072e-01

-1.6071e-01

1.6071e-01

-1.6100e-01

1.6100e-01

-2.8000e-04

2.8000e-04

-2.9162e-04

2.9162e-04

0.9

-8.0360e-02

8.0360e-02

-8.0354e-02

8.0354e-02

-8.0500e-02

8.0500e-02

-1.4000e-04

1.4000e-04

-1.4581e-04

1.4581e-04

1.0

0.0000e+00

0.0000e+00

0.0000e+00

0.0000e+00

0.0000e+00

0.0000e+00

0.0000e+00

0.000e+00

0.0000e+00

0.000e+00

S. Indrakumar, K. Kanagarajan / Journal of Applied Nonlinear Dynamics 7(2) (2018) 131–146 1

143

1 Ŧ Exact o Euler

Ŧ

Exact

.

o

Euler

.

RKŦ4

RKŦ4 0.8

0.5

Alpha

u(t)

0.6

0

0.4

Ŧ0.5 0.2

Ŧ1

0

0.5

1 t

1.5

0 Ŧ0.8

2

Ŧ0.4

0

0.4

0.6

u(t)

Fig. 4 Comparison between the exact, Euler and RK4 solutions for (2)-differentiability at t = 2. Table 3 Approximate value of Euler and RK4 solution for (1)-differentiability at t = 2. α

Euler

RK4

Difference for RK4-Euler

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

u1 (tn ; α )

0.0

0.0000e+00

7.6406e-01

0.0000e+00

7.6757e-01

0.0000e+00

1.4419e-02

0.1

3.1403e-02

7.1905e-01

3.1481e-02

7.2229e-01

3.2093e-04

1.3298e-02

0.2

6.3521e-02

6.7477e-01

6.3688e-02

6.7774e-01

6.8379e-04

1.2219e-02

0.3

9.6355e-02

6.3119e-01

9.6620e-02

6.3392e-01

1.0886e-03

1.1182e-02

0.4

1.2990e-01

5.8834e-01

1.3027e-01

5.9082e-01

1.5355e-03

1.0187e-02

0.5

1.6416e-01

5.4619e-01

1.6466e-01

5.4844e-01

2.0245e-03

9.2340e-03

0.6

1.9914e-01

5.0477e-01

1.9977e-01

5.0679e-01

2.5556e-03

8.3232e-03

0.7

2.3484e-01

4.6406e-01

2.3560e-01

4.6587e-01

3.1289e-03

7.4546e-03

0.8

2.7125e-01

4.2406e-01

2.7216e-01

4.2568e-01

3.7443e-03

6.6281e-03

0.9

3.0838e-01

3.8479e-01

3.0945e-01

3.8621e-01

4.4020e-03

5.8439e-03

1

3.4623e-01

3.4623e-01

3.4747e-01

3.4747e-01

5.1018e-03

5.1018e-03

6 Conclusions In this paper, fourth order Runge-Kutta method for solving FDDE under generalized differentiability concept, we translate the FDDE into two systems of ODDE and then solve numerically Runge-Kutta method of order four. From the results obtained we conclude that these proposed methods are well suited for finding the numerical solution of FDDE. Higher order Runge-Kutta methods will be considered in our future research work.

144


Table 4 Approximate value of Euler and RK4 solution for (2)-differentiability at t = 2. α

Euler

RK4

Difference for RK4-Euler

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

u2 (tn ; α )

0.0

2.4438e-01

5.0563e-01

2.5671e-01

5.0622e-01

1.2333e-02

5.9676e-04

0.1

2.5188e-01

4.8701e-01

2.6336e-01

4.8792e-01

1.1476e-02

9.1577e-04

0.2

2.5997e-01

4.6897e-01

2.7062e-01

4.7023e-01

1.0650e-03

1.2628e-03

0.3

2.6865e-01

4.5152e-01

2.7850e-01

4.5316e-01

9.8515e-03

1.6380e-03

0.4

2.7791e-01

4.3466e-01

2.8699e-01

4.3670e-01

9.0814e-03

2.0413e-03

0.5

2.8775e-01

4.1838e-01

2.9609e-01

4.2085e-01

8.3395e-03

2.4727e-03

0.6

2.9818e-01

4.0268e-01

3.0581e-01

4.0561e-01

7.6258e-03

2.9324e-03

0.7

3.0919e-01

3.8757e-01

3.1613e-01

3.9099e-01

6.9403e-03

3.4202e-03

0.8

3.2079e-01

3.7304e-01

3.2708e-01

3.7698e-01

6.2830e-03

3.9363e-03

0.9

3.3298e-01

3.5910e-01

3.3863e-01

3.6358e-01

5.6539e-03

4.4806e-03

1

3.4575e-01

3.4575e-01

3.5080e-01

3.5080e-01

5.0531e-03

5.0531e-03

1

0.8

Alpha

0.6

0.4

0.2

0 0.8 2

0.6 1.5

0.4 1 0.2

0.5 0

0

u(t)

t

Fig. 5 The approximation of fuzzy solution by RK4 for (1)-differentiability at t = 2.

1

0.8

Alpha

0.6

0.4

0.2

0 0.7 0.6 2 0.4

1.5 1

0.2 0.5 0 u(t)

0 t

Fig. 6 The approximation of fuzzy solution by RK4 for (2)-differentiability at t = 2.


145

References [1] Zadeh, L.A. (1965), Fuzzy Sets, Information and Control, 8, 338-353. [2] Dubois, D. and Prade, H. (1982), Towards fuzzy differential calculus part 3: differentiation, Fuzzy Sets and Systems, 8, 225-233. [3] Seikkala, S. (1987), On the fuzzy initial value problem, Fuzzy Sets and Systems, 24(3), 319-330. [4] Kaleva, O. (1987), Fuzzy differential equations, Fuzzy Sets and Systems, 24, 301-317. [5] Friedman, M., Ma, M., and Kandel, A. (1999), Numerical solutions of fuzzy differential and integral equations, Fuzzy Sets and Systems, 106 35-48. [6] Ma, M., Friedman, M., and Kandel, A. (1999), Numerical solutions of fuzzy differential equations, Fuzzy Sets and Systems, 105, 133-138. [7] Abbasbandy, S. and Allahviranloo, T. (2002), Numerical solutions of fuzzy differential equations by Taylor Method, Computational Methods in Applied Mathematics, 2, 113-124. [8] Abbasbandy, S. and Allahviranloo, T. (2004), Numerical solution of Fuzzy differential equation by RungeKutta method, Nonlinear Studies, 11, 117-129. 199-211. [9] Allahviranloo, T., Ahmady, N., and Ahmady, E. (2007), Numerical solutions of fuzzy differential equations by predictor-corrector method, Information Sciences, 177(7), 1633-1647. [10] Khastan, A. and Ivaz, K. (2009), Numerical solution of fuzzy differential equations by Nystrom method, Chaos, Solitons and Fractals, 41, 859-868. [11] Bede, B. and Gal, S.G. (2004), Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems, 147, 385-403. [12] Bede, B. and Gal, S.G. (2005), Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151, 581-599. [13] Bede, B., Rudas, I.J., and Bencsik, A.L. (2007), First order linear fuzzy differential equations under generalized differentiability, Information Sciences, 177, 1648-1662. [14] Chalco-Cano, Y. and Roman-Flores, H. (2008), On new solutions of fuzzy differential equations, Chaos Solitons Fractals, 38, 112-119. [15] Khastan, A., Bahrami, F., and Ivaz, K. (2009), New results on multiple solutions for Nth-order fuzzy differential equations under generalized differentiability, Boundary value Problems, 13, Article ID 395714. [16] Nieto, J.J., Khastan, A., and Ivaz, K. (2009), Numerical solution of fuzzy differential equations under generalized differentiability, Nonlinear Analysis: Hybrid system, 3, 700-707. [17] Khastan, A., Nieto, J.J., and Rodriguez-Lopez, R. (2014), Fuzzy delay differential equations under generalized differentiability, Information Sciences, 275, 145-167. [18] Bede, B. (2008), Note on “Numerical solutions of fuzzy differential equations by predictor corrector method”, Information Sciences, 178, 1917-1922. [19] Diamond, P. and Kloeden, P. (1994), Metric Spaces of Fuzzy sets, World Scientific, Singapore. [20] Puri, M.L. and Ralescu, D.A. (1983), Differentials of fuzzy functions, Journal of Mathematical Analysis Application, 91, 552-558. [21] Lupulescu, V. (2009), On a class of fuzzy functional differential equations, Fuzzy Sets and Systems, 160, 1547-1562. [22] Hale, J.K. (1997), Theory of Functional Differential Equations, Springer, New York. [23] Khastan, A., Nieto, J.J., and Rodriguez-Lopez, R. (2011), Variation of constant formula for first order fuzzy differential equations, Fuzzy Sets and Systems, 177, 20-33. [24] Kuang, Y. (1993), Delay differential equations with Applications in population dynamics, Academic Press, Boston. [25] Schuster, R. and Schuster, H. (1995), Reconstruction models for the Ehrlich ascites tumor of the mouse in O. Arino, D. Axelrod, M. Kimmel, (Eds), Mathematical Population Dynamics, Volume 2, Wuertz, Winnipeg, Canada, 335-348. [26] Mackey, M.C. (1977), Leon Glass, Oscillation and chaos in physiological control systems, Science, 197 287289. [27] Akira Shibata. and Nobuhiko Saito. (1980), Time delays and chaos in two competing species, Mathematical Biosciences, 51, 199-211. [28] May, R.M. (1980), Nonlinear phenomena in ecology and epidemiology, The New York Academy of Sciences, 357 267-281. [29] Doney Farmer, J. (1982), Choatic attractors of an infinite-dimensional dynamical system, Physica, 4D, 366393. [30] Grassberger, P. (1984), Dimensions and entropies of strange attractors from a fluctuating dynamic approach, Physica, 13D, 34-54. [31] Lakshmikantham, V. and Mohapatra, R.N. (2003), Theory of Fuzzy Differential Equations and Inclusions,

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Existence of Positive Solutions for System of Second Order Integro-differential Equations with Multi-point Boundary Conditions on Time Scales V. Krishnaveni†, K. Sathiyanathan Department of Mathematics, SRMV College of Arts and Science, Coimbatore-641020, India Submission Info Communicated by J.A.T. Machado Received 21 February 2017 Accepted 27 May 2017 Available online 1 July 2018

Abstract In this paper, we have investigated the existence of positive solutions for system of nonlinear itegro-differential equations with multi(m)point boundary conditions on time scales. Existence of positive solutions are established via Guo-Krasnosel’skii fixed point theorem for operators on a cone in a Banach space. An example is given to illustrate the effectiveness of our proposed result.

Keywords Integro-differential equations Multi-point boundary conditions Green’s function Time scale


1 Introduction Integrodifferential equations are experienced in numerous regions of science and innovation. It is outstanding that the idea of eventual outcome presented in physics is essential. To model procedures with delayed consequences then again defer it is not adequate to utilize ordinary or partial differential equations. A way to deal with resolve this issue is to utilize integrodifferential equations. Particularly one generally depicts a model which has inherited properties by integrodifferential equations by and by. The time scale hypothesis shows a structure where, a built up consequence of a general time scale is applied to special cases. On the off chance that T = R and T = Z then we have the outcomes for differential and difference equations respectively. A lot of work has been done since 1988 binding together and amplifying the theories of differential and difference equations [1]. Some basic definitions and theorems on time scales can be found in the standard books [2,3]. Dynamic equation on time scale has a huge possible application in the fields of population dynamics [2], quantum mechanics, electrical engineering, neural networks, heat transfer analysis and combinatorics [2,4]. Gupta [5] obtained results for certain three-point boundary value problems for nonlinear ordinary differential equations. Since then there are, many authors have investigated the existence of positive solutions and solutions to m-point boundary value problems on time scales [2, 6–11]. Very few authors have studied the results of integro-differential equations on time scales [12–16]. † Corresponding


148

V. Krishnaveni, K. Sathiyanathan / Journal of Applied Nonlinear Dynamics 7(2) (2018) 147–163

Motivated by the results of the above studies, we are planning to extend these results to a system of integro-differential equations on time scales satisfying the multi-point boundary conditions. In this paper, we consider the system of nonlinear second order dynamic equations on time scales ˆ t ⎫ ΔΔ ¯ ⎪ λ h(t, s, x(s), y(s))Δs = 0, t ∈ [a, σ (b)]T ,⎪ x + λ p(t) f (y, y(t)) + ⎬ a (1) ˆ t ⎪ ⎪ ΔΔ μ¯ k(t, s, x(s), y(s))Δs = 0, t ∈ [a, σ (b)]T ,⎭ y + μ q(t)g(y, y(t)) + a

satisfying the multi point boundary conditions Δ

x(a) = 0, α1 x(σ (b)) + β1 x (σ (b)) =

m−1

∑

k=2 n−1

⎫ ⎪ ⎪ x (ξk ), m ≥ 3,⎪ ⎪ ⎬ Δ

(2)

⎪ ⎪ ⎪ y(a) = 0, α2 y(σ (b)) + β2 y (σ (b)) = ∑ y (ηk ), n ≥ 3, ⎪ ⎭ Δ

Δ

k=2

where T is the time scale with a, σ 2 (b) ∈ T, 0 ≤ a < ξ1 < · · · < ξm−1 < σ (b), 0 ≤ a < η1 < · · · < ηn−1 < σ (b). We shall give sufficient conditions on λ , λ¯ , μ , μ¯ , f , h, g and k such that the BVP (1)-(2) has positive solutions. By a positive solution of the BVP (1)-(2), we mean a pair (x, y) ∈ C 2 ([a, σ (b)]T ) × C 2 ([a, σ (b)]T ) satisfying (1)-(2) with y ≥ 0, y(t) ≥ 0 for all t ∈ [a, σ (b)]T and (x, y) = (0, 0). We assume the following conditions that hold, good throughout the paper: (A1) the functions f , h, g, k : R+ × R+ → R+ are continuous, (A2) the functions p, q : [a, σ (b)]T → R+ are continuous p, q do not vanish identically on any closed interval [a, σ (b)]T . (A3) α1 , α2 , β1 and β2 are positive constants such that α1 ≥

β1 , α2 ξ2 −a

≥

β2 η2 −a , β1

> m − 2 and β2 > n − 2,

(A4) each of these f (x, y) s g(x, y) , h0 = , lim sup x+y x+y (x,y)→(0+ ,0+ ) (x,y)→(0+ ,0+ ) f (x, y) i g(x, y) , h0 = , lim+ + inf lim+ + inf f0i = x+y x+y (x,y)→(0 ,0 ) (x,y)→(0 ,0 ) f (x, y) f (x, y) , hs∞ = , lim sup lim sup f∞s = x+y x+y (x,y)→(∞,∞) (x,y)→(∞,∞) f (x, y) f (x, y) , hi∞ = , lim inf lim inf f∞i = x+y x+y (x,y)→(∞,∞) (x,y)→(∞,∞)

f0s =

lim

sup

exist as positive real numbers. Similarly we define gs0 , k0s , gi0 , k0i , gs∞ , k∞s , gi∞ , k∞i . 2 Green’s function and bounds In this section, we construct the Green’s functions for the homogeneous problems corresponding to (1)-(2) and estimate bounds for the Green’s functions. Let G(t, s) be the Green’s function for the homogeneous BVP, −xΔΔ = 0,

t ∈ [a, σ (b)]T ,

x(a) = 0, α1 x(σ (b)) + β1 xΔ (σ (b)) =

(3) m−1

∑ xΔ (ξk ),

k=2

m ≥ 3.

(4)


149

Lemma 1. Let d1 = α1 (σ (b)− a)+ β1 − m + 2 = 0. Then the Green’s function G(t, s) for the homogeneous BVB (1)-(2) is given by ⎧ G1 (t, s), a ≤ s ≤ σ (s) ≤ ξ2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ G2 (t, s), ξ2 ≤ s ≤ σ (s) ≤ ξ3 , .. (5) G(t, s) = . ⎪ ⎪ ⎪ (t, s), ξm−2 ≤ s ≤ σ (s) ≤ ξm−1 , G ⎪ ⎪ ⎩ m−2 Gm−1 (t, s), ξm−1 ≤ s ≤ σ (s) ≤ σ (b), where

⎧ 1 ⎪ ⎪ ⎨ [(α1 (σ (b) − t) + β1 − m + j + 1)(σ (s) − a) + ( j − 1)(t − σ (s))], σ (s) ≤ t, d1 G j (t, s) = 1 ⎪ ⎪ ⎩ (t − a)[α1 (σ (b) − σ (s)) + β1 − m + j + 1], t ≤ s, d1

for all j = 1, 2, · · · , m − 1. Proof. It is easy to see that, if b(t) ∈ C ([a, σ (b)]T , R+ ), then the following problem −xΔΔ = b(t),

t ∈ [a, σ (b)]T ,

satisfying the boundary conditions (4) has a unique solution ˆ σ (b) ˆ t m−1 ˆ ξk 1 (α1 (σ (b) − σ (s)) + β1 )b(s)Δs − ∑ b(s)Δs] − (t − σ (s))b(s)Δs. y = (t − a)[ d1 a a k=2 a Rearranging the terms, it can be written as ˆ σ (b) ˆ ξ2 ˆ ξ j+1 m−2 1 (α1 (σ (b) − σ (s)) + β1 )b(s)Δs − (m − 2) b(s)Δs − ∑ (m − j − 1) b(s)Δs] y = (t − a)[ d1 a a ξj j=2 ˆ t + (σ (s) − t)b(s)Δs. a

Case 1. Let a ≤ s ≤ σ (s) ≤ ξ2 and σ (s) ≤ t. Then we have 1 (t − a)[α1 (σ (b) − σ (s)) + β1 − (m − 2)] + σ (s) − t d1 1 = (α1 (σ (b) − t) + β1 − m + 2)(σ (s) − a). d1

G(t, s) =

Case 2. Let a ≤ s ≤ σ (s) ≤ ξ2 and t ≤ s. Then we have G(t, s) =

1 (t − a)[α1 (σ (b) − σ (s)) + β1 − m + 2]. d1

Case 3. Let ξ j ≤ s ≤ σ (s) ≤ ξ j+1 , for j = 2, 3, · · · , m − 2 and σ (s) ≤ t. Then we have 1 (t − a)[α1 (σ (b) − σ (s)) + β1 − (m − j − 1)] + σ (s) − t d1 1 = [(α1 (σ (b) − t) + β1 − m + j + 1)(σ (s) − a) + ( j − 1)(t − σ (s))]. d1

G(t, s) =

150


Case 4. Let ξ j ≤ s ≤ σ (s) ≤ ξ j+1 for j = 2, 3...m − 2 and t ≤ s. Then we have G(t, s) =

1 (t − a)[α1 (σ (b) − σ (s)) + β1 − m + j + 1]. d1

Case 5. Let ξm−1 ≤ s ≤ σ (s) ≤ σ (b), and σ (s) ≤ t. Then we have 1 (t − a)[α1 (σ (b) − σ (s)) + β1 ] + σ (s) − t d1 1 = [(α1 (σ (b) − t) + β1 )(σ (s) − a) + (m − 2)(t − σ (s))]. d1

G(t, s) =

Case 6. Let ξm−1 ≤ s ≤ σ (s) ≤ σ (b) and t ≤ s. Then we have G(t, s) =

1 (t − a)[α1 (σ (b) − σ (s)) + β1 ]. d1

Lemma 2. Assume that the condition (A3) is satisfied. Then the Green’s function G(t, s) of (1)-(2) is positive, for all (t, s) ∈ (a, σ (b))T × (a, b)T . Proof. By simple algebraic calculations, we can easily establish the positivity of the Green’s function. Lemma 3. Assume that the condition (A3) is satisfied. Then the Green’s function G(t, s) in (5) satisfies the following inequality: g(t)G(σ (s), s) ≤ G(t, s) ≤ G(σ (s), s), for all (t, s) ∈ [a, σ (b)]T × [a, σ (b)]T ,

(6)

where g(t) = min{

t −a σ (b) − t , }. σ (b) − a σ (b) − a

(7)

Proof. The Green’s function G(t, s) is given in (5). In each case, we prove the inequality as in (6). Case 1. Let s ∈ [a, b]T and σ (s) ≤ t. Then ((α1 (σ (b) − t) + β1 − m + j + 1)(σ (s) − a) + ( j − 1)(t − σ (s)) G(t, s) = G(σ (s), s) (α1 (σ (b) − σ (s)) + β1 − m + j + 1)(σ (s) − a) α1 (σ (b) − t) + β1 − m + j + 1 + α1(t − σ (s)) = 1, ≤ α1 (σ (b) − σ (s)) + β1 − m + j + 1 and also ((α1 (σ (b) − t) + β1 − m + j + 1)(σ (s) − a) + ( j − 1)(t − σ (s)) G(t, s) = G(σ (s), s) (α1 (σ (b) − σ (s)) + β1 − m + j + 1)(σ (s) − a) σ (b) − t . ≥ σ (b) − a Case 2. Let s ∈ [a, b]T and t ≤ s. Then t −a G(t, s) = ≤ 1, G(σ (s), s) σ (s) − a and also t −a t −a G(t, s) = ≥ . G(σ (s), s) σ (s) − a σ (b) − a Hence the result.


151

Lemma 4. Assume that the condition (A3) is satisfied and s ∈ [a, b]T . Then the Green’s function G(t, s) in (5) satisfies min

t∈[ξm−1 ,σ (b)]T

G(t, s) ≥ k1 G(σ (s), s),

where k1 =

β1 − m + 2 < 1. α1 (σ (b) − a) + β1 − m + 2

(8)

Proof. By Lemma 3, we can easily establish the result. We also formulate the same results as Lemmas 1-4 above for the following BVP, −yΔΔ = 0,

t ∈ [a, σ (b)]T ,

y(a) = 0, α2 y(σ (b)) + β2 yΔ (σ (b)) =

(9) m−1

∑ yΔ (ξk ),

m ≥ 3.

(10)

k=2

where 0 ≤ a < η2 < ... < ηn−1 < σ (b). If d2 = α2 (σ (b) − a) + β2 − n + 2 = 0, we denote by H(t, s), the Green’s function for the homogeneous BVP (9) - (10) and define in a similar manner as G(t, s). Under similar assumptions as those from Lemmas 2-4, we have (B1) the Green’s function H(t, s) is positive , for all (t, s) ∈ (a, σ (b))T × (a, b)T , (B2) g(t)H(σ (s), s) ≤ H(t, s) ≤ H(σ (s), s), for all (t, s) ∈ [a, σ (b)]T × [a, b]T , where g(t) is given in 7, (B3) min

t∈[ηn−1 ,σ (b)]T

H(t, s) ≥ k2 H(σ (s), s), s ∈ [a, b]T

where k2 =

β2 − n + 2 < 1. α2 (σ (b) − a) + β2 − n + 2

(11)

To establish criteria for the existence of positive solutions for the BVP (1) -(2), we will employ the following Guo Krasnosel’skii fixed point theorem. Theorem 5. Let X be a Banach space, k ⊆ X be a cone and suppose that Ω1 , Ω2 are open subsets of X ¯ 2 \ Ω1 ) → k is completely continuous operator with 0 ∈ Ω1 and Ω¯1 ⊂ Ω2 . Suppose further that T : k ∩ (Ω such that either (i) Tu ≤ u , u ∈ k ∩ ∂ Ω1 and Tu ≥ u , u ∈ k ∩ ∂ Ω2 , or (ii) Tu ≥ u , u ∈ k ∩ ∂ Ω1 and Tu ≤ u , u ∈ k ∩ ∂ Ω2 holds. ¯ 2 \ Ω1 ) Then T has a fixed point in k ∩ (Ω

152


3 Existence of positive solutions In this section, we shall give sufficient condition on λ , μ , λ¯ , μ¯ , f , h, g and k such that the BVP (1) -(2) has positive solutions in a cone. We consider the Banach space E = {x/x ∈ C[a, σ (b)]T } with the supremum norm · , and the Banach space Z = E × E with the norm (x, y) Z = x + y , where x = sup |y|, t ∈ [a, σ (b)]T . Define a cone P ⊂ Z by min (y + y(t)) ≥ k (x, y) Z },

P = {(x, y) ∈ Z|y ≥ 0, y(t) ≥ 0 on [a, σ (b)]T and

t∈[l,σ (b)]T

where l = max{ξm−1 , ηn−1 }, k = min{k1 , k2 } and k1 , k2 are defined in (8) and (11) respectively. We shall present some existence results for the positive solutions of the BVP (1)-(2) under various assumptions on f0s , gs0 , hs0 , k0s , f0i , gi0 , hi0 , k0i , f∞s , gs∞ , hs∞ , k∞s , f∞i , gi∞ , hi∞ and k∞i . Now, we define the positive numbers U1 , U2 , U3 , U4 , V1 , V2 , V3 and V4 by ˆ ˆ α i σ (b) α s σ (b) −1 G(σ (s), s)p(s)Δs] , U2 = [ f0 G(σ (s), s)p(s)Δs]−1 , U1 = [k k1 f∞ 2 2 l a ˆ ˆ β i σ (b) β s σ (b) −1 H(σ (s), s)q(s)Δs] , U4 = [g0 H(σ (s), s)q(s)Δs]−1 , U3 = [k k2 g∞ 2 2 l a ˆ σ (b) ˆ s ˆ ˆ α i α s σ (b) s −1 G(σ (s), s)Δτ Δs] , V2 = [h0 G(σ (s), s)Δτ Δs]−1 , V1 = [k k1 h∞ 2 2 l a a a ˆ σ (b) ˆ s ˆ σ (b) ˆ s β β H(σ (s), s)Δτ Δs]−1 , V4 = [k0s H(σ (s), s)q(s)Δτ Δs]−1 , V3 = [k k2 k∞i 2 2 l a a a where α > 0 and β > 0 are two positive real numbers such that α + β = 1. Theorem 6. Assume that the conditions (A1)-(A4) hold. (i) If f0s , gs0 , hs0 , k0s , f∞i , gi∞ , hi∞ , k∞i ∈ (0, ∞), U1 < U2 , U3 < U4 , V1 < V2 and V3 < V4 , then for each λ ∈ (U1 ,U2 ), λ¯ ∈ (V1 ,V2 ), μ ∈ (U3 ,U4 ) and μ¯ ∈ (V3 ,V4 ) there exists a positive solution (y, y(t)) on [a, σ (b)]T for (1)-(2). (ii) If f0s = gs0 = hs0 = k0s = 0, f∞i , gi∞ , hi∞ , k∞i ∈ (0, ∞), then for each λ ∈ (U1 , ∞), λ¯ ∈ (V1 , ∞), μ ∈ (V3 , ∞) and μ¯ ∈ (V3 , ∞) there exists a positive solution (y, y(t)) on [a, σ (b)]T for (1)-(2). (iii) If f0s , gs0 , hs0 , k0s ∈ (0, ∞), f∞i = gi∞ = hi∞ = k∞i = ∞, then for each λ ∈ (0,U2 ), λ¯ ∈ (0,V2 ), μ ∈ (0,U4 ) and μ¯ ∈ (0,V4 ) there exists a positive solution (y, y(t)) on [a, σ (b)]T for (1)-(2). (iv) If f0s = gs0 = hs0 = k0s = 0, f∞i , gi∞ , hi∞ , k∞i = ∞, then for each λ ∈ (0, ∞), λ¯ ∈ (0, ∞), μ ∈ (0, ∞) and μ¯ ∈ (0, ∞) there exists a positive solution (y, y(t)) on [a, σ (b)]T for (1)-(2). Proof. (i) Let T1 , T2 : P → E and T : P → Z be the operators defined by ˆ T1 (x, y)(t) = λ

σ (b)

a

ˆ T2 (x, y)(t) = μ

ˆ

a

σ (b)

ˆ

σ (b)

G(t, s)p(s) f (x(s), y(s))Δs + λ¯

s

G(t, s) ˆ

a

ˆ

σ (b)

H(t, s)q(s)g(x(s), y(s))Δs + μ¯

h(s, τ , x(τ ), y(τ ))Δτ Δs,

a s

H(t, s) a

k(s, τ , x(τ ), y(τ ))Δτ Δs

a

and T (x, y)(t) = (T1 (x, y)(t), T2 (x, y)(t)),

for (x, y) ∈ Z,


153

where G(t, s) and H(t, s) are the Green’s functions for homogeneous BVPs (1)-(2) and (9)-(10) respectively. It is obvious that a fixed point of T is a solution of the BVP (1)-(2). Now we show that T : P → P. Let (x, y) ∈ P. From Lemma 2 and (B1), T1 (x, y)(t) ≥ 0 and T2 (x, y)(t) ≥ 0 on [a, σ (b)]T . Also, for (x, y) ∈ P, by Lemma 3, we have ˆ T1 (x, y)(t) = λ ˆ

σ (b)

ˆ a

σ (b)

s

G(t, s)

a

≤λ

ˆ

σ (b)

G(t, s)p(s) f (x(s), y(s))Δs + λ¯

ˆ

σ (b)

G(σ (s), s)p(s) f (x(s), y(s))Δs + λ¯

a

h(s, τ , x(τ ), y(τ ))Δτ Δs,

a

ˆ

s

G(σ (s), s)

a

h(s, τ , x(τ ), y(τ ))Δτ Δs

a

so that ˆ

T1 (x, y) ≤ λ

σ (b)

ˆ

σ (b)

G(σ (s), s)p(s) f (x(s), y(s))Δs + λ¯

a

ˆ

s

G(σ (s), s)

a

h(s, τ , x(τ ), y(τ ))Δτ Δs.

a

Next if (x, y) ∈ P, then by Lemma 4, we have min

t∈[l,σ (b)]T

T1 (x, y)(t) ≥ =

min

t∈[ξm−1 ,σ (b)]T

min

t∈[ξm−1 ,σ (b)]T

ˆ

T1 (x, y)(t) ˆ

G(t, s)p(s) f (x(s), y(s))Δs a

ˆ

σ (b)

+λ¯

σ (b)

{λ

s

G(t, s) a

ˆ ≥ λ k1

h(s, τ , x(τ ), y(τ ))Δτ Δs}

a

σ (b)

G(σ (s), s)p(s) f (x(s), y(s))Δs

a

ˆ

+λ¯ k1

σ (b)

ˆ

s

G(σ (s), s)

a


a

≥ k1 T1 (x, y) . In a similar manner, we conclude that min

t∈[l,σ (b)]T

T2 (x, y)(t) ≥ k2 T2 (x, y) .

Therefore, min (T1 (x, y)(t) + T2 (x, y)(t)) ≥

t∈[l,σ (b)]T

min

t∈[l,σ (b)]T

T1 (x, y)(t) +

min

t∈[l,σ (b)]T

T2 (x, y)(t)

≥ k1 T1 (x, y) + k2 T2 (x, y)

≥ k (T1 (x, y), T2 (x, y)) Z = k (T (x, y) Z .

Hence T (x, y) ∈ P and so T : P → P. By standard arguments, we can easily show that T1 and T2 are completely continuous and so, T is completely continuous operator. Now, let λ ∈ (U1 ,U2 ), λ¯ ∈ (V1 ,V2 ), μ ∈ (U3 ,U4 ), μ¯ ∈ (V3 ,V4 ) and let ε > 0 be a positive number such that ε > f∞i , gi∞ , hi∞ , k∞i and

154


ˆ

α [kk1 ( f∞i − ε ) 2

l

−1

G(σ (s), s)p(s)Δs]

α ≤ λ , [( f0s + ε ) 2

ˆ

σ (b)

G(σ (s), s)p(s)Δs]−1 ≥ λ ,

a

ˆ σ (b) ˆ s α s ¯ G(σ (s), s)Δτ Δs] ≤ λ , [(h0 + ε ) G(σ (s), s)Δτ Δs]−1 ≥ λ¯ , 2 l a a a ˆ σ (b) ˆ σ (b) β β s i −1 [kk2 (g∞ − ε ) H(σ (s), s)q(s)Δs] ≤ μ , [(g0 + ε ) H(σ (s), s)q(s)Δs]−1 ≥ μ , 2 2 l a ˆ σ (b) ˆ s ˆ σ (b) ˆ s β β s i −1 ¯ [kk2 (k∞ − ε ) H(σ (s), s)Δτ Δs] ≤ μ , [(k0 + ε ) H(σ (s), s)Δτ Δs]−1 ≥ μ¯ . 2 2 l a a a

α [kk1 (hi∞ − ε ) 2

ˆ

σ (b)

σ (b) ˆ s

−1

By the definitions of f0s , gs0 , hs0 and k0s , there exists J1 > 0 such that f (x, y) ≤ ( f0s + ε )(x + y), g(x, y) ≤ (gs0 + ε )(x + y), h(x, y) ≤ (hs0 + ε )(x + y), k(x, y) ≤ (k0s + ε )(x + y), 0 < x + y ≤ J1 . By (A1), the above inequalities are also valid for x = y = 0. Let (x, y) ∈ P with (x, y) Z = J1 i.e., x + y = J1 . Then, from Lemma 3, for a ≤ t ≤ σ (b), we have ˆ s ˆ σ (v) ˆ σ (b) ¯ G(t, s)p(s) f (x(s), y(s))Δs + λ G(t, s) h(s, τ , x(τ ), y(τ ))Δτ Δs, T1 (x, y)(t) = λ a

ˆ

a

σ (b)

≤λ

a

ˆ

G(σ (s), s)p(s)( f0s + ε )(x(s) + y(s))Δs

σ (b)

+λ¯ a

≤

ˆ

s

G(σ (s), s) a

ˆ

λ ( f0s + ε )

σ (b)

(hs0 + ε )(x(τ ) + y(τ ))Δτ Δs,

G(σ (s), s)p(s)( x + y )Δs

a

ˆ

+λ¯ (hs0 + ε )

σ (b)

ˆ

s

G(σ (s), s)

a

( x + y )Δτ Δs,

a

≤ α ( x + y ) = α (x, y) Z . Hence,

T1 (x, y) ≤ α (x, y) Z In a similar manner, we conclude that

T2 (x, y) ≤ β (x, y) Z Therefore,

T (x, y) Z = (T1 (x, y), T2 (x, y)) Z = T1 (x, y) + T2 (x, y)

≤ α (x, y) Z + β (x, y) Z = (α + β ) (x, y) Z = (x, y) Z .

ˆ

s

G(σ (s), s)

a

σ (b)

≤λ

σ (b)

G(σ (s), s)p(s) f (x(s), y(s))Δs + λ¯

a

ˆ

a

ˆ

a

h(s, τ , x(τ ), y(τ ))Δτ Δs,


155

Hence, T (x, y) Z ≤ (x, y) Z . If we set Ω1 = {(x, y) ∈ Z | (x, y) Z < J1 } . then

T (x, y) Z ≤ (x, y) Z , for (x, y) ∈ P ∩ ∂ Ω1

(12)

By the definitions of f∞i , gi∞ , hi∞ and k∞i , there exists J¯2 > 0 such that f (x, y) ≥ ( f∞i − ε )(x + y), g(x, y) ≥ (gi∞ − ε )(x + y), h(x, y) ≥ (hi∞ − ε )(x + y), k(x, y) ≥ (k∞i − ε )(x + y), x + y ≥ J¯2 . Let

J¯2 . J2 = max 2J1 , k Choose (x, y) ∈ P with (x, y) Z = J2 . Then min (y + y(t)) ≥ k (x, y) Z ≥ J¯2 .

t∈[l,σ (b)]T

From Lemma 4, we have ˆ T1 (x, y)(t) = λ

σ (b)

ˆ

a

σ (b)

ˆ

+λ¯ k1 ˆ ≥ λ k1


a

l

+λ¯ k1

σ (b)

a σ (b)

ˆ

ˆ

s

G(σ (s), s)

h(s, τ , x(τ ), y(τ ))Δτ Δs,

a


σ (b) l

≥ λ k1 ( f∞i − ε )

ˆ

s

G(σ (s), s) ˆ

h(s, τ , x(τ ), y(τ ))Δτ Δs,

a

σ (b)

l

+λ¯ k1 (h∞i − ε )

ˆ

G(σ (s), s)p(s)k (x, y) Z Δs

σ (b)

l

ˆ

s

G(σ (s), s)

α α

(x, y) Z + (x, y) Z 2 2 = α (x, y) Z .

s

G(t, s) a

ˆ

≥ λ k1

ˆ

σ (b)

G(t, s)p(s) f (x(s), y(s))Δs + λ¯

(x, y) Z Δτ Δs,

a

≥

Hence,

T1 (x, y) ≥ α (x, y) Z . In a similar manner, we conclude that

T2 (x, y) ≥ β (x, y) Z .

a

h(s, τ , x(τ ), y(τ ))Δτ Δs,

156


Therefore,


≥ α (x, y) Z + β (x, y) Z = (α + β ) (x, y) Z = (x, y) Z . Hence, T (x, y) Z ≥ (x, y) Z . If we set Ω2 = {(x, y) ∈ Z | (x, y) Z < J2 } . then

T (x, y) Z ≥ (x, y) Z , for (x, y) ∈ P ∩ ∂ Ω2

(13)

¯ 2 \ Ω1 ) and Applying Theorem 5 to (12) and (13), we obtain that T has a fixed point (x, y) in P ∩ (Ω hence the BVB (1)-(2) has a positive solution such that J1 ≤ x + y ≤ J2 . (ii) Let λ ∈ (U1 , ∞), λ¯ ∈ (V1 , ∞), μ¯ ∈ (U3 , ∞) and μ¯ ∈ (V3 , ∞) and let ε > 0 be a positive number such that f∞i , gi∞ , hi∞ , k∞i > ε and

α [kk1 ( f∞i − ε ) 2 α [kk1 (hi∞ − ε ) 2 β [kk2 (gi∞ − ε ) 2

.

ˆ

σ (b)

l

ˆ ˆ

G(σ (s), s)p(s)Δs]−1 ≤ λ ,

σ (b) ˆ s

l

G(σ (s), s)Δτ Δs]−1 ≤ λ¯ ,

a

σ (b)

H(σ (s), s)q(s)Δs]−1 ≤ μ ,

l

ˆ σ (b) ˆ s β [kk2 (k∞i − ε ) H(σ (s), s)Δτ Δs]−1 ≤ μ¯ , 2 l a ˆ σ (b) α [ G(σ (s), s)p(s)Δs]−1 ≥ ε , 2λ a ˆ σ (b) ˆ s α [ G(σ (s), s)Δτ Δs]−1 ≥ ε , 2λ¯ a a ˆ σ (b) β [ H(σ (s), s)q(s)Δs]−1 ≥ ε , 2μ a ˆ σ (b) ˆ s β [ H(σ (s), s)Δτ Δs]−1 ≥ ε , 2μ¯ a a

By the definitions of f0s = gs0 = hs0 = k0s = 0, there exists J1 > 0 such that f (x, y) ≤ ε (x + y), g(x, y) ≤ ε (x + y), h(x, y) ≤ ε (x + y), k(x, y) ≤ ε (x + y), 0 ≤ x + y ≤ J1 . Let (x, y) ∈ P with (x, y) Z = J1 , i.e., x + y = J1 . Then, from Lemma 3, for a ≤ t ≤ σ (b), we have


ˆ T1 (x, y)(t) = λ

σ (b)

ˆ G(t, s)p(s) f (x(s), y(s))Δs + λ¯

a

ˆ ˆ

ˆ

σ (b)

σ (b)

ˆ

ˆ

s

h(s, τ , x(τ ), y(τ ))Δτ Δs,

a

G(σ (s), s)p(s)ε (x(s) + y(s))Δs

σ (b)

+λ¯

ˆ

G(σ (s), s)

a

a

≤ λε

h(s, τ , x(τ ), y(τ ))Δτ Δs,

a

G(σ (s), s)p(s) f (x(s), y(s))Δs + λ¯

a

≤λ

s

G(t, s) a

σ (b)

≤λ

ˆ

σ (b)

157

a σ (b)

ˆ

s

G(σ (s), s)

ε (x(τ ) + y(τ ))Δτ Δs,

a

ˆ

σ (b)

G(σ (s), s)p(s)( x + y )Δs + λ¯ ε

a

ˆ

s

G(σ (s), s)

a

( x + y )Δτ Δs,

a

≤ α ( x + y ) = α (x, y) Z . Hence,

T1 (x, y) ≤ α (x, y) Z . In a similar manner, we conclude that

T2 (x, y) ≤ β (x, y) Z . Therefore,


≤ α (x, y) Z + β (x, y) Z = (α + β ) (x, y) Z = (x, y) Z . Hence, T (x, y) Z ≤ (x, y) Z . Define the set Ω1 = {(x, y) ∈ Z | (x, y) Z < J1 } . then

T (x, y) Z ≤ (x, y) Z , for (x, y) ∈ P ∩ ∂ Ω1

(14)

By the definitions of f∞i , gi∞ , hi∞ , k∞i ∈ (0, ∞) there exists J¯2 > 0 such that f (x, y) ≥ ( f∞i − ε )(x + y), g(x, y) ≥ (gi∞ − ε )(x + y), h(x, y) ≥ (hi∞ − ε )(x + y), k(x, y) ≥ (k∞i − ε )(x + y), x + y ≥ J¯2 . Define the set {Ω2 = (x, y) ∈ Z, | (x, y) Z < J2 } and proceeding in a similar manner of proof (i), we get

T (x, y) Z ≥ (x, y) Z , for (x, y) ∈ P ∩ ∂ Ω2 .

(15)

¯ 2 \ Ω2 ) and Applying Theorem 5 to (14) and (15), we obtain that T has a fixed point (x, y) in P ∩ (Ω hence the BVP (1)-(2) has a positive solution such that J1 ≤ x + y ≤ J2 . Similarly, we can prove remaining.

158


Prior to our next result, we define the positive numbers Q1 , Q2 , Q3 , Q4 , R1 , R2 , R3 and R4 by ˆ σ (b) ˆ σ (b) ˆ s γ γ G(σ (s), s)p(s)Δs]−1 , R1 = [kk1 hi0 G(σ (s), s)Δτ Δs]−1 , Q1 = [kk1 f0i 2 2 l l a ˆ ˆ σ (b) ˆ s γ s σ (b) γ G(σ (s), s)p(s)Δs]−1 , R2 = [hs∞ G(σ (s), s)Δτ Δs]−1 , Q2 = [ f ∞ 2 2 a a a ˆ σ (b) ˆ σ (b) ˆ s δ δ H(σ (s), s)q(s)Δs]−1 , R3 = [kk2 k0i H(σ (s), s)Δτ Δs]−1 , Q3 = [kk2 gi0 2 2 l l a ˆ ˆ σ (b) ˆ s δ s σ (b) δ H(σ (s), s)q(s)Δs]−1 , R4 = [k∞s H(σ (s), s)Δτ Δs]−1 , Q4 = [g∞ 2 2 a a a where γ > 0 and δ > 0 are two positive real numbers such that γ + δ = 1. Theorem 7. Assume that the conditions (A1)-(A4) hold. (i) If f0i , gi0 , hi0 , k0i , f∞s , gs∞ , hs∞ , k∞s ∈ (0, ∞), Q1 < Q2 , Q3 < Q4 , R1 < R2 and R3 < R4 , then for each λ ∈ (Q1 , Q2 ), λ¯ ∈ (R1 , R2 ), μ ∈ (Q3 , Q4 ) and μ¯ ∈ (R3 , R4 ) there exists a positive solution (y, y(t)) on [a, σ (b)]T for (1)-(2). (ii) If f∞s = gs∞ = hs∞ = k∞s = 0, f0i , gi0 , hi0 , k0i ∈ (0, ∞), then for each λ ∈ (Q1 , ∞), λ¯ ∈ (R1 , ∞), μ ∈ (Q3 , ∞) and μ¯ ∈ (R3 , ∞) there exists a positive solution (y, y(t)) on [a, σ (b)]T for (1)-(2). (iii) If f∞s , gs∞ , hs∞ , k∞s ∈ (0, ∞), f0i = gi0 = hi0 = k0i = ∞, then for each λ ∈ (0, Q2 ), λ¯ ∈ (0, R2 ), μ ∈ (0, Q4 ) and μ¯ ∈ (0, R4 ) there exists a positive solution (y, y(t)) on [a, σ (b)]T for (1)-(2). (iv) If f∞s = gs∞ = hs∞ = k∞s = 0, f0i = gi0 = hi0 = k0i = ∞, then for each λ ∈ (0, ∞), λ¯ ∈ (0, ∞), μ ∈ (0, ∞) and μ¯ ∈ (0, ∞) there exists a positive solution (y, y(t)) on [a, σ (b)]T for (1)-(2). Proof. Let λ ∈ (Q1 , Q2 ), λ¯ ∈ (R1 , R2 ), μ ∈ (Q3 , Q4 ) and μ¯ ∈ (R3 , R4 ) and let ε > 0 be a positive number such that ε < f0i , ε < gi0 , ε < hi0 , ε < k0i and

γ [kk1 ( f0i − ε ) 2

ˆ l

σ (b)

G(σ (s), s)p(s)Δs]−1 ≤ λ ,

ˆ σ (b) ˆ s γ i [kk1 (h0 − ε ) G(σ (s), s)Δτ Δs]−1 ≤ λ¯ , 2 l a ˆ σ (b) γ s [( f + ε ) H(σ (s), s)q(s)Δs]−1 ≥ λ , 2 ∞ a ˆ σ (b) ˆ s γ s [(h + ε ) H(σ (s), s)Δτ Δs]−1 ≥ λ¯ , 2 ∞ a a ˆ σ (b) δ i [kk2 (g0 − ε ) H(σ (s), s)q(s)Δs]−1 ≤ μ , 2 l ˆ σ (b) ˆ s δ i [kk2 (k0 − ε ) H(σ (s), s)Δτ Δs]−1 ≤ μ¯ , 2 l a ˆ σ (b) δ i [(g + ε ) H(σ (s), s)q(s)Δs]−1 ≥ μ , 2 ∞ a ˆ σ (b) ˆ s δ i [(k + ε ) H(σ (s), s)Δτ Δs]−1 ≥ μ¯ . 2 ∞ a a


159

By the definitions of f0i , gi0 ∈ (0, ∞) there exists J3 > 0 such that f (x, y) ≥ ( f0i − ε )(x + y), g(x, y) ≥ (gi0 − ε )(x + y), h(x, y) ≥ (hi0 − ε )(x + y), k(x, y) ≥ (k0i − ε )(x + y), 0 < x + y ≤ J3 . By (A1), the above inequalities are also valid for x = y = 0. Let (x, y) ∈ P with (x, y) Z = J3 i.e., x + y = J3 . Then, from Lemma 4, for a ≤ t ≤ σ (b), we have ˆ T1 (x, y)(t) = λ

σ (v)

a

≥ λ k1

σ (b) l

+λ¯ k1 ˆ ≥ λ k1

ˆ

σ (b)

l

+λ¯ k1

ˆ

ˆ

h(s, τ , x(τ ), y(τ ))Δτ Δs,

a


σ (b)

l σ (b)

s

G(t, s) a

ˆ

≥


ˆ

s

G(σ (s), s)

h(s, τ , x(τ ), y(τ ))Δτ Δs,

a

G(σ (s), s)p(s)( f0i − ε )(x(s) + y(s))Δs

σ (b) l

ˆ ˆ

λ k1 ( f0i − ε )

s

G(σ (s), s)

l

+λ¯ k1 (hi0 − ε )

a

σ (b)

ˆ

(hi0 − ε )(x(τ ) + y(τ ))Δτ Δs,

G(σ (s), s)p(s)k (x, y) Z Δs

σ (b)

l

γ γ ≥ x, y Z + x, y Z 2 2 = γ (x, y) Z .

ˆ

s

G(σ (s), s)

k (x, y) Z Δτ Δs,

a

Hence,

T1 (x, y) ≥ γ (x, y) Z . In a similar manner, we conclude that

T2 (x, y) ≥ δ (x, y) Z . Therefore,


≥ γ (x, y) Z + δ (x, y) Z = (γ + δ ) (x, y) Z = (x, y) Z . Hence, T (x, y) Z ≥ (x, y) Z . If we set Ω3 = {(x, y) ∈ Z | (x, y) Z < J3 } . then

T (x, y) Z ≥ (x, y) Z , for (x, y) ∈ P ∩ ∂ Ω3 .

(16)


160

Now we define the functions f ∗ , g∗ , h∗ , k∗ : R+ → R+ by f ∗ (w) = max

0≤x+y≤w

f (x, y), g∗ (w) = max g(x, y) 0≤x+y≤w

∗

∗

for all w ∈ R+ .

h (w) = max h(x, y), k (w) = max k(x, y), 0≤x+y≤w

Then

0≤x+y≤w

f (x, y) ≤ f ∗ (w), g(x, y) ≤ g∗ (w), h(x, y) ≤ h∗ (w), k(x, y) ≤ k∗ (w), x + y ≤ w.

It follows that the functions f ∗ , g∗ , h∗ and k∗ are nondecreasing and satisfy the conditions f ∗ (w) g∗ (w) h∗ (w) = f∞s , lim sup = gs∞ , lim max sup = hs∞ , w→∞ w→∞ w→∞ t∈[u,σ (b)] w w w k∗ (w) lim max sup = k∞s . w→∞ t∈[a,σ (b)] w lim sup

Next, by the definitions of f∞s , gs∞ , hs∞ , k∞s ∈ (0, ∞) there exists J¯4 > 0 such that f ∗ (w) ≤ ( f∞s + ε )w, g∗ (w) ≤ (gs∞ + ε )w, h∗ (w) ≤ (hs∞ + ε )w, k∗ (w) ≤ (k∞s + ε )w, w ≥ J¯4 . Let J4 = max{2J3 , J¯4 }, choose (x, y) ∈ P with (x, y) Z = J4 . Then, by the definitions of f ∗ , g∗ , h∗ and k∗ we have f (y, y(t)) ≤ f ∗ (y, y(t)) ≤ f ∗ ( y + y(t) ) = f ∗ ( (x, y) Z ), h(x(τ ), y(τ )) ≤ h∗ (x(τ ), y(τ )) ≤ h∗ ( x(τ ) + y(τ ) ) = h∗ ( (x, y) Z ), g(y, y(t)) ≤ g∗ (y, y(t)) ≤ g∗ ( y + y(t) ) = g∗ ( (x, y) Z ), k(x(τ ), y(τ )) ≤ k∗ (x(τ ), y(τ )) ≤ k∗ ( x(τ ) + y(τ ) ) = k∗ ( (x, y) Z ). From Lemma 3, for a ≤ t ≤ σ (a), we have ˆ T1 (x, y)(t) = λ ˆ

σ (b) a

ˆ


a

ˆ

σ (b)

G(σ (s), s)p(s) f (x(s), y(s))Δs + λ¯

ˆ

∗

ˆ

a

a

σ (b) a

ˆ

+λ¯

G(σ (s), s)p(s)( f∞s + ε ) (x, y) Δs

σ (b)

ˆ

s

G(σ (s), s)

a

a


a

σ (b)

G(σ (s), s)p(s) f ( (x, y) Z )Δs + λ¯

s

G(σ (s), s)

a

σ (b)

≤λ

s

G(t, s)

a

≤λ

ˆ

σ (b) a

σ (b)

≤λ ˆ


(hs∞ + ε ) (x, y) Δτ Δs

= γ (x, y) Z . Hence,

T1 (x, y) ≤ γ (x, y) Z . In a similar manner, we conclude that

T2 (x, y) ≤ δ (x, y) Z .

ˆ

s

G(σ (s), s) a

h∗ ( (x, y) Z )Δτ Δs


161

Therefore,


≤ γ (x, y) Z + δ (x, y) Z = (γ + δ ) (x, y) Z = (x, y) Z . Hence, T (x, y) ≤ (x, y) Z . If we set Ω4 = {(x, y) ∈ Z | (x, y) Z < J4 } , then

T (x, y) Z ≥ (x, y) Z , for (x, y) ∈ P ∩ ∂ Ω4 .

(17)

¯ 4 /Ω3 ) and Applying Theorem 5 to (16) and (17), we obtain that T has a fixed point (x, y) in P ∪ (Ω hence the BVP (1)-(2) has a positive solution such that J3 ≤ (x, y) ≤ J4 . The proof of the remaining cases (ii)-(iv) are similar that of (i) and we shall omit them.

4 Example Let us consider an example to illustrate the above result. Let T = {(1/2) p : p ∈ N0 } ∪ [1, 2]. Take m = 3, n = 4, a = 1/2, b = 2, ζ2 = 3/2, η2 = 1, n3 = 3/2, α1 = 4, α2 = 7, β1 = 2, β2 = 3. Now, consider the BVP, ˆ s ⎫ ΔΔ ¯ ⎪ λ h(t, s, y, y(t))Δt = 0, t ∈ [a, σ (2)]T ,⎪ x + λ p(t) f (y, y(t)) + ⎬ a ˆ s (18) ⎪ ⎭ λ¯ k(t, s, y, y(t))Δt = 0, t ∈ [a, σ (2)]T ,⎪ yΔΔ + μ q(t)g(y, y(t)) + a

x(1/2) = 0, 4x(σ (2)) + 2xΔ (σ (2)) = xΔ (3/2), y(1/2) = 0, 7y(σ (2)) + 3yΔ (σ (2)) = yΔ (1) + yΔ (3/2),

where (x + y)(700(x + y) + 1)(10 + cos x) (x + y)(1600(x + y) + 1)(5 + sin y) , g(x, y) = x+y+1 x+y+1 2(x+y) 2(x+y) a(x + y) + e a(x + y) + e , k(x, y) = . h(x, y) = (x+y) 2(x+y) c+e +e c + e(x+y) + e2(x+y) f (x, y) =

here p(t) = q(t) = 1, a = 800, c = 500. The Green’s function G(t, s) is G1 (t, s), 1/2 ≤ s ≤ σ (s) ≤ 3/2, G(t, s) = G2 (t, s), 3/2 ≤ s ≤ σ (s) ≤ σ (2), where

⎧ 1 ⎪ ⎨ [4(σ (2) − t) + 1](σ (s) − 1/2), σ (s) ≤ t, 7 G1 (t, s) = ⎪ ⎩ 1 (t − 1/2)[4(σ (2) − σ (s)) + 1], t ≤ s, 7

(19)

162


and ⎧ 1 ⎪ ⎨ [4(σ (2) − t) + 2](σ (s) − 1/2) + t − σ (s), σ (s) ≤ t, G1 (t, s) = 7 ⎪ ⎩ 1 (t − 1/2)[4(σ (2) − σ (s)) + 2], t ≤ s, 7 The Green’s function H(t, s) is ⎧ ⎪ ⎨H1 (t, s), 1/2 ≤ s ≤ σ (s) ≤ 1, H(t, s) = H2 (t, s), 1 ≤ s ≤ σ (s) ≤ 3/2, ⎪ ⎩ H3 (t, s), 3/2 ≤ s ≤ σ (s) ≤ σ (2), where ⎧ 2 ⎪ ⎨ [7(σ (2) − t) + 1](σ (s) − 1/2), σ (s) ≤ t, H1 (t, s) = 23 ⎪ ⎩ 2 (t − 1/2)[7(σ (2) − σ (s)) + 1], t ≤ s, 23 ⎧ 2 ⎪ ⎨ [(7(σ (2) − t) + 2)(σ (s) − 1/2) + t − σ (s)], σ (s) ≤ t, H2 (t, s) = 23 ⎪ ⎩ 2 (t − 1/2)[7(σ (2) − σ (s)) + 2], t ≤ s, 23 and ⎧ 2 ⎪ ⎨ [(7(σ (2) − t) + 3)](σ (s) − 1/2) + 2(t − σ (s)), σ (s) ≤ t, H3 (t, s) = 23 ⎪ ⎩ 2 (t − 1/2)[7(σ (2) − σ (s)) + 3], t ≤ s, 23 After simple calculations, we get k1 = 1/7, k2 = 2/23, k = 2/23, f0s = f0i = 5, f∞s = 9600, f∞i = 6400, gs0 = i s i s gi0 = 11, gs∞ = 7700, gi∞ = 6300, hi0 = hs0 = koi = k0s = 800 502 , h∞ = h∞ = k∞ = k∞ = 800, U1 = 0.04802556818α , U2 = 0116262975α , U3 = 0.08336416581β , U4 = 0.1095672886β , V1 = 0.1550917431192α , V2 = 0.4183333333α , V3 = 0.2656550218β , V4 = 0.42036407767β where α > 0, β > 0 are two positive real numbers such that α + β = 1. Employing Theorem 6 of (i), for each λ ∈ (U1 ,U2 ), λ¯ ∈ (V1 ,V2 ), μ ∈ (U3 ,U4 ) and μ¯ ∈ (V3 ,V4 ), there exist a positive solution (y, y(t)) of the BVP (18)-(19).

References [1] Hilger, S. (1990), Analysis on measure chains-a unified approach to continuous and discreare calcuus, Result. Math., 18, 18-56. [2] Bohner, M. and Peterson, A. (2001), Dynamic Equations on Time Scales: An Introduction with Applications, Birkh¨ auser, Boston. [3] Bohner, M. and Peterson, A.(Eds.)(2003), Advances in Dynamic Equations on Time Scales, Birkh¨ auser, Boston. [4] Spedding, V. (2009), Taming Nature’s Numbers, New Scientist, 28-31. [5] Gupta, C.P.(1992), Solvability of a three-point nonlinear boundary value problem for a second-order ordinary differential equations, J. Math. Anal. Appl., 168, 540-551. [6] Benchohra, M., Berhoun, F., Hamani, S., Henderson, J., Ntouyas, S.K., Ouahab, A., and PurnarasI, K.(2009), Eigenvalues for iterative of systems of nonlinear boundary value problems on time scales, Nonlinear Dyn. Sys. Theory, 9, 11 -22


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[7] Erbe, L. and Peterson, A. (1999), Green’s functions and comparison theorems for differential equations on measure chains, Dynam. Contin. Discrete Impuls. Systems, 6, 121-137. [8] Franco, D. (2001), Green’s functions and comparison results for impulsive integrodifferential equations, Nonlin. Anal. Th. Meth. Appl., 47, 5723-5728. [9] Guo, D., Lakshmikantham, V., and Liu, X. (1996), Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, Dordrecht. [10] Prasa, K.R., Sreedhar, N., and Narasimhulu, Y. (2014), Eigenvalue intervals for iterative systems of nonlinearm-point boundary value problems on time scales, Diff. Eqns. Dyn. Sys., 22, 353-368. [11] Prasad, K.R., Sreedhar, N., and Srinivas, M.A.S. (2015), Existence of positive solutions for systems of second order multi-point boundary value problems on time scales, Pro. Indian. Sci., 125, 353-370. [12] Aneta, S.N. (2010), Integrodifferential Equations on Time Scales with Henstock-Kurzweil-Pettis Delta Integrals, Abst. Appl. Anal. [13] Guo, D. (1999), Initial value problems for second-order integro-differential equations in Banach spaces, Nonlin. Anal. Th. Meth. Appl., 37, 289-300. [14] Guo, D. and Liu, X. (1993), Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces, J. Math. Anal. Appl., 177, 538-553. [15] Xing, Y., Han, M., and Zheng, G.(2005), Initial value problem for first-order integrodifferential equation of Volterra type on time scales, Nonlin. Anal. Th. Meth. Appl., 60, 429-442. [16] Xing, Y., Ding, W., and Han, M. (2008), Periodic boundary value problems of integrodifferential equation of Volterra type on time scales, Nonlin. Anal. Th. Meth. Appl., 68, 127-138.



Spatiotemporal Patterns of a Pursuit-evasion Generalist Predator-prey Model With Prey Harvesting Lakshmi Narayan Guin, Benukar Mondal, Santabrata Chakravarty† Department of Mathematics, Visva-Bharati, Santiniketan-731 235, West Bengal, India Submission Info Communicated by A.C.J. Luo Received 1 April 2017 Accepted 27 May 2017 Available online 1 July 2018 Keywords Predator-prey model Reaction-diffusion equations Prey harvesting Turing instability Spatiotemporal pattern

Abstract The present investigation deals with a diffusive predator-prey model in order to study the dynamic response of a reaction-diffusion model with linear prey harvesting. The governing equations of the proposed model system subject to the homogeneous Neumann boundary condition provide some qualitative interpretations of solutions to the reaction-diffusion system. The conditions of diffusion-driven instability and the Turing bifurcation region in two parameter space are explored. From the outcome of the present mathematical analysis carried out followed by the numerical simulations based on the model parameters, it reveals that for unequal diffusive coefficients, prey harvesting may induce that diffusion-driven instability resulting in stationary Turing patterns. The choice of parameter values is important to study the effect of prey harvesting and diffusion, while it depends more on the non-linearity of the model system. Moreover, the model dynamics exhibits the influence of both prey harvesting and diffusion controlled pattern formation growth to holes, stripes-holes mixture, stripes, labyrinthine, stripes-spots mixture and spots replication. All these features illustrate that the dynamics of the proposed model with the control of prey harvesting is not straightforward, but rich and complex in nature. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction When more than one species interact via reaction mechanisms such as predator-prey, mutualism, competition, reaction kinetics etc. and they all diffuse usually with different diffusivity constants, a situation emerges to be modeled via reaction-diffusion system. In 1952, Alan M Turing, one of the key scientists of 20th century, suggested that under certain conditions, a combination of chemical / morphogen reaction and diffusion produces steady state spatial patterns of chemical / morphogen concentration [1]. Pattern formation in nonlinear complex systems is one of the central problems of the natural, social, and technological sciences. Turing’s idea is a simple but profound one. He said that, if in the absence of diffusion, chemical species tend to a linearly stable uniform steady state then, under certain conditions, spatially inhomogeneous patterns can develop by diffusion-driven instability. Diffusion is † Corresponding


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usually believed to be a stabilizing process that is why this was such a novel concept. Such patterns are of interest because they give a possible explanation for the development of pattern and form in developmental biology [2–5] and experimental chemical systems [6]. The spatial patterns are omnipresent in natural world, which modify the temporal dynamics and stability properties of population density at a range of spatial scales, whose effects must be included in temporal ecological models that do not signify space explicitly. The spatial component of ecological interactions has been identified as an essential factor in how ecological communities are shaped [7]. Most probably, the significance of spatial aspect was first visualized by Gause [8] in his laboratory experiments with Paramecium and Didinum. A parallel effect of space on species persistence or extinction was observed and studied by Luckinbill [9, 10]. Reaction-diffusion model systems were first used to clarify ecological pattern formation by Segel and Jackson [11] supporting on the novel concept by Turing. Similar ideas were later adopted to explain patterns in plankton systems by Levin and Segel [12]. Over the last three decades, several research articles have been published on reaction-diffusion predator-prey models and various kinds of spatial patterns have been reported for these reaction-diffusion model systems [13–18]. An attempt is made in the present investigation to study Turing pattern formation in a two-species reaction-diffusion predator-prey model with Holling type II functional response. Population dynamics has drawn interest from the commercial / economical harvesting industry and from many scientific societies including biology, ecology, economics etc. From the point of view of human needs, both harvesting and predation are processes in which species members are taken away by an external agency, sometimes for population management, but more often for the benefit of the harvester. Dynamics of predator-prey models with constant-yield harvesting (harvested biomass is independent of the population size) and constant-effort harvesting (a constant proportion of the population is harvested) are studied extensively by several authors [19–23]. The exploitation of biological resources and the harvesting of some biological populations are commonly practiced in fishery, forestry, and wildlife management. Mathematical models have been used extensively and successfully to gain insight into the scientific management of renewable resources like fisheries and forestries [20, 24, 25]. Therefore, it is necessary to study the appropriate population model with prey harvesting. Predator-prey models play an important role in studying the management of renewable resources [26, 27]. The effect of harvesting on the dynamics of predator-prey systems and the role of harvesting in the management of renewable resources has been the centre of attraction [19,22,23,28]. Systematic analysis of harvested predator-prey models points out the possibility of extinction of one or both the species due to uncontrolled harvesting. In addition, it finds out the permissible stage of harvesting to ensure the long term survival of the renewable resource. It is an unquestionable fact that a non-constant harvesting method considering prey or predator species is more efficient and economic than the constant harvesting method. That is why, it is needed to take into account non-constant harvesting rate. However, the effect of linear prey harvesting on the spatiotemporal dynamics of a reaction-diffusion predator-prey system has not been reported there. To the best of our knowledge, little attention has been paid to the dynamics of a reaction-diffusion predator-prey model incorporating prey harvesting. Assuming the importance of linear prey harvesting and spatiotemporal pattern formation on predator-prey model in ecology, an attempt has been made here to investigate the influence of prey harvesting and diffusion in a Holling type II predator-prey model. Consequently, the objective of this investigation is to study systematically the dynamical properties of a reaction-diffusion predator-prey model with linear prey harvesting. The rest of this paper is organized as follows: In Section 2, the reaction-diffusion model is formulated and due emphasis is paid on the existence and feasibility criteria of the unique interior equilibrium points. The Turing bifurcation analysis around the interior equilibrium point for the model system is carried out in Section 3. In Section 4, the results of Turing pattern formation via numerical simulations are exhibited with discussion at length. Finally, conclusions and remarks are adequately presented in Section 5.


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2 The reaction-diffusion model and its analysis In real world, the spatial distributions of the predator and prey are inhomogeneous within a fixed bounded domain and each species has a common tendency to diffuse to the regions of smaller species concentration. For this reason, the reaction-diffusion equations do appear to describe spatial dispersal of each species. In the predator-prey models, the interaction between the predator and the prey acts as the reaction item while the diffusion item appears due to pursuit-evasion phenomenon, that means, predators pursuing prey and prey escaping predators. In such a system, there is a tendency that the preys stay away from the predators and the escape velocity of the preys may be taken to be proportional to the dispersive velocity of the predators. In the same manner, there is a tendency that the predators will get closer to the preys and the chase velocity of predators may be considered to be proportional to the dispersive velocity of the preys [29]. Normally, there are two major categories of predators - generalist and specialist. Specialist predators feed almost exclusively on one species of prey. But, the generalist predator is quite different from specialist predator. Generalist predators feed on many types of species. When a focal prey population is threatened by extinction, the predator is capable of changing its diet to another species and may continue to persist; as a result, their dynamics is not influenced by the activities of a specific prey population [3, 30, 31]. In the present investigation, an attempt is made to analyze the following 2D continuous pursuitevasion generalist predator-prey model with linear prey harvesting as u puv ∂u = au 1 − − Hu + D1∇2 u, − ∂t K1 u+c v epuv ∂v = bv 1 − − dv + D2 ∇2 v, + ∂t K2 u+c u(0, x, y) > 0, v(0, x, y) > 0,

(1a) (1b) (1c)

where u and v denote prey and generalist predator population size respectively at (t, x, y) in a given spatial domain of R2 ; D1 and D2 signify the natural dispersive force of movement for u(t, x, y) and v(t, x, y), respectively and ∇2 is the Laplacian operator in 2D space, describing the random moving. Generally, diffusion is believed as a spatial transmission way, which moves from high species concentration to low species concentration to get more food in a good living environment. All the parameters a, b, c, d, e, p, K1 , K2 , H, D1 , D2 of the system (1), assume only positive values and will be considered as constants throughout the investigation. The parameters a and K1 represent intrinsic rate of growth (per capita rate of change) and carrying capacity of the prey population respectively while b and K2 are the corresponding parameters to the generalist predator. d stands for the natural mortality rate for generalist predator population. p and e(0 < e < 1) are the maximal relative increase of predation and conversion factor denoting the newly born generalist predators for each captured prey. The quantities H(0 ≤ H < 1) and c, represent linear harvesting rate of prey population and the interference coefficient of the generalist predator population respectively. System (1) arises as a Holling type II reaction-diffusion generalist predator-prey model of interacting species in the same spatial domain of R2 . Assuming that the system parameters do not depend on space or time, that is, the environment is uniform. In this investigation, we assume that the predator species in the model (1) is not of commercial / economical importance. The prey species is continuously being harvested with non-constant rate in time by a harvesting agency. The harvesting activity does not affect the predator species directly. Although there are numerous work on predator-prey system incorporating the harvesting which have been carried out till date but little attention has been paid to the reaction-diffusion system (1) under the assumption of the linear prey harvesting. ˜ K2 v, ˜ at˜ , xL, ˜ yL), ˜ where L denotes We make a change of variables and time rescaling: (u, v,t, x, y) = (K1 u, the characteristic length of the system in spatial domain Ω. The non-dimensional form of the system

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(1) (after dropping tildes) is given by

∂u = f1 (u, v) + d1 ∇2 u, (x, y) ∈ Ω, t > 0, ∂t ∂v = f2 (u, v) + d2 ∇2 v, (x, y) ∈ Ω, t > 0, ∂t u(0, x, y) > 0, v(0, x, y) > 0, (x, y) ∈ Ω,

(2a) (2b) (2c)

uvγ uvε where f1 (u, v) = u(1 − u) − u+ α − hu, f2 (u, v) = β v(1 − v) + u+α − δ v and the dimensionless parameters pK2 D1 D2 d H , β = ba , γ = ep d1 = aL and d2 = aL are α = Kc1 , ε = aK 2. a , δ = a, h = a , 1 2 u = 0, (x, y) ∈ ∂ Ω, t > 0 where ∂ Ω is the closed Believing Neumann boundary condition as ∂∂η v (x,y) smooth boundary of the reaction-diffusion domain Ω and η is the unit outward normal to ∂ Ω. The homogeneous Neumann boundary condition means that the system (2) is self-contained and has no population flux across the boundary ∂ Ω. To identify Turing patterns of the system (2), one may think about a spatially homogeneous system. Consequently, one initially finds the steady state in R2+ in this fashion: (i) e0 (0, 0), total extinction of both the species; (ii) e1 (0, 1 − βδ ), extinction of the prey species; (iii) e2 (1 − h, 0), extinction of the predator species; 3 −h) ; u3 ∈ (iv) interior equilibrium point e3 (u3 , v3 ), coexistence of both the species where v3 = (α +u3 )(1−u ε (0, 1 − h), h ∈ [0, 1) and u3 be the positive root of the cubic equation

Au3 + Bu2 +Cu + D = 0,

(3)

with the coefficients A = β > 0, B = 2β α − β (1 − h), C = β α 2 − 2β α (1 − h) + ε (γ + β − δ ), D = αε (β − δ ) − β α 2 (1 − h). Using Descartes’ rule of signs, equation (3) has one and only one positive root if any one of the following inequalities holds: (a) B < 0, C < 0, D < 0, (b) B > 0, C < 0, D < 0, (c) B > 0, C > 0, D < 0. 2 δ) The first condition (a) holds if h < 1 − M1 where M1 =max [2α , β α +ε2(βγα+β −δ ) , ε (ββ− α ]. The second condition (b) holds if 1 − 2α < h < 1 − M2 where M2 =max [ β α

2 + ε (γ + β − δ )

2β α β α 2 + ε (γ + β − δ ) ]. 2β α

δ) , ε (ββ − α ]. The third condition

δ) where M3 =max [2α , Thus, one observes that different (c) holds if 1 − M3 < h < 1 − ε (ββ− α equilibria exist for different levels of the harvesting efforts designed for a set of parameters of the nonspatial model of (2). Due to the ecological importance, primary attention has been focused to study the dynamic behaviour about the positive interior equilibrium point e3 (u3 , v3 ).

3 Turing bifurcation analysis of the model system (2) In this section, we expose spatiotemporal patterns through diffusion-driven instability. It is an well known fact owing to Routh-Hurwitz criterion that linear stability of e3 (u3 , v3 ) for the non-spatial model


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of (2) is guaranteed if tr(J) = J11 + J22 < 0 and det J = J11 J22 − J12 J21 > 0, where ⎤ ⎡ εu v u3 3 3 − u3 − (αε+u J11 J12 (α +u3 )2 3) ⎦= . J=⎣ αγ v3 − β v J J 3 21 22 2 (α +u ) 3

Under certain conditions on the parameter values, such a steady state could be linearly stable without the presence of diffusion but unstable with diffusion, which is the famous phenomenon of Turing instability. Now we focus our attention to the conditions of Turing instability occurring at the interior equilibrium e3 (u3 , v3 ) of the spatial model system (2). The diffusive model (2) is called a Turing system on 2D domain with Neumann boundary condition if the following conditions are satisfied around e3 (u3 , v3 ) for some wave number k = 0, see [3, 13] for details: (i) J11 + J22 < 0, (ii) J11 J22 − J12 J21 > 0, (iii) d1 J22 + d2 J11 > 0, (d1 J22 + d2 J11 )2 > (J11 J22 − J12 J21 ), (iv) 4d1 d2

⇒ d1 J22 + d2 J11 > 2 d1 d2 (J11 J22 − J12 J21 ). One observes that the constraints (i) − (iv) are necessary for Turing pattern formation given the proper small heterogeneous perturbation around the homogeneous steady state and system length scale [32,33]. The parametric space which satisfies the above four Turing conditions (i) − (iv) is referred to as “Turing space”. The spatial patterns through diffusion-driven instability are characterized by the dynamic responsibility that diffusion plays in destabilizing the homogeneous steady state of the system. They emerge spontaneously as the model system is driven into a state where it is unstable towards the growth of finite-wavelength stationary perturbations. It is interesting to note that any homogeneous coexistence steady state via linear stability analysis will be always stable or unstable when k = 0, depending on the sign of J11 and consequently Turing patterns cannot develop. We have plotted C = det J + k4 d1 d2 − k2 (d1 J22 + d2 J11 ) < 0 (see [3,13] for details) versus k ranging over [0, 2] and especially when k ∈ (0.268883, 1.432717), k ∈ (0.328991, 1.118928) and k ∈ (0.489056, 0.723139) for three different values of h = 0.0, 0.06 and 0.11 respectively. Although the conditions for Turing instability were obtained analytically, it is yet to prove that the corresponding set of parameter values is not empty. To prove it, we have plotted the polynomial C for different values of h and other parameter values are mentioned in the caption of the Figure 1. Figure 1 illustrates that the sufficient condition for the Turing instability is satisfied for different values of h and length of the interval of wave number k decreases with the increasing values of h. From the mathematical viewpoint, the Turing bifurcation occurs when Im(μ (k)) = 0, Re(μ (k)) = 0 12 J21 ) . In Figure 2, the Turing space at k = kT = 0 where the wave number kT satisfies kT2 = (J11 J22d1−J d2 is shown for the model (2) with parameters α = 0.64, β = 0.0327, γ = 0.28, δ = 0.05, ε = 1.0, h = 0.1 in d1 d2 - plane. The equilibria that can be found in the Turing space area, are stable with respect to homogeneous perturbations but they lose their stability with respect to perturbations of specific wave number k.

4 Turing pattern structures: numerical simulations In this section, numerical simulations are carried out for the spatiotemporal model (2) in 2D square domain Ω = [0, 100] × [0, 100]; discretizing through x → (x0 , x1 , x2 , ...xn ) and y → (y0 , y1 , y2 , ...yn ) with

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n = 100. The numerical integration of the model (2) is executed through forward Euler integration, with a time step of Δt = 0.01 satisfying the CFL (Courant-Friedrichs-Lewy) stability criterion for diffusion equations [34] and using the standard five-point approximation for the 2D Laplacian with n+1 the Neumann boundary condition. More specifically, the concentrations (un+1 i, j , vi, j ) at the moment (n + 1)Δt and at the mesh point (xi , y j ) are given by n n n 2 n un+1 i, j = ui, j + Δt[ f1 (ui, j , vi, j ) + d1 ∇ ui, j ], n n n 2 n vn+1 i, j = vi, j + Δt[ f2 (ui, j , vi, j ) + d2 ∇ ui, j ],

with the Laplacian defined by ∇2 uni, j = ∇2 vni, j =

uni+1, j + uni−1, j + uni, j+1 + uni, j−1 − 4uni, j h2

vni, j+1 + vni, j−1 + vni+1, j + vni−1, j − 4vni, j

where the space step size h(= Δx = Δy) = 1.0.

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In the numerical simulations, different features of dynamics are observed and it is found that the distributions of both the species are of the same type; accordingly one can restrict the analysis of spatial pattern formation to one distribution. One can only show the distribution of prey species (u) as an instance. Figure 3 shows the development of spatial pattern formation for reaction-diffusion predatorprey model (2) with α = 0.64, β = 0.0327, γ = 0.6, δ = 0.05, ε = 1.0, h = 0.1, d1 = 0.01, d2 = 20.0. It is an observable fact that the steady-state pattern takes a long time to settle down, starting with a homogeneous state e3 (u3 , v3 ), and the random perturbation leads to the formation of stripes (cf. Figure 3(a)), and ending with hot spots (red spots on a blue background) only (cf. Figure 3(d)). Figure 4 shows the evolution of the stationary Turing pattern of the prey species at h = 0.0, 0.06, 0.08, 0.11 with small random perturbation of the stationary solution u3 and v3 of the spatially homogeneous system (2) with α = 0.64, β = 0.0327, γ = 0.28, δ = 0.05, ε = 1.0, d1 = 0.01, d2 = 20.0. In this case, one can observe that as h is increased, cold spots (blue spots on a red background) prevail over the whole domain finally, starting with the spots and labyrinthine mixture patterns, and the dynamics of the system does not undergo any further changes. Figure 5 shows six typical spatiotemporal patterns of the prey species for the system (2) for different values of the parameter γ . From the Figure 5, one can see that the values for the prey concentrations are represented in a colour scale varying from blue to red and on increasing the significant parameter γ , the sequence “cold spots (cf. Figure 5(a)) → cold spots and stripe-like mixture patterns (cf. Figure 5(b)) → labyrinthine patterns (cf. Figure 5(c)) → almost stripe patterns (cf. Figure 5(d)) → hot spots and stripe-like mixture patterns (cf. Figure 5(e)) → hot spots (cf. Figure 5( f ))” is observed. Similarly, from Figure 6, one can observe that as the value of the significant parameter ε increases, the sequence “cold spots (cf. Figure 6(a)) → cold spots and stripe-like mixture patterns (cf. Figure 6(b)) → labyrinthine patterns (cf. Figure 6(c)) → hot spots and labyrinthine mixture patterns (cf. Figure 6(d))” is found to appear. Figure 7 exhibits five typical spatial patterns of the prey species (u) for the system (2) for different


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values of the parameter β . From Figure 7, one can observe a variety of patterns on increasing the parameter β , the sequence “cold spots (cf. Figure 7(a)) → cold spots and stripe mixture patterns (cf. Figure 7(b)) → labyrinthine patterns (cf. Figure 7(c)) → hot spots and stripe mixture patterns (cf. Figure 7(d)) → hot spots (cf. Figure 7(e))” is formed. In the same way, from the patterns of Figure 8, one can examine that as the value of the significant parameter δ increases, the sequence “hot spots (cf. Figure 8(a)) → hot spots and stripe mixture patterns (cf. Figure 8(b)) → labyrinthine patterns (cf. Figure 8(c)) → cold spots and labyrinthine mixture patterns (cf. Figure 8(d)) → cold spots (cf. Figure 8(e))” is observed. Figure 9 shows stationary patterns through diffusion-driven instability of prey species at α = 0.59, 0.60, 0.62 with β = 0.0327, γ = 0.28, δ = 0.05, ε = 1.0, h = 0.1, d1 = 0.01, d2 = 20.0. From Figure 9, one observes that the coexistence of hot spot and stripe-like patterns (cf. Figure 9(a)) changes to coexistence of cold spot and stripe-like patterns (cf. Figure 9(c)) through labyrinthine patterns (cf. Figure 9(b)) and these patterns prevail over the whole domain at last. A three-dimensional (3D) view of the spatial pattern for both the prey and predator species at stationary level is presented in Figure 10. 5 Conclusions and remarks The present study is mainly dealt with an in-depth analysis of the spatial pattern formation of a diffusive generalist predator-prey model system with linear prey harvesting within 2D space. The reaction-diffusion model is duly investigated both analytically and numerically. The stable and unstable regions in Turing parametric space for the spatiotemporal model (2) are demarketed (cf. Figure 2).

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The necessary conditions for diffusion-driven instability in terms of system parameters are mentioned. Besides, according to these parameter regions, the corresponding dispersion relations (cf. Figure 1) and the relevant patterns are shown (cf. Figures 3-10). These results indicate that Turing patterns can emerge through the interaction between the diffusion and prey harvesting as well as other significant parameters in the system (2). Based on both the mathematical analysis and suitable numerical simulations, one may realize that its Turing pattern includes holes, stripes-holes mixture, stripes, labyrinthine, stripes-spots mixture, and spots patterns. From the biological point of view, the results of the present study have some clear meaning. The effect of diffusion with prey harvesting on the stability of the coexistence equilibrium point e3 (u3 , v3 ) has been estimated carefully with special attention in the reaction-diffusion predator-prey model (2). The distinguished feature is that the uniform steady state of a reaction-diffusion equation is stable for temporal model, but it becomes unstable for the corresponding spatiotemporal model resulting in the emergence of diffusion-driven instability. For the reaction-diffusion model (2) without harvesting, a picture of the effect of diffusion is provided, including Turing instability of the positive equilibrium e3 (u3 , v3 ) (cf. Figure 4(a)). Due to the small variation of the significant parameters viz. h, γ , ε , β , δ , and α , one may find that the qualitative dynamics of the spatial model (2) are fundamentally different. By

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varying the value of the prey harvesting parameter h, one finds different classic types of steady-state pattern. The results of numerical simulation point out that the effect of the prey harvesting with other significant parameters for spatial pattern formation is remarkable (cf. Figures 4-9). It may be concluded that the present results related to the consequence of prey harvesting in a reaction-diffusion predator-prey interactions model under homogeneous Neumann boundary condition may enrich the experimental research work on pattern formation in the interacting models and may well explain the field observations on some areas.

Acknowledgement The authors are thankful to the learned referees for their valuable comments and suggestions towards an improvement of the present paper. The authors also gratefully acknowledge the financial support in part from Special Assistance Programme (SAP-III) sponsored by the University Grants Commission (UGC), New Delhi, India (Grant No. F.510 / 3 / DRS-III / 2015 (SAP-I)).

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SIAM Journal on Applied Mathematics, 58, 193-210. [22] Beddington, J.R. and May, R.M. (1980), Maximum sustainable yields in systems subject to harvesting at more than one trophic level, Mathematical Biosciences, 51, 261-281. [23] Beddington, J.R. and Cooke, J.G. (1982), Harvesting from a prey-predator complex, Mathematical Biosciences, 14, 155-177. [24] Makinde, O.D. (2007), Solving ratio-dependent predator-prey system with constant effort harvesting using Adomian decomposition method, Applied Mathematics and Computation, 186, 17-22. [25] Solow, R.M. and Clark, C.W. (1977), Mathematical Bioeconomics: The Optimal Management of RENEWABLE Resources, JSTOR. [26] Hill, S.L., Murphy, E.J., Reid, K., Trathan, P.N., and Constable, A. J. (2006), Modelling Southern Ocean ecosystems: krill, the food-web, and the impacts of harvesting, Biological Reviews, 81, 581-608. [27] Christensen, V. (1996), Managing fisheries involving predator and prey species, Reviews in fish Biology and Fisheries, 6, 417-442. [28] Chen, L. and Li, Y., and Xiao, D. (1972), Bifurcations in a ratio-dependent predator-prey model with prey harvesting, Canadian Applied Mathematics Quarterly, 19, 293-318. [29] Shukla, J.B. and Verma, S. (1981), Effects of convective and dispersive interactions on the stability of two species, Bulletin of Mathematical Biology, 43, 593-610. [30] Schreiber, S.J. (1997), Generalist and specialist predators that mediate permanence in ecological communities, Journal of Mathematical Biology, 36, 133-148. [31] Lv, Y., Yuan, R., and Pei, Y. (2014), Effect of harvesting, delay and diffusion in a generalist predator-prey model, Applied Mathematics and Computation, 226, 348-366. [32] Ermentrout, Bard (1991), Stripes or spots? Non-linear effects in bifurcation of reaction-diffusion equations on the square, Proceedings of Royal Society, London, 434, 413-417. [33] Nagorcka, B.N. and Mooney, J.R. (1992), From stripes to spots: prepatterns which can be produced in the skin by a reaction-diffusion system, Mathematical Medicine and Biology, 9, 249-267. [34] Shen, J. and Jung, Y. M. (2005), Geometric and stochastic analysis of reaction-diffusion patterns, International Journal of Pure and Applied Mathematics, 19, 195-244.



Dynamics and Stability Results of Fractional Pantograph Equations with Complex Order D. Vivek†, K. Kanagarajan, S. Harikrishnan Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore - 641 020, Tamilnadu, India Submission Info Communicated by J.A.T. Machado Received 2 March 2017 Accepted 20 June 2017 Available online 1 July 2018

Abstract In this paper, we study the existence, uniqueness and stability of solutions for fractional pantograph equations with complex order. The Krasnoselkii’s fixed point theorem and Banach contraction principle are used to obtain the desired results.

Keywords Pantograph equation Nonlocal condition Complex order Existence


1 Introduction The pantograph type is one of the special types of delay differential equations, and growing attention is given to its analysis and numerical solution. Pantograph type always has the delay term fall after the initial value but before the desire approximation being calculated. The pantograph equation is used in different fields of pure and applied mathematics such as number theory, dynamical systems, probability, quantum mechanics and electrodynamics (see [1–4] and references therein). Pantograph equation has been studied by many researchers and solved by several numerical methods. Nowadays remarkable contributions have been made to the theory and applications of the fractional differential equations (FDEs). Many problems can be modelled with the help of the FDEs in many areas such as seismic analysis, viscous damping, viscoelastic materials and polymer physics(see [5, 6]). The Ulam stability of functional equation, which was invented by Ulam on a talk given to a conference at Wisconsin University in 1940, is one of the essential subjects in the mathematical analysis area. The finding of Ulam stability plays a pivotal role in regard to this subject. For extensive study on the advance of Ulam type stability, readers refer can to [7–9] and the references therein. The credit † Corresponding


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of solving this problem partially goes to Hyers. To study Hyers-Ulam stability of FDEs, different researchers presented their works with different methods, (see [10–12] and references therein). Recently, A. Neamaty et. al. [13] studied existence and uniqueness results for fractional differential equation with complex order. At this writing, only one paper in the literature has been devoted to fractional pantograph equations with complex order. In this paper, we consider the following complex order nonlocal pantograph equation (Dθ0+ x)(t) = f (t, x(t), x(λ t)), t ∈ J := [0, 1], θ = m + iα , (1) x(0) + g(x) = x0 , where Dθ0+ is the Caputo fractional derivative of order θ ∈ C. Let α ∈ R+ , 0 < λ < 1, m ∈ (1, 2] and f : J × R × R → R, g : C([0, 1], R) → R are given continuous functions. It is seen that system (1) is equivalent to the following nonlinear integral equation(see [13–15] for more details). x(t) = x0 − g(x) +

1 Γ(θ )

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Let C(J, R) be the Banach space of continuous function x(t) with x(t) ∈ R for t ∈ J and xC(J,R) = supt∈J x(t). In passing, we note that the application of nonlinear condition x(0) + g(x) = x0 in physical problems yeilds better effect than the initial condition x(0) = x0 [15]. The outline of the paper is as follows. In Section 2, we give some basic definitions and results concerning the complex derivative. In Section 3, we present our main results by Krasnoselkii’s fixed point theorem. In section 4, we discuss the stability results.

2 Preliminaries In this section, we introduce notation, definitions, and preliminary facts that we need in the sequel. Definition 1. [6] The Riemann-Liouville fractional integral of order q ∈ C, (Re(q) > 0) of a function f : (0, ∞) → R is ˆ t 1 q (t − s)q−1 f (s)ds. I0+ f (t) = Γ(q) 0 Definition 2. [6] The Riemann-Liouville fractional derivative of order q ∈ C, (Re(q) > 0) of a function f : (0, ∞) → R has the form q D0+

dn 1 f (t) = Γ(n − q) dt n

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1 f )(t) = Γ(n − q)

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Definition 4. [16] The Stirling asymptotic formula of the Gamma function for z ∈ C is following 1

Γ(z) = (2π ) 2 z

z−1 2

1 e−z [1 + O( )], z

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(v → ∞).

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Lemma 1. (see Lemma 7.1.1, [17]) Let z, w : [0, T ) → [0, ∞) be continuous functions where T ≤ ∞. If w is nondecreasing and there are constants k ≥ 0 and 0 < q < 1 such that ˆ t z(t) ≤ w(t) + k (t − s)q−1 z(s)ds, t ∈ [0, T ), 0

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t

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Remark 1. Under the hypothesis of Lemma 1, let w(t) be a nondecreasing function on [0, T ). Then we have z(t) ≤ w(t)Eq,1 (kΓ(q)t q ). Following fixed point theorem is used to establish the existence results. Theorem 2. (Krasnoselkii’s fixed point theorem) Let K be a closed convex and nonempty subset of a Banach space X . Let T and S, be two operators such that • T x + Sy ∈ K for any x, y ∈ K; • T is compact and continuous; • S is contraction mapping. Then there exists z1 ∈ K such that z1 = T z1 + Sz1 . We are ready to present our results. We adopt same ideas in [18].

3 Main results Let us list some hypotheses to prove our existence results. (A1) f : J × R × R → R is continuous function. (A2) there exists a positive constant L > 0 such that | f (t, x, u) − f (t, y, v)| ≤ L(|x − y| + |u − v|),

t ∈ J,

u, v, x, y ∈ C(J, R).

(A3) g : C(J, R) → R is continuous and 0 < b < 1 such that |g(x) − g(y)| ≤ b |x − y| ,

for all x, y ∈ C(J, R).

(A4) there exists a function μ ∈ L1 (J, R+ ) such that | f (t, x, y)| ≤ μ (t), for all t ∈ J, x, y ∈ R.


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The main results are based on Theorem 2. Theorem 3. Assume the hypotheses (A1)-(A3) are fulfilled. If b
0 such that for each ε > 0 and for each solution z ∈ C(J, R) of the inequality θ D + z(t) − f (t, z(t), z(λ t)) ≤ ε , t ∈ J, 0 there exists a solution x ∈ C(J, R) of equation (1) with |z(t) − x(t)| ≤ C f ε , t ∈ J. Definition 6. The equation (1) is generalized Ulam-Hyers stable if there exists ψ f ∈ C ([0, ∞), [0, ∞)), ψ f (0) = 0 such that for each solution z ∈ C(J, R) of the inequality θ D + z(t) − f (t, z(t), z(λ t)) ≤ ε , t ∈ J, 0 there exists a solution x ∈ C(J, R) of equation (1) with |z(t) − x(t)| ≤ ψ f ε ,

t ∈ J.

Definition 7. The equation (1) is Ulam-Hyers-Rassias stable with respect to ϕ ∈ C(J, R) if there exists a real number C f > 0 such that for each ε > 0 and for each solution z ∈ C(J, R) of the inequality θ D + z(t) − f (t, z(t), z(λ t)) ≤ εϕ (t), t ∈ J, 0 there exists a solution x ∈ C(J, R) of equation (1) with |z(t) − x(t)| ≤ C f εϕ (t),

t ∈ J.

Definition 8. The equation (1) is generalized Ulam-Hyers-Rassias stable with respect to ϕ ∈ C(J, R) if there exists a real number C f ,ϕ > 0 such that for each solution z ∈ C(J, R) of the inequality θ D + z(t) − f (t, z(t), z(λ t)) ≤ ϕ (t), t ∈ J, 0 there exists a solution x ∈ C(J, R) of equation (1) with |z(t) − x(t)| ≤ C f ,ϕ ϕ (t),

t ∈ J.

Remark 2. A function z ∈ C(J, R) is a solution of (1) if and only if there exists a function g ∈ C(J, R) (which depend on z) such that 1. |g(t)| ≤ ε , t ∈ J; 2. Dθ0+ z(t) = f (t, z(t), z(λ t)) + g(t), t ∈ J. Remark 3. Clearly, 1. Definition 5⇒ Definition 6. 2. Definition 7⇒ Definition 8.


185

We ready to prove our stability results for problem (1). The arguments are based on the Banach contraction principle. First we list the following hypothesis: (A5) There exists an increasing function ϕ ∈ C[J, R] and there exists λϕ > 0 such that for any t ∈ J I0θ+ ϕ (t) ≤ λϕ ϕ (t). Theorem 5. Let conditions (A1)-(A3), 0 < b < 1 and Ωb,L,m,θ < 1 hold, then the problem (1) is UlamHyers stable. Proof. Let ε > 0 and let z ∈ C(J, R) be a function which satisfies the inequality: θ D + z(t) − f (t, z(t), z(λ t)) ≤ ε , for any t ∈ J, 0

(5)

and let x ∈ C(J, R) be the unique solution of the following pantograph equation Dθ0+ x(t) = f (t, x(t), x(λ t)), t ∈ J := [0, 1], θ = m + iα , z(0) + g(x) = x0 , where α ∈ R+ , m ∈ (1, 2] and 0 < λ < 1. Using equation (2), we obtain

x(t) = x0 − g(x) + I0θ+ f (s, x(s), x(λ s)) (t).

By integration of the inequality (5), we obtain ˆ t 1 ε . (t − s)θ −1 f (s, z(s), z(λ s))ds| ≤ |z(t) − x0 + g(z) − Γ(θ ) 0 Γ(θ + 1) We have |z(t) − x(t)|

ˆ t 1 (t − s)θ −1 f (s, z(s), z(λ s))ds| Γ(θ ) 0 ˆ t 1 (t − s)θ −1 [ f (s, z(s), z(λ s)) − f (s, x(s), x(λ s))]ds| + |g(x) − g(z) + Γ(θ ) 0 ˆ t 1 ε + |g(z) − g(x)| + |(t − s)θ −1 || f (s, z(s), z(λ s)) − f (s, x(s), x(λ s))|ds ≤ Γ(θ + 1) |Γ(θ )| 0 ˆ t 2L ε + b|z(t) − x(t)| + |(t − s)θ −1 ||z(s) − x(s)|ds. ≤ Γ(θ + 1) |Γ(θ )| 0

≤ |z(t) − x0 + g(z) −

Thus,

ˆ t 2L ε + |(t − s)θ −1 ||z(s) − x(s)|ds (1 − b)Γ(θ + 1) (1 − b)|Γ(θ )| 0 ˆ t 2L ε + (t − s)m−1 |z(s) − x(s)|ds. ≤ (1 − b)Γ(θ + 1) (1 − b)|Γ(θ )| 0

|z(t) − x(t)| ≤

Using Lemma 1(Gronwall inequality) and Remark 1, we obtain |z(t) − x(t)| ≤

2L εT θ Em,1 ( Γ(m)). Γ(θ + 1) (1 − b)|Γ(θ )|

Thus, the equation (1) is Ulam-Hyers stable.

(6)

186


Theorem 6. Let conditions (A1)-(A3), (A5), 0 < b < 1 and Ωb,L,m,θ < 1 hold. Then, the problem(1) is generalized Ulam-Hyers-Rassias stable. Proof. Let z ∈ C(J, R) be solution of the inequality θ D + z(t) − f (t, z(t), z(λ t)) ≤ εϕ (t), 0

t ∈ J, ε > 0,

(7)

and let x ∈ C(J, R) be the unique solution of the following pantograph equation Dθ0+ x(t) = f (t, x(t), x(λ t)), t ∈ J := [0, 1], θ = m + iα , z(0) + g(x) = x0 , where α ∈ R+ , m ∈ (1, 2] and 0 < λ < 1. Using equation (2), we obtain

x(t) = x0 − g(x) + I0θ+ f (s, x(s), x(λ s)) (t).

By integration of the inequality (7), we obtain ˆ t 1 (t − s)θ −1 f (s, z(s), z(λ s))ds| ≤ ελϕ ϕ (t). |z(t) − x0 + g(z) − Γ(θ ) 0 On the other hand, we have |z(t) − x(t)|

ˆ t 1 (t − s)θ −1 f (s, z(s), z(λ s))ds| ≤ |z(t) − x0 + g(z) − Γ(θ ) 0 ˆ t 1 (t − s)θ −1 [ f (s, z(s), z(λ s)) − f (s, x(s), x(λ s))]ds| + |g(x) − g(z) + Γ(θ ) 0 ˆ t 1 |(t − s)θ −1 || f (s, z(s), z(λ s)) − f (s, x(s), x(λ s))|ds ≤ ελϕ ϕ (t) + |g(z) − g(x)| + |Γ(θ )| 0 ˆ t 2L |(t − s)θ −1 ||z(s) − x(s)|ds. ≤ ελϕ ϕ (t) + b|z(t) − x(t)| + |Γ(θ )| 0

Thus,

ˆ t 2L 1 ελϕ ϕ (t) + |(t − s)θ −1 ||z(s) − x(s)|ds |z(t) − x(t)| ≤ (1 − b) (1 − b)|Γ(θ )| 0 ˆ t 2L 1 ελϕ ϕ (t) + (t − s)m−1 |z(s) − x(s)|ds. ≤ (1 − b) (1 − b)|Γ(θ )| 0

(8)

Using Lemma 1(Gronwall inequality) and Remark 1, we obtain |z(t) − x(t)| ≤

ελϕ ϕ (t) 2L Em,1 ( Γ(m)). (1 − b) (1 − b)|Γ(θ )|

Thus, the equation (1) is generalized Ulam-Hyers-Rassias stable. Remark 4. Theorem 3, 4, 5 and 6 can easily be extended to the generalized multi-pantograph equation of generalized fractional derivative of the form (Dθ0+ x)(t) = f (t, x(t), x(λ1 t) · · · x(λm t)), x(0) + g(x) = x0 , where α ∈ R+ and λ ∈ (0, 1), m ∈ (1, 2].

t ∈ [0, 1],

θ = m + iα ,


187

Acknowledgements This work was financially supported by the Tamilnadu State Council for Science and Technology, Dept. of Higher Education, Government of Tamilnadu.The authors are grateful to the referees for their careful reading of the manuscript and valuable comments. The authors thank the help from editor too.

References [1] Balachandran, K., Kiruthika, S., and Trujillo, J.J. (2013), Existence of solutions of Nonlinear fractional pantograph equations, Acta Mathematica Scientia, 33B, 1-9. [2] Derfel, G.A. and Iserles, A. (1997), The pantograph equation in the complex plane, Journal Mathematical Analysis and Applications, 213, 117-132. [3] Iserles, A. (1993), On the generalized pantograph functional-differential equations, Eur. Journal of Applied Mathematics, 4, 1-38. [4] Liu, M.Z. and Li, D. (2004), Runge-Kutta methods for the multi-pantograph delay equation, Applications and Mathematical Computations, 155, 853-871. [5] Hilfer, R. (1999), Application of fractional Calculus in Physics, World Scientific, Singapore. [6] Podlubny, I. (1999), Fractional differential equations, Academic Press, San Diego. [7] Andras, S. and Kolumban, J.J. (2013), On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Analysis, 82, 1-11. [8] Jung, S.M. (2004), Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17, 1135-1140. 218(3), (2011), 860-865. [9] Muniyappan, P. and Rajan, S. (2015), Hyers-Ulam-Rassias stability of fractional differential equation, International Journal of pure and Applied Mathematics, 102, 631-642. [10] Ibrahim, R.W. (2012), Generalized Ulam-Hyers stability for fractional differential equations, International Journal of mathematics, 23, doi:10.1142/S0129167X12500565. [11] Wang, J., Lv, L., and Zhou, Y. (2011), Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electronic Journal of Qualitative Theory of Differential Equations, 63, 1-10. [12] Wang, J. and Zhou, Y. (2012), New concepts and results in stability of fractional differential equations, Communications on Nonlinear Science and Numerical Simulations, 17, 2530-2538. [13] Neamaty, A., Yadollahzadeh, M., and Darzi, R. (2015), On fractional differential equation with complex order, Progress in fractional differential equations and Apllications, 1(3), 223-227. [14] Balachandran, K. and Trujillo, K.J.J. (2010), The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Analysis Theory Methods and Applications, 72, 4587-493. [15] Bashir, A. and Sivasundaram, S. (2008), Some existence results for fractional integro-differential equations with nonlocal conditions, Communications in Applied Analysis, 12, 107-112. [16] Bai, Z. and Lu, H. (2005), Positive solutions for a boundary value problem of nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 311, 495-505. [17] Hyers, D.H., Isac, G., and Rassias, T.M. (1998), Stability of functional equation in several variables, 34, Progress in nonlinea differential equations their applications, Boston (MA): Birkhauser. [18] Vivek, D., Kanagarajan, K., and Harikrishnan, S. (2007), Existence and uniqueness results for pantograph equations with generalized fractional derivative, Journal of Nonlinear Analysis and Applications (ISPACS), Accepted article-2017. Id: jnaa-00370. [19] Rus, I.A. (2010), Ualm stabilities of ordinary differential equations in a Banach space, Carpathian Journal Mathematics, 26, 103-107. [20] Ye, H., Gao, J., and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, Journal of Mathematics and Applications, 328, 1075-1081.



Variational Iteration Method in the Fractional Burgers Equation A. R. Gómez Plata†, E. Capelas de Oliveira Department of Mathematics, Cajica, Universidad Militar Nueva Granada, 250247, Colombia Imecc, Campinas-SP, University of Campinas, 13083-859, Brazil Submission Info Communicated by J.A.T. Machado Received 4 September 2016 Accepted 20 June 2017 Available online 1 July 2018 Keywords Nonlinear Fractional Dynamics Variational iteration method Caputo derivative Fractional Burgers equation

Abstract The variational iteration method (VIM) is a analysis tool efficient for approximate non-linear fractional differential equations. Recently differents investigators are used this method in your works and we study the Lagrange multipliers of the variational iteration method for the time fractional Burgers equation and apply those in differents particular cases. In this conference we present approximations of the solutions for a particular case of the time fractional Burgers equation (BF), with the use of the variational iteration method, the Caputo derivate for 0 < α ≤ 1, after make an comparation with the Adomian descomposition method (ADM). ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Preliminaries In this section we will present definitions and results that we use in the paper, a short review of FC and the VIM for the linear and nonlinear equations. 1.1

Fractional calculus

First of all, we introduce the Riemann-Liouville (RL) fractional integral, considered in the left, only [1]. Let n be a positive integer and α ∈ C such that Re(α ) > 0, we define the RL fractional integral by means of Itα [ f (t)] =

1 Γ(n − α )

ˆ

t

f (τ )(t − τ )α +1−ndτ ,

n − 1 < α < n.

(1)

0

The Caputo derivatives has been used by many authors in several physical applications [2–8]. One reason for this choice is the fact that the initial conditions associated with the fractional differential equation are usually expressed in terms of integer order derivatives. Let n be a positive integer and α ∈ C such that Re(α ) > 0. We introduce the fractional derivative of order α in the Caputo sense, † Corresponding



190

A. R. G´ omez Plata, E. Capelas de Oliveira /Journal of Applied Nonlinear Dynamics 7(2) (2018) 189–196

denoted by c Dtα [ f (t)], by means of the integral

α c Dt [ f (t)]

=

 ˆ t  1 f (n) (τ )     Γ(n − α ) 0 (t − τ )α +1−n dτ ,    dn   Dn f (t) ≡ f (t) , dt n

n − 1 < α < n, (2)

α = n.

The relation involving the Caputo derivatives and the RL fractional integral is given by α c Dt [ f (t)]

= Itn−α ·c Dtn f (t)

(3)

with n − 1 < α < n. As we have already said, the Laplace transform methodology is an efficient tool to discuss a fractional differential equation. Then, we introduce the Laplace transform of the derivative in the Caputo sense. Denoting by L the Laplace integral operator, we can write the Laplace transform of the Caputo derivatives as follows n−1

(L [c Dtα f (t)])(s) = sα (L [ f (t)])(s) − ∑ sα −k−1 (Dk f )(0+ ) ,

(4)

k=0

with s is the parameter of the Laplace transform. As we have mentioned above, this expression shows that the Laplace transform of the fractional derivative in the Caputo sense involves only the derivative of integer order evaluated in t = 0+ , conversely the corresponding RL derivative. In our particular problems, as we will be seen in the sequence, we take the parameter α as a real number such that 0 < α ≤ 1 in problems involving (anomalous) diffusion and 1 < α ≤ 2 in problems associated with wave propagation. To close this subsection, as an example, we consider the fractional integral and the fractional derivative of a power function t λ , with λ a real parameter. For the fractional integral we have Γ(λ + 1) λ +α , (5) t Itα [t λ ] = Γ(λ + 1 + α ) with α ≥ 0, λ > −1 and t > 0. On the other hand, for the fractional derivative we have α λ c Dt [t ] =

Γ(λ + 1) λ −α t , Γ(λ + 1 − α )

(6)

with α > 0, λ > −1 and t > 0. 1.2

Variational iteration method

The VIM [9] was extended to fractional differential equations and has been one of the methods frequently used. Classical and fractional differential equations are studied using VIM. On the order hand classical and fractional partial differential equations are studied in [10–18] particularly, nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow is discussed in [19]. Here we consider a more general fractional differential equation α c Dt [u] + R[u] + N[u] =

f (t),

where c Dtα [u] is the Caputo derivative, R[u] is a linear term, N[u] is a nonlinear one and f (t) is a function associated with the non homogeneous term. Odibat and Momami in [20] applied the VIM to the above equation and suggested a variational correction formula


 ˆ t   λ (t, τ )(c Dtα un + R[u] + N[u] − f (τ ))dτ ,  un+1 = un +

191

0 0,

(9)

where L is a linear operator, N is a non-linear operator in x,t and c Dtα is the Caputo derivative of order α , subject to the initial conditions u(k) (x, 0) = ck (x) , (10) k = 0, 1, 2, . . . , m − 1, m − 1 < α ≤ m. Applying the operator J α and the inverse of the operator c Dtα the both side of Eq.(9), yield m−1

u (x,t) =

∑

k=0

k ∂ ku + t x, 0 + J α g (x,t) − J α [Lu (x,t) + Nu (x,t)] . ∂ tk k!

(11)

The Adomian’s decomposition method [23–26] suggest that the solution u(x,t) should be decomposed into the infinite series of components as ∞

u (x,t) =

∑ un (x,t) ,

(12)

n=0

and the nonlinear function in Eq.(9), be decomposed as follows: ∞

Nu =

∑ An ,

(13)

n=0

where An are the so-called the Adomian’s polynomials. Substituting the decomposition series equations Eq.(12), and Eq.(13), into both sides of Eq.(11), gives ∞

m−1

n=0

k=0

∑ un (x,t) = ∑

∞ ∞ k ∂ ku + t α α An ]. x, 0 + J u (x,t) + g(x,t) − J [L n ∑ ∑ ∂ tk k! n=0 n=0

(14)

192


from this equation, the iterates are determined by folowing recursive way: m−1

u0 (x,t) =

∑

k ∂ ku + t x, 0 + J α g (x,t) , ∂ tk k!

k=0 α

u1 (x,t) = −J (Lu0 + A0 ) , u2 (x,t) = −J α (Lu1 + A1 ) , .. .=

(15)

un+1 (x,t) = −J α (Lun + An ) . The Adomian’s polynomial An can be calculated for all forms of nonlinearity according to specific algorithms constructed by Adomian [25]. The Adomian polynomials can be easily calculated by the homotopy perturbation method. The general formulation for an Adomian’s polynomials is An =

∞ 1 dn [ n N( ∑ λ k uk )]λ =0 . n! d λ n=0

(16)

Finally, we approximate the solution u(x,t) by the truncated series as N−1

∑ un (x,t) ,

φN (x,t) =

(17)

n=0

with solution lim φN (x,t) = u (x,t) .

N→∞

(18)

2 ADM in the fractional Burgers equation Consider the fractional Burgers equation given by α c Dt [u] + ε uux

= vuxx , t > 0, 0 < α ≤ 1,

(19)

subject to the initial condition u (x, 0) = g (x) . write the Eq.(19), where Lx = obtain

∂ 2u . ∂ x2

α c Dt u (x,t)

= vLx u − ε uux ,

(20) (21)

Using the operator J α in both sides of the Eq.(21), and using the initial condition u (x,t) = u (x, 0) + J α [vLx u − εφ (u)] ,

(22)

with φ (u) = uux . The decomposition method for solution Adomian assumes the form of a series given by ∞

u (x,t) =

∑ un (x,t) .

(23)

n=0

Of the Eq.(23), We have the set of recursive relationships given by u0 = g (x) ,

un+1 = J α [vLx un − ε An ] ,

(24)


193

where φ (u) = ∑∞ n=0 An (u0 , u1 , . . . , un ) are polynomials Adomian. Using the recursive relationship we can get the first three terms of the decomposition so u1 = J α [vLx u0 − ε A0] = vg′′ − ε gg′

u0 = u (x, 0) = g (x) ,

tα , Γ (α + 1)

u2 = J α [vLx u1 − ε A1 ] = [2ε 2 gg′2 + ε 2 g2 g′′ − 4ε vg′ g′′ − 2ε vgg′′′ + v2 g(4) ]

t 2α . Γ (2α + 1)

(25)

The approximation for the solution in serial form is u(x,t) = g(x) + [vg′′ − ε gg′ ]

tα + [2ε 2 gg′2 + ε 2 g2 g′′ − 4ε vg′ g′′ − 2ε vgg′′′ + v2 g(4) ] Γ(α + 1) ×

t 2α + . . . . (26) Γ(2α + 1)

To verify the validity of this approximation in the next section we will compare the approximation of the fractional equation Burger with ADM and VIM. Considering the case ε = 1 and the following initial condition µ + σ + (σ − µ ) exp (γ ) , t ≥ 0, (27) u (x, 0) = g (x) = 1 + exp (γ ) with γ = µv (x − λ ), µ ,σ parameters,λ and v are arbitrary constants. The exact solution for the entire case α = 1 [27], it is given by u (x,t) = where ζ = 2.1

µ v

µ + σ + (σ − µ ) exp (ζ ) , 1 + exp (ζ )

(28)

(x − σ t − λ ).

VIM by ADM in the fractional Burgers equation

We study the VIM and ADM and your approximations for the BF. For this, we considered two intervals different in the variable x for the BF thus   c Dtα u + uux − uxx = 0,t > 0, 0 < α ≤ 1, 2ex (29)  u(x, 0) = , 0 ≤ x ≤ 1, 0 < x ≤ 5. x 1+e First recording the approximation for the solution of the BF with ADM, where your Adomian polynomials for non-linear term uux and iteration formula is calculated for an initial condition arbitrary. We adjust yourself for your in [28] and get u(x,t) =

2ex tα ′ ′′ − [gg + g ] 1 + ex Γ(α + 1) t 2α . + [2g(g ) + g g + 4g g + 2gg + g ] Γ(2α + 1) ′ 2

2 ′′

′ ′′

′′′

(30)

(4)

On the other hand we have the second approximation of the VIM, where the Lagrange multipliers and correction functional iteration is give in [28] for this case by tα t 2α + (2g(g′ )2 + g2 g′′ + 2gg3 + 4g′ g′′ + g(4) ) Γ(α + 1) Γ(1 + 2α ) 3 α Γ(1 + 2α ) t − (gg′ + g′′ )((g′ )2 + gg′′ + g(3) ) 2 . Γ (1 + α ) Γ(1 + 3α )

u2 (x,t) =g − [gg′ + g′′ ]

(31)

194


Table 1 Comparation VIM by ADM in the BF. x

t

ADM

VIM

Exacta

0.1

0.1

0.999959

0.999963

1.00000

0.1

0.5

0.797468

0.797986

0.802625

0.1

0.9

0.59099

0.59401

0.620051

0.5

0.1

1.19734

1.19736

1.19738

0.5

0.5

0.995526

0.99778

1.00000

0.5

0.9

0.775291

0.788438

0.802625

0.9

0.1

1.37993

1.37996

1.37995

0.9

0.5

1.19472

1.19769

1.19738

0.9

0.9

0.981771

0.999089

1.00000

Fig. 1 Approximation, Left: ADM. Middle: Exact. Right: VIM. For BF with 0 < x ≤ 1,α = 1.0.

Fig. 2 Approximation, Left: ADM. Middle: Exact. Right: VIM. For BF with 0 < x ≤ 5,α = 1.0.

The exact solution is know when α = 1 by u(x,t) =

2 exp(x − t) . 1 + exp(x − t)

(32)

In the Table 1. we can see that the VIM get best approximations of the solutions when 0 < x ≤ 1. The Fig.1 show that the approximation of the ADM and VIM is very nearby for the exact solution when α = 1, 0 < x ≤ 1. The Fig.2 evidence that the approximation by the VIM is most nearby the exact solution when 0 < x ≤ 5.


195

3 Conclusions In this paper, we studied the variational iteration method and the Adomian descomposition method for fractional Burgers equation. The variational iteration and Adomian descomposition method are efficient tools for the calculates approximations of this fractional partial differential equations when 0 < x ≤ 1, but the variational iteration method is most efficient in the calculate of this approximations when 0 < x ≤ 5 and α = 1. References [1] Diethelm, K. (2010), The Analysis of Fractional Differential Equations: An Application- Oriented Exposition using Differential Operator of Caputo Type, Springer, Braunschweig. [2] Mainardi, F., Luchko, Y., and Pagnini, G. (2001), The fundamental solution of the space-time fractional diffusion equation, Frac. Calc. Appl. Anal., 4, 153-192. [3] Costa, F.S., Marao, J.A.P.F., Soares, J.C.A., and Capelas de Oliveira, E. (2015), Similarity solution to fractional nonlinear space-time, diffusion-wave equation, J. Math. Phys., 56, 033507. [4] Capelas de Oliveira, E., Mainardi, F., and Vaz Jr, J. (2014), Fractional models of anomalous relaxation based on the Kilbas and Saigo function, Meccanica (Milano. Print), 49, 2049-2060. [5] Figuereido Camargo, R., Capelas de Oliveira, E., and Vaz, Jr. J. (2009), On anamalous diffusion and the fractional generalized Langevin equation for a harmonic oscillator, J. Math. Phys., 50, 123-518. [6] Diethelm, K. and Ford, N.J. (2004), Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154, 621-640. [7] Liu, F.W., Anh, V., and Turner, I. (2004), Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, 209-219. [8] Wazwaz, A.M. (2007), The variational iteration method for solving linear and nonlinear systems of PDEs, Comput. Math. Appl., 54, 895-902. [9] Ramos. J.I. (2008), On the variational iteration method and the other iterative techniques for nonlinear differential equations, Appl. Math. Comput., 199, 39-69. [10] Mainardi, F. (1996), Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals, 7, 1461-1477. [11] Wazwaz, A.M., (2007), The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations, Comput. Math. Appl., 54, 926-932. [12] Wazwaz, A.M. (2007), The Variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations, Comput. Math. Appl., 54, 933-939. [13] Ozer, H. (2007), Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics, Int. J. Nonlinear Sci. Numer. Simul., 8, 513-518. [14] Sheng, H. and Chen, Y.Q. (2011), Application of numerical inverse Laplace transform algorithms in fractional calculus, J. Franklin Ins., 348, 315-330. [15] Ruiz-Medina, M.D., Angulo, J.M., and Anh, V.V. (2001), Scaling limit solution of a fractional Burgers’ equation, Stochastic Process. Appl., 93, 285-300. [16] Hayat, T., Khan, M., and Asghar, S. (2007), On the MHD flow of fractional generalized Burgers’ fluid with modified Darcy’s law, Acta. Mech. Sin., 23, 257-261. [17] Shah, S.H.A.M. (2010), Some helical flows of a Burgers’ fluid with fractional derivative, Meccanica, 45, 143-151. [18] Chen, Y. and H. L. (2008), An Numerical solutions of coupled Burgers’ equations with time and spacefractional derivatives, Appl. Math. Comput., 200, 87-95. [19] Jun, Y.J., Tenreiro Machado, J.A., and Hristov, J. (2015), Nonlinear dynamics for local fractional Burgers’ equation arising in fractal flow, Nonlinear Dyn., DOI 10.1007/s11071-015-2085-2. [20] Odibat, Z. and Momami, S. (2009), The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Comp. Math. Appl., 58, 2199-2208. [21] Inokuti, M., Sekine, H., and Mura, T. (1978), General use of the Lagrange multiplier in non-linear mathematical physics, in: S Nemat-Nasser (Ed), Variational Method in the Mechanics of Solid, Pergamon Press, Oxford, 156-162. [22] Wu, G.C. and Baleanu, D. (2013), Variational iteration method for Burgers’ flow with fractional derivativesNew Lagrange multipliers, App. Math. Mod., 37, 6183-6190. [23] Adomian, G. (1998), A review of the Decomposition method in applied mathematics, J. Math. Anal. App., 135, 501-544.

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[24] Adomian, G. (1994), Solving Frontier Problems of Physics: Decomposition Method, Springer: Athens, Georgia. [25] Wazwaz, A.M. (2000), A New algorithm for calculating Adomian polynomials for nonlinear operators, App. Math. Comp., 3, 33-51. [26] Wazwaz, A.M. and El-Sayed, S.M. (2001), A new Modification of the Adomian decomposition method for linear and nonlinear operators, Appl. Math. Comput., 122, 393-405. [27] Momani, S. (2006), Non-perturbative analytical solutions of the space-and time-fractional Burgers equations, Chaos, Solitons and Fractals, 28, 930-937. [28] Gómez Plata, A.R. (2016), Non-linear fractional differential equations, PhD. Thesis in applied Mathematics, Imecc-Unicamp, Campinas.



Influence of Round-off Errors on the Reliability of Numerical Simulations of Chaotic Dynamic Systems Shijie Qin1 , Shijun Liao1,2† 1

2

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai, 200240, China Ministry-of-Education Key Lab for Scientific and Engineering Computing, Shanghai, 200240, China Submission Info Communicated by J.A.T. Machado Received 1 June 2017 Accepted 24 June 2017 Available online 1 July 2018 Keywords Round-off error Reliability of numerical simulations Chaotic dynamic system Butterfly-effect

Abstract We illustrate that, like the truncation error, the round-off error has a significant influence on the reliability of numerical simulations of chaotic dynamic systems. Due to the butterfly-effect, all numerical approaches in double precision cannot give a reliable simulation of chaotic dynamic systems. So, in order to avoid man-made uncertainty of numerical simulations of chaos, we had to greatly decrease both of the truncation and round-off error to a small enough level, plus a verification of solution reliability by means of an additional computation using even smaller truncation and round-off errors. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction The sensitive dependence on initial conditions (SDIC) was first found in 1890 by Henri Poincaré [1] in a particular case of the three-body problem, who later proposed that such phenomena could be common, say in meteorology [2]. In 1963, using a digit computer (Royal McBee LGP-30) to solve a set of coupled ordinary differential equations (ODEs) ⎧ ⎨ x˙ = −σ x + σ y, y˙ = rx − y − x z, (1) ⎩ z˙ = x y − bz, where σ , b and 0 < r < +∞ are physical parameters, Edward N. Lorenz [3] found the so-called butterflyeffect: a tiny change in initial condition might result in large difference in a later state. The tiny difference in initial condition of such kind of chaotic dynamic systems enlarges exponentially [4], which can be characterized by a positive Lyapunov exponent λ . In other words, the maximum Lyapunov exponent of a chaotic dynamic system must be positive. However, Lorenz [5] also reported that, by † Corresponding


198

Shijie Qin, Shijun Liao / Journal of Applied Nonlinear Dynamics 7(2) (2018) 197–204

means of the Runge-Kutta method with data in double precision, the maximum Lyapunov exponents of numerical simulations of a chaotic dynamic system given by different values of time-step may fluctuate around zero, say, its value constantly changes between positive and negative ones, even if the initial condition is exactly the same and the time step becomes rather small. Thus, the computer-generated numerical simulations of chaotic dynamic systems are sensitive not only to initial condition but also to numerical algorithms. This is easy to understand, since there always exist the truncation and roundoff errors at any steps of numerical simulations of chaos, which enlarge exponentially due to the socalled butterfly-effect. In addition, Teixeira et al. [6] investigated the time-step sensitivity of nonlinear atmospheric models and found that “different time steps may lead to different model climates and even different regimes”, thus “for chaotic systems, numerical convergence cannot be guaranteed forever ”. Hoover et al. [7] illustrated the Lyapunov’s instability by comparing numerical simulations of a chaotic Hamiltonian system given by two Runge-Kutta and five symplectic integrators [8–10] in double precision, and found that “all numerical methods are susceptible to Lyapunov instability, which severely limits the maximum time for which chaotic solutions can be accurate”, although “all of these integrators conserve energy almost perfectly” and “they also reverse back to the initial conditions even when their trajectories are inaccurate”. As reported by Hoover et al. [7] , “the advantages of higher-order methods are lost rapidly for typical chaotic Hamiltonians”, and “there is little distinction between the symplectic and the Runge-Kutta integrators for chaotic problems, because both types lose accuracy at the very same rate, determined by the maximum Lyapunov exponent.” Even for some dynamic systems without Lyapunov’s instability, it is rather hard to gain accurate prediction, too. For example, let us consider the famous Lorenz equation (1) in the case of σ = 10, b = 8/3, which is chaotic only when r ≥ 24.74. It is well-known that, when 1 < r < 24.74, the long-term solution of the Lorenz equations should finally tend to one of the two stable fixed points C( b(r − 1), b(r − 1), r − 1), and

C (−

b(r − 1), −

b(r − 1), r − 1).

However, in the case of r = 22, Li et al. [11] studied the sensitive dependance of the fixed point on the time-step Δt, which are calculated by means of many explicit/implicit numerical approaches (such as Euler’s method, Runge-Kutta methods of orders from 2 to 6, Taylor series methods of orders from 2 to 10, Adams methods of orders from 2 to 6, and so on) in double precision, but found that the long-term results of numerical simulations are rather sensitive to the step size Δt, say, they always fluctuate between the two fixed points, no matter how small the time-step Δt is. Thus, they made the conclusion that “numerical solution obtained by any stepsize is unrelated to exact solution” [11]. These numerical facts lead to some intense arguments. Some even believed that “all chaotic responses are simply numerical noise and have nothing to do with the solutions of differential equations” [12]. On the other side, using double precision data and a few examples based on the 15th-order Taylor-series procedure with decreasing time-step, Lorenz [13] was optimistic and believed that “numerical approximations can converge to a chaotic true solution throughout any finite range of time, although, if the range is large, confirming the convergence can be utterly impractical.” Is it possible to gain a convergent solution of a chaotic dynamic system in a long enough interval of time? This question is of critical importance. Note that convergent chaotic simulations cannot be guaranteed even if different high-order numerical methods were used, as illustrated by many researchers [6, 7, 11]. So, it is useless to reduce truncation errors only. Note that all of them used data in double precision, which leads to round-off error at each step, which also enlarges exponentially due to the butterfly-effect of chaos, just like truncation errors. So, to guarantee the convergence of chaotic solution, both of the truncation and round-off errors must be controlled to be much smaller than physical variables under investigation. In 2009, Liao [14]


199

Fig. 1 Final value of x(t) of the Lorenz equations in case of r = 23 with the initial value (5,5,10) versus stepsize h, obtained by the 4th-order Runge-Kutta method in double precision. The stepsize h varies from (a) 10−6 to 10−1 , (b) from 2.11349 × 10−2 to 3.16228 × 10−2 and (c) from 2.41107 × 10−2 to 2.45086 × 10−2, respectively.

suggested the so-called “Clean Numerical Simulation” (CNS) [15, 16] for chaotic dynamic systems and turbulence, which is based on the arbitrary order of Taylor expansion method [17, 18] and the use of all data in arbitrary precision (i.e. multiple precision [19]), plus a verification of solution reliability. By means of the CNS using the 3500th-order Taylor expansion method and data in 4180-digit precision, the convergent, reliable chaotic solutions of Lorenz equation were obtained even in [0, 10000], a rather long interval of time [20]. Its solution reliability was further verified by means of the CNS using the 3600th-order Taylor series method and data in 4515-digit precision [20]. This work supports Lorenz’s optimistic viewpoint that “numerical approximations can converge to a chaotic true solution throughout any finite range of time” [13]. Here, we further illustrate the effects of round-off error on the reliability of numerical simulations of chaotic dynamic systems, and show the importance of reducing both of truncation and round-off errors.

2 Influence of round-off errors To investigate the influence of round-off error on the reliability of numerical simulations of chaotic dynamic systems, let us consider the Lorenz equation (1) in the case of σ = 10, b = 8/3 and r = 23, with the exact initial condition x(0) = 5, y(0) = 5, z(0) = 10. (2) Since its solution is chaotic when r ≥ 24.74, the long-term numerical simulation should finally tend to one of the two stable fixed points in he case of r = 23. Let h = Δt denote the step-size. It is found that the final values of the numerical simulations given by the 4th-order Runge-Kutta method using data in double precision indeed rather sensitive to the step-size h, as shown in Fig. 1. No matter how small the stepzise h, the final values always fluctuate between the two fixed points. The same phenomenon was reported by Li et al. [11]. In order to reduce the truncation error, we use the Mth-order Taylor expansion method and the stepsize h = 0.01 to gain numerical simulations of Lorenz equation (1) in the case of σ = 10, b = 8/3 and r = 23 with the initial condition (2). Obviously, as M becomes large, the truncation error could be

200


np=4 np=8

10

Final value

5

0

-5

-10

5

10

15

20

25

30

35

40

M

Fig. 2 Final values of numerical simulation of x(t) of the Lorenz equations in case of r = 23 with the initial value (5,5,10) versus M (i.e. the truncated Mth-order Taylor’s expansion method), given by the same laptop (Thinkpad L440 with Intel Core i7-4712MQ) using data in double-precision but the different np (number of processes).

15

np=4 np=8

10

x

5

0

-5

-10

-15

0

50

100

150

200

250

t

Fig. 3 The numerical simulations of x(t) of the Lorenz equations in case of r = 23 with the initial value (5,5,10), given by the same laptop (Thinkpad L440 with Intel Core i7-4712MQ) using the 200th-order Taylor’ expansion method (i.e. M = 200) and data in double precision but the different np (number of processes).

rather small. To increase the computation efficiency, we use parallel computation by means of different numbers of processes, denoted by np. It is found that, using the double precision, we gain different final values by means of high-order Taylor expansion method even using the same laptop (Thinkpad L440 with Intel Core i7-4712MQ) but different numbers of processes, as shown in Fig. 2. Even if the order of Taylor expansion method is rather high, such as M = 200, corresponding to rather small truncation error, we still gain different final values using the same laptop but different numbers of processes, as shown in Fig. 3. It is found that, using the high-order Taylor series method in double precision, we would gain different final values using the same number of processes np but different computers, as shown in Fig. 4. This kind of man-made uncertainty of numerical simulations cannot be avoided even by means of rather high order of Taylor series method such as M = 200, as shown in Fig. 5. All of these illustrate that decreasing truncation error alone cannot avoid the man-made uncertainty of numerical simulations for the considered problem. As pointed out by Monniaux [21], even using the same programmes with the same compiler, which


201

Laptop Tianhe-II

10

Final value

5

0

-5

-10

5

10

15

20

25

30

35

40

M

Fig. 4 Final value of numerical simulations of x(t) versus M (i.e., the truncated Mth-order Taylor’s expansion), given by means of the same number of processes (np=4) and data in double precision but the different computers, i.e. the laptop (Thinkpad L440 with Intel Core i7-4712MQ) and the supercomputer Tianhe-II, respectively.

20

Laptop Tianhe-II

15 10

x

5 0 -5 -10 -15 -20

0

50

100

150

200

250

t

Fig. 5 The numerical simulations of x(t), given by the 200th-order Taylor’s expansion method (i.e. M = 200) using the same number of processes (np=4) and data in double precision but the different computers, i.e. the laptop (Thinkpad L440 with Intel Core i7-4712MQ) and the supercomputer Tianhe-II, respectively.

have exactly the same expression, the same values in the same variables and so on, different working platforms may exhibit subtle differences with respect to floating-point computations. Thus, both of the different number of processes operating on the same computer and the different computers with the same number of processes can generate rather tiny difference of round-off errors, which unfortunately would be enlarged so greatly (due to the butterfly-effect of chaos) that completely different numerical simulations might be obtained! In practice, round-off error sometimes indeed might lead to some serious problems, such as the system failure in the military: on February 25, 1991, a loss of significance in a MIM-104 Patriot missile battery prevented it from intercepting an incoming Scud missile in Dhahran, Saudi Arabia, contributing to the death of 28 soldiers from the U.S. Army’s 14th Quartermaster Detachment [22]. Such kind of man-made uncertainty of numerical simulations of the Lorenz equation can be avoided by decreasing both of the truncation and round-off errors at the same time! It is found that, using the Mth-order Taylor series method (M ≥ 130) in the 512-digit precision, all numerical simulations

202


10

Final value

5

0

-5

-10

0

50

100

150

200

M

Fig. 6 Final value of the numerical simulation of x(t) of the Lorenz equations in case of r = 23 with the initial value (5,5,10) given by means of the Mth-order Taylor’s expansion using data in 512-digit precision. For all given M, the same final values of x(t) are obtained, which are independent of the computers (i.e. the laptop and the supercomputer) and the number of precesses (np = 4 or 8). As M ≥ 130, the final value becomes to be independent of M, too.

agree quite well in the whole interval of time and besides tend to the same fixed point, even if different numbers of processes are used on different computers, as shown in Figs. 6-8. All of these examples illustrate the importance of decreasing both of truncation and round-off errors to the reliability of numerical simulations of chaotic dynamic systems, such as Lorenz equation, three-body problem and so on. Therefore, it clearly indicates that, using numerical approaches in double precision, one can not avoid the man-made uncertainty of numerical simulations for chaotic dynamic systems, as shown by many researchers [3, 5–7, 11, 12]. The above-mentioned examples also explain why one had to use the 3500th-order Taylor expansion method in the 4180-digit precision [20] so as to gain a convergent numerical simulation of a chaotic solution of Lorenz equation in a rather long interval of time [0,10000], whose reliability was further verified by means of the 3600th-order Taylor series method and data in 4515-digit precision. Nowadays, the importance of using multiple-precision data to gain reliable numerical simulations of chaotic dynamic systems receives recognition by more and more researchers [23–28].

3 Concluding remarks and discussions We confirm that, due to the butterfly effect, the traditional numerical approaches in double precisions indeed cannot give reliable numerical simulations of chaotic dynamic systems. Thus, decreasing the truncation error alone cannot avoid the man-made uncertainty of numerical simulations of chaos. However, such kind of man-made uncertainty of numerical simulations for chaotic dynamic systems can be completely avoided by decreasing both of the truncation and round-off errors at the same time, plus a verification of solution reliability by means of additional computations using even smaller truncation and round-off errors.


203

15 np=4 np=8

10

x

5

0

-5

-10

-15 380

400

420

440

460

480

500

520

540

560

t

Fig. 7 The numerical simulations of x(t) given by the 200th-order Taylor’s expansion method (i.e. M = 200) and data in 512-digit precision using the same laptop (Thinkpad L440 with Intel Core i7-4712MQ) but different np (number of processes).

15

10

Laptop Tianhe-II

x

5

0

-5

-10

-15 380

400

420

440

460

480

500

520

540

560

t

Fig. 8 The numerical simulations of x(t) given by means of the 200th-order Taylor’s expansion method (i.e. M = 200) and data in 512-digit precision using the same number of processes (np=4) but the different computers, i.e. the laptop (Thinkpad L440 with Intel Core i7-4712MQ) and the supercomputer Tianhe-II, respectively.

In this paper we illustrate that, due to the butterfly-effect, even the very tiny difference of roundoff error caused by different numbers of processes or different computers might lead to significant variation of numerical simulations of a chaotic dynamic system. So, the butterfly-effect is indeed a huge obstruction for us to gain reliable numerical simulations of chaos in a long interval of time. Note that a few current numerical investigations suggest that turbulent flows might be sensitive even to micro-level thermal fluctuation [28]. Then naturally, the turbulent flows should be also sensitive to numerical noises. Thus, it should be of benefit to study the influence of numerical noises to numerical simulations of turbulence.

204


References [1] Poincaré, J.H. (1890), Sur le probléme des trois corps et les équations de la dynamique, Divergence des séries de m. Lindstedt, Acta Math, 13, 1-270. [2] Wolfram, S. (2002), A New Kind of Science, Wolfram Media Inc.: Champaign. [3] Lorenz, E.N. (1963), Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20(2), 130-141. [4] Sprott, J.C. (2010), Elegant Chaos: Algebraically Simple Chaotic Flows, World Scientific: Singapore. [5] Lorenz, E.N. (2006), Computational periodicity as observed in a simple system, Tellus-A, 58, 549-559. [6] Teixeira, J., Reynolds, C.A., and Judd, K. (2007), Time step sensitivity of nonlinear atmospheric models: numerical convergence, truncation error growth, and ensemble design, Journal of the Atmospheric Sciences, 64(1), 175-188. [7] Hoover, W. and Hoover, C. (2015), Comparison of very smooth cell-model trajectories using five symplectic and two runge-kutta integrators, Computational Methods in Science and Technology, 21, 109-116. [8] Yoshida, H. (2008), Construction of higher order symplectic integrators, Physics Letters A, 150(5), 262-268. [9] Farrés, A., Laskar, J., Blanes, S., Casas, F., Makazaga, J., and Murua, A. (2013), High precision symplectic integrators for the solar system, Celestial Mechanics and Dynamical Astronomy, 116(2), 141-174. [10] Mclachlan, R.I., Modin, K., and Verdier, O. (2014), Symplectic integrators for spin systems, Physical Review E, 89(6), 247-285. [11] Li, J., Zeng, Q., and Chou, J. (2001), Computational uncertainty principle in nonlinear ordinary differential equations (i) numerical results, Science China Technological Sciences, 44(1), 55-74. [12] Yao, L. and Hughes, D. (2008), Comment on “computational periodicity as observed in a simple system” by Edward N. Lorenz (2006), Tellus-A, 60(4), 803´lC805. [13] Lorenz, E.N. (2008), Reply to comment by L.-S. Yao and D. Hughes, Tellus-A, 60(4), 806´lC807. [14] Liao, S. (2009), On the reliability of computed chaotic solutions of non-linear differential equations, Tellus-A, 61(4), 550-564. [15] Liao, S. (2013), On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems, Chaos, Solitons and Fractals, 47(1), 1-12. [16] Liao, S. (2014), Physical limit of prediction for chaotic motion of three-body problem, Communications in Nonlinear Science and Numerical Simulation, 19(3), 601-616. [17] Corliss, G. and Chang, Y. (1982), Solving ordinary differential equations using taylor series, ACM Trans. Math. Software, 8(2), 114-144. [18] Barrio, R., Blesa, F., and Lara, M. (2005), VSVO formulation of the taylor method for the numerical solution of ODEs, Computers and Mathematics with Applications, 50(1), 93-111. [19] Oyanarte, P. (1990), MP al a multiple precision package, Computer Physics Communications, 59(2), 345-358. [20] Liao, S. and Wang, P. (2014), On the mathematically reliable long-term simulation of chaotic solutions of lorenz equation in the interval [0,10000], Science China - Physics, Mechanics and Astronomy, 57(2), 330-335. [21] Monniaux, D. (2008), The pitfalls of verifying floating-point computations, ACM Transactions on Programming Languages and Systems, 30(3), 1-41. [22] Office, U.S.G.A. (1992), Government Accountability, Patriot Missile Defense: Software Problem Led to System Failure at Dhahran, Saudi Arabia, Technical Report, GAO/IMTEC-92-26. [23] Kehlet, B. and Logg, A. (2010), A Reference Solution for the Lorenz System on [0, 1000], American Institute of Physics, 1281, 1635-1638. [24] Sarra, S. and Meador, C. (2011), On the numerical solution of chaotic dynamical systems using extend precision floating point arithmetic and very high order numerical methods, Nonlinear Analysis: Modelling and Control, 16(3), 340-352. [25] Wang, P., Li, J.P., and Li, Q. (2012), Computational uncertainty and the application of a high-performance multiple precision scheme to obtaining the correct reference solution of lorenz equations, Numer Algorithms, 59, 147-159. [26] Wang, P. (2016), Forward period analysis method of the periodic hamiltonian system, PLoS ONE, 11(10), e0163303. [27] Barrio, R., Dena, A., and Tucker, W. (2015), A database of rigorous and high-precision periodic orbits of the lorenz model, Computer Physics Communications, 194, 76-83. [28] Lin, Z., Wang, L., and Liao, S. (2017), On the origin of intrinsic randomness of Rayleigh-Bénard turbulence, Science China - Physics, Mechanics and Astronomy, 60(1), 14712.



Dynamics of a Non-uniform Euler-Bernoulli Beam: Sensitivity Study in the Parameter Space Lilian M. Ribeiro, Gilson V. Soares, Alexandre C. L. Almeida, Adélcio C. Oliveira† Departamento de F´ısica e Matemática, Universidade Federal de S˜ ao Jo˜ ao Del Rei C.P. 131,Ouro Branco, MG, 36420-000, Brazil Submission Info Communicated by A.C.J. Luo Received 25 April 2017 Accepted 27 June 2017 Available online 1 July 2018 Keywords parametric oscillations Landau approximation differential transform Method attractor

Abstract The dynamics and spectrum of vibrations of non-uniform EulerBernoulli beam were investigated. The beam cross section varies linearly in direction of its length. Spacial and temporal solutions were investigated, spacial by Differential transform Method and temporal by four order Runge Kutta. A comparison between temporal numerical solutions and generalized Landau analytical approximations was conducted. There is a significant region, in parameter spaces, that the generalized Landau solution works well, and that there is a region, in parameter space, where the system behaves as it was periodic for practical reasons, thus we name it as almost stable attractor. It was shown that the solutions are strongly sensible to parameters that are related to geometrical and physical proprieties of the beam, it means that a small deviation on parameters can change the dynamics from convergent to divergent, and between those regimes, there is a region on parameter space that the model imitates a periodic system. ©2018 L&H Scientific Publishing, LLC. All rights reserved.

1 Introduction There is a large application scope to Mechanical Vibrations, from Mechanics to Biology, also, any advance in this area can reflect in other interdisciplinary applications, this justify the existence of a study area related to it. An important question in dynamical systems is the time scale. A periodic system has the period as its main time scale, while a chaotic system has the inverse of the largest Lyapunov exponent as its main time scale, and integrable dissipative systems have the dumping time as its main time scale [1]. This issue goes far beyond from mechanical dynamical system, see [2] and references in there. Dissipative dynamical systems have, in general, attractors, they can integrable or chaotic, there may be more than one for the same set of parameters [3, 4]. The attractors are defined as a point or a set of points that form the permanent solution of the system. Trajectories that are close to an attractor tends to be captured by them. The effect of weak dissipation is a relevant investigation area, with many applications [5,6]. The parametric dissipative oscillator has the origin as an attractor, † Corresponding



206

Lilian M. Ribeiro et al. / Journal of Applied Nonlinear Dynamics 7(2) (2018) 205–221

the oscillations either converge to zero as time goes to infinity or they grow to infinite amplitudes, see [7, 8]. In this work, it was investigated the dynamics of a non-uniform beam [9–12]. A dynamic investigation of geometric structures has a lot of practical applications and consequently are subject of many scientific works [9, 10, 12–14], also, uniform and non-uniform beams also have been extensively studied in literature with many good results, see [9–12, 15]. They have a large application range, such as mechanical projects present in aerospace devices, machines, ships, civil structures and others. Stress tension is one of the most important motivation, since it can be the source of fails and accidents [16]. A particular case where parametric beam has been used is to model cell micro-tubule vibrations [17,18] in this case, the driven force is stochastic. Another interesting application was to modeling vibrational effects of timber beams of Eucalyptus using a probabilistic approach [19]. The investigated model of non-uniform beam of this work has a cross section that varies linearly with the length, this beam can be considered as a simplified aircraft wing model. The spatial solution was obtained using the Differential Transform Method (DTM) [11, 12, 20, 21], and it was observed a good agreement with known results available in the literature [16], see also [11, 12]. The time solution was obtained using fourth order Runge Kutta, and a generalization of Landau approximate solution [7] was derived for dissipative parametric oscillations. It was shown that there is a set of unstable orbits that occurs, in parameter space, in a region near the attractor of generalized Landau solution, these orbits converge slowly to origin or diverge slowly and they dominate the system dynamics for a large range of parameters. 2 The model The beam was modeled in terms of Bernoulli-Euler equation, i.e.:

∂2 ∂2 ∂ ∂ ∂ 2Ψ ρ A(x) (p(x,t) )]Ψ + kΨ + [EI(x) ]Ψ − [ = f (x,t), ∂ x2 ∂ x2 ∂x ∂x ∂ t2

(1)

where Ψ is the deformation measured from neutral line, L is the length of the beam, x is the coordinate in the x direction, E is the Young modulus of the material, I(x) is the moment of inertia cross section of the beam around the y axis passing through the centroid, p(x,t) is the axial traction force, A(x) is the cross-sectional area, ρ is the material density and f (x,t) is an external load per unit length. It will be considered the case where the solution can be written as Ψ(x,t) = y(x)u(t), and constant density ρ (x) = ρ for p(x,t) = 0, k = 0, and the Equation (1) can be writen as

∂2 ∂2 ∂ 2Ψ (EI(x) )Ψ + = f (x,t). ρ A(x) ∂ x2 ∂ x2 ∂ t2

(2)

Since it is common for the load to be evenly distributed on the beam, and proportional to the beam deflection, it is natural to expect f to have the form f (x,t) = A(x)Ψ(x,t)h(t) = A(x)y(x)g(t), where g(t) is a continuous function of time. The boundary conditions taken into account are for x = L Ψ(L,t) = 0 and for x = 0 are

2.1

∂ 2 Ψ(0,t) =0 ∂ x2

and

and

∂ Ψ(L,t) = 0, ∂x ∂ 3 Ψ(0,t) = 0. ∂ x3

Space solution

It was considered a beam with a cross section that varies linearly in the x direction.

(3)

(4)


207

The beam’s geometrical model is shown in Figure 1, and it is defined

σ=

b1 , b0

Ic =

π a4 , 64

Ir =

π a3 b0 , I0 = Ir + Ic , 12

Ar = ab0 ,

Ac =

π a2 , 4

and

A0 = Ar + Ac .

(5)

The parameter σ quantifies the beam asymmetry. The others parameters are

α = (1 − σ )

Ir , I0

β=

σ Ir + Ic , I0

γ = (1 − σ )

Ar , A0

λ=

σ Ar + Ac , A0

(6)

where α and β are geometric parameters γ and λ are related to momentum of the cross section (see Figure 1). Then, new parameters are defined as η = x/L and y(η ) = Ψ(Lη ,t)/g(t). Now, the equation of motion is obtained as following form: (αη + β )

d3 d4 η ) + 2 α y( y(η ) − Ω4 (γη + λ )y(η ) = 0, dη 4 dη 3

(7)

where Ω is the dimensionless natural frequency of the beam given by Ω4 =

ρ A0 ω 2 L4 , EI0

(8)

and ω is the constant that appears in the separation of variables and physically means the natural beam frequency (the same that appears in Equation 14). The cross-sectional area is calculated by: A(η ) = [ and the moment of inertia by

σ Ar + Ac Ar (1 − σ )η + ], A0 A0

Ir σ Ir + Ic I(η ) = [ (1 − σ )η + ]. I0 I0

(9)

(10)

The differential transform method (DTM) was used to solve Equation 7. The method (DTM) is discussed in Appendix A. The transform of y(η ) is Y [k] and it is defined by Y [k] = (

1 dk y(η ))η =0 . k! d η k

(11)

For Equation 7 it is found a recurrence relation as Y [k + 4] =

Ω4 ( λ Y [k] + γY [k − 1] ) −α (k − 2) Y [k + 3] + , for k ≥ 0. β (k + 4) β (k + 4)(k + 3)(k + 2)(k + 1)

(12)

From the boundary conditions in Equation (4) it was found Y [2] = 0 and Y [3] = 0 (see Tab. 1), and from Equation (3) it is made Y [0] = A and Y [1] = B, witch result in a linear system in the variables A and B, as explained in the Appendix A. The boundary conditions determine the acceptable values for Ω (from the determinant of the system) and consequently the natural frequencies. Figure 2 shows the first four normal vibration modes, spatial solution, for asymmetry σ = 0, 5.

208


Fig. 1 Geometric model of the beam. 1

×10 -3 Ω1 Ω2 Ω3

0.5

y(η)

Ω4

0 0.2

0.4

0.6

0.8

η -0.5

-1

Fig. 2 Numerical solutions of Equation (7). The parameters are α = 0.472, β = 0.528, γ = 0.464 and λ = 0.536. Fundamental vibration (first excited) mode Ω1 = 2.056 (black solid line), second excited mode Ω2 = 4.828 (red dashed line), third excited mode Ω3 = 7.928 (blue dot dashed line) and fourth excited mode Ω4 = 11.041 (pink dotted line).

2.2

Temporal solution

Now, it was considered a periodic harmonic external force and a dissipative force, then f (x,t) = A(x)y(x)g(t), where g(t) is given by g(t) = −δ

d u(t) + ϕ u(t) cos(θ t), dt

(13)

where the first term represents damping and the second is a linear coupling with an external exciting source. Here δ is the dissipation straight, ϕ is the amplitude of disturbing force, θ the frequency of disturbance and ω is the natural frequency of the beam. Now the temporal part of Equation (2) can be written as


1015

|u(t)| +1

u(t)

5

209

0

1010

105

-5 0

10

20

30

40

50

100 10-1

100

101

102

103

t

t

Fig. 3 Numerical solution of Equation (14). The parameters are u(0) = 1, δ = 0.02, ω = 48.824, ϕ = 1, θ = 2ω , α = 0.472, β = 0.528, γ = 0.464 and λ = 0.536. (left) Short times for u(t), (right) Long times in logarithmic scale for |u(t)| + 1.

d2 d u(t) + δ u(t) + ω 2 − ϕ cos(θ t) u(t) = 0. 2 dt dt

(14)

The numerical solutions of Equation (14) were obtained by four order Runge Kutta method. This equation for δ = 0 is a Mathieu equation. Landau [7] solved it approximately for small oscillations in parametric resonance regime, i.e. θ ≈ 2ω . Figure 3 shows a typical solution of (14). For small dissipation, the parametric term dominates and the amplitude increases indefinitely. For δ = 0, the Equation (14) is given by [7]: d2 q(t) + ω 2 [1 − ξ cos(θ t)] q(t) = 0, dt 2 where ξ =

ϕ . ω2

(15)

For ξ ≪ 1, then it is shown that an approximate solution is of the type

ε ε q(t) = a(t) cos[(ω + )t] + b(t) sin[(ω + )t], 2 2

(16)

where ε = θ − 2ω . After some manipulations, it is possible to solve (approximately) the Equation (15) as ε ε (17) q(t) ≈ eκ t [a0 cos[(ω + )t] + b0 sin[(ω + )t]], 2 2 where κ 2 = 14 [( ξ ω2 )2 − ε 2 ], a0 and b0 are determined by initial conditions, see [8]. In Figure 4-a there is the RK solution and Landau solution (17) for short time. In this scale, it is difficult to distinguish these curves. For longer times, there is a small difference between RK and Landau solutions, see Figure 4-b. We call attempting for the difference in time intervals of both axes in Figures 4-a and 4-b. In Figure 5 it is shown the numerical (u, u) ˙ and Landau (q, q) ˙ phase space solution, it is clear that, for larger times, that Landau solution does not reproduce the u behavior. For small ϕ there is a better agreement between the solutions, as can be observed on Figure 6. This is due to the fact that Landau solution is valid for small oscillations and its amplitude is directly related to ϕ . 2

210


1.5

2

RK method Landau

1

1

u(t)

0.5 0

0 -0.5

-1 -1 -1.5 0

0.2

0.4

0.6

0.8

-2 119

1

119.2

119.4

t

119.6

119.8

120

t

(a)

(b)

Fig. 4 Numerical solutions of Equation (14) for δ = 0 using RK method (blue continuous line) and Landau solution (red dashed line). The parameters are u(0) = 1, ω = 48.824, ϕ = 1, θ = 2ω , ξ = −4.195 × 10−4, ε = 0, κ = 5.12 × 10−3. (a) t ∈ [0, 1], (b) t ∈ [100, 101].

50

dq(t)/dt

du(t)/dt

50

0

0

-50

-50

-1.5

-1

-0.5

0

u(t)

0.5

1

1.5

-1.5

-1

-0.5

0

0.5

1

1.5

q(t)

Fig. 5 Phase space solution (RK), u(t) and Landau solution q(t). The used parameters are u(0) = q(0) = 1, u(0) ˙ = q(0) ˙ = 0, ω = 48.824, ϕ = 1 and θ = 2ω (consequently , ξ = −4.195 × 10−4, ε = 0 and κ = 5.12 × 10−3).

60

60

40

40

20

20

dq(t)/dt

du(t)/dt


0

0

-20

-20

-40

-40

-60 -1.5

211

-60 -1

-0.5

0

0.5

1

1.5

-1.5

-1

-0.5

u(t)

0

0.5

1

1.5

q(t)

Fig. 6 Phase space solution (RK), u(t) and Landau solution q(t). The used parameters are u(0) = q(0) = 1, u(0) ˙ = q(0) ˙ = 0, ω = 48.824, ϕ = 1×10−3 , and θ = 2ω (consequently ξ = −4.195×10−7, ε = 0 and κ = 5.12×10−6).

3 Extension of the Landau approach including dissipation The Landau solution is for a parametric oscillator without dissipation and for θ = 2ω + ε , where ε /ω ≪ 1 , thus his solution obeys d e 2 − ϕ cos(θ t)]uL (t) ≈ 0, vL (t) + [ω (18) dt where dtd uL (t) = vL (t). It was considered the case where δ 6= 0, now, an approximate solution is obtained in the form ueL = uL e−µ t , then d d ueL + µ ueL = e−µ t uL . dt dt

(19)

Considering Landau approximation (Landau) and after some algebraic calculations the ueL is approximately given by 2 d d2 e + µ 2 − ϕ cos(θ t) ueL . (20) ueL ≈ −2µ ueL − ω 2 dt dt p e = ω 2 − µ 2 , and µ = δ /2, then the equation (20) is given by e 2 + µ 2 = ω 2 follows ω Defining ω d d2 ueL ≈ −δ ueL − ω 2 − ϕ cos(θ t) ueL , 2 dt dt

(21)

that is equivalent to Equation (14). Thus, the approximate solution for δ 6= 0 is δ ε ε e + )t]], e + )t] + b0 sin[(ω q(t) = e(κ − 2 )t [a0 cos[(ω 2 2

(22)

e + ε , here ε /ω e ≪ 1 , and the constants a0 and b0 are determined where κ 2 = 14 [( 2ϕωe )2 − ε 2 ], and θ = 2ω by q(0) = a0 , (23) and b0 =

− δ2 ) [a0 ] . e + ε2 ω

dq dt (0) − (κ

(24)

212


Fig. 7 Numerical solution of ∆uq(τ ), Equation (25), as function of τ = 2ωπ t. The used parameters are q(0) = u(0) = 1, ω = 48.824, φ = 10−3, θ = 97.648624. The red line is for δ = 1.274 × 10−3, blue line δ = 0.032, green line δ = 0.064, pink line δ = 0.096 and the black line δ = 0.127.

Fig. 8 Numerical solution of ∆uq(τ ), Equation (25), as function of τ = 2ωπ t. The used parameters are q(0) = u(0) = 1, ω = 48.824, δ = 1.274 × 10−3, θ = 97.648624. The red line is for φ = 10−3 , blue line φ = 6 × 10−3, green line φ = 0.011.

The extended Landau approximation works well for a large region, in parameter space. In Figures 7, 8 and 9 it is shown the plot of the mean deviation (∆uq(t)) between q(t) and u(t) as function of time, it is mathematically defined as 1 ∆uq(τ ) = τ

ˆ

τ

|q(t) − u(t)|dt.

(25)

0

As can observed in Figure 7, the Landau approximation (17) works better when the dissipation increases. This is due to fact that the trajectories are attracted to the origin attractor. When φ is increased, its approximation becomes worse, see in Figure 8, when the parametric force was increased. The dissipation changes the parametric resonance frequency, as can be observe in Figure 9, in fact this is more complicated as can be observed in parameters phase space, see Figure 10 where it has been shown ∆l function defined as (26) ∆l = − log[∆uq(t, φ , δ )]. The orange region in Figure 10 is where the approximation works better, and blue where quality became


213

Fig. 9 Numerical solution of ∆uq(τ ), Equation (25), as function of τ = 2ωπ t. The used parameters are q(0) = u(0) = 1, ω = 48.824, φ = 10−3 , δ = 1.274 × 10−3. The red line is for θ = 97.648624, blue line θ = 97.648629, green line θ = 97.648634, pink line θ = 97.648619 and the brown line θ = 97.648614.

Fig. 10 Parameter space for numerical solution of ∆l for t = 20ωπ , see Equation (25), as function of φ and δ . The used parameters are q(0) = u(0) = 1, ω = 48.824, φ = [1, 6] × 10−3, δ = [1.274, 1.350] × 10−3.

214


Fig. 11 Numerical solution of u(t). The used parameters are u(0) = 5, ω = 48.824,δ = 7.95464×−4, ϕ = 0.125, ξ = 5.235 × 10−5, ε = 10−3 and κ = 3.97854 × 10−4. (a) Short time, (b) long time, (c) whole range, (d) |u(t)| + 1 in logarithmic scale.

worse, although the mean deviation is very small in all area of this figure a . This model has an attractor that is the origin, the u(t) goes to zero or grows indefinitely. The approximate solution q(t) has an other attractor that occurs when κ = δ2 . Near that region in parameter space, the system spends a longer time before it reaches the equilibrium or diverges asymptotically. In Figure 11 it was shown plot of the u(t) evolution for κ − δ2 = 1.216 × 10−7 and in Figure 12 for κ − δ2 = −4.104 × 10−4 . In first case, the system diverges asymptotically, while in the second case it slowly converges to origin. We call attempting to the fact that the collapse or divergence occurs in a time of 104 greater than the system characteristic time ω −1 , this is the slow convergence or divergence, the almost stable orbit occurs in a time greater, t/ω −1 ≫ 104 . In Figure 13 the parameter space is presented where the axes are dissipation δ and φ . This parameter space was constructed in terms of the function dcr (decay rate), that was defined as dcr = log(u(T 1)u(T 2)),

(27)

where the the parameters were setted as T 1 = 200 and T 1 = 800 in units of 1/ω , and u(0) = 1. Then the colors can be associated with different dynamic characteristics. Red is rapid divergence, dcr≫ 0, The numerical simulations indicate that this parameter phase space is qualitatively independent of the choice of the final time t, we used t ≫ ω −1 . a


215

Fig. 12 Numerical solution of u(t). The used parameters are u(0) = 5, ω = 48.824,δ = 3.79668 × 10−3, ϕ = 0.313, ξ = 1.314 × 10−4, ε = 1.2 × 10−3 and κ = 1.48793 × 10−3. (a) Short time, (b) long time, (c) whole range, (d) |u(t)| + 1 in logarithmic scale.

yellow slow divergence, dcr> 0 , green almost steady dcr≈ 0, blue slow convergence to origin, dcr< 0 and black fast convergence to origin, dcr≪ 0. Since there is a significant area in parameters phase space with a almost stable attractors, then it can be conjectured that it must be physically relevant. In Figure 14 the parameter space is presented, there the axes are dissipation δ and θ , as it can be observed, the model is more sensible to the parameter θ , the parametric resonance is clearly defined by the peak shown on this figure. In Figure 15 it is shown ∆l, the parameters are same as Figure 13, as observed, the almost stable orbits lay in a region where the Landau approximation works well, thus this regime is characterized by κ ≈ δ2 . Since the system is linear , then the general solution of (1) is ∞

w(x,t) = ∑ an un (t)yn (x),

(28)

1

where an is a weigh solution coefficient in the subspace defined by the set {un (t)yn (x)}. In a situation where limt→∞ un (t) = 0 for ∨n ∈ N, if uk (t) is in a space parameter region as the green part of figure 13 or 14 then w(x,t) ≈ uk (t)yk (x), for t ≫ ω1k . In this situation, for practical proposes, the k mode dominate the dynamics. In Figures 16 and 17 it has been shown the relative weigh (ϑ ) of the solutions uk (t) that corresponds to ωk . The weigh ϑ1 is almost stable, i.e. it converges for very large

216


Fig. 13 Parameter space for the numerical solution of dcr, Equation (27). The used parameters are u(1) = 5, ω = 48.824, ϕ = 10−3 , ξ = −4.195 × 10−4, ε = 0.1 and κ = 0.099. In red it has fast divergent trajectories, yellow slow divergent trajectories, green almost stable trajectories, blue slow convergent trajectories and black fast y x convergent trajectories. The axis are defined by: x axis δ = 600 and y axis φ = 60 + 10−3.

1

-1

dcr

0

-2 -3 100

δ

50

0

(a)

0

20

40

60

80

θ

(b)

Fig. 14 Parameter space for the numerical solution of dcr, Equation (27). The used parameters are u(0) = 1, ω = 48.824, ϕ = 1, ξ = −4.195 × 10−4, ε = 0.1 and κ = 0.099, φ = 0.668. In red it is the fast divergent trajectories, blue represents the slow divergent trajectories, in green it is the almost stable trajectories. The axis are defined by: x axis δ = x · 1.023 × 10−4 and y axis θ = y−35 350 + 97.649.


Fig. 15 Parameter space for numerical solution of ∆l for t = The parameters are the same of Figure 13.

20π ω ,

217

see Equation (25), as function of φ and δ .

time t ≫ 1/ω1 . The weigh is defined as

ϑk =

|uk | . |(u1 , u2 , u3 )|

(29)

From those Figures 16 and 17, it can be concluded that the almost stable orbit dominate the dynamics, in this case ϑ1 ≈ 1, (30) for t ≫ 1/ω1 , and ϑk ≈ 0 for k 6= 1. With the used parameters of Figures 16 and 17, u1 is in parametric resonance, while the others uk , k 6= 1 are not. As k is increased , the corresponding ωk is more distant from resonance thus converges faster to zero. This is a practical important result, since all parameters depend on geometrical and physical beams properties thus any small variation on length, on inertia momentum or any manufacturing defect will produce a very different dynamical behavior. It is noted that the system is very sensitive to small deviation on physical and geometrical parameters, this feature of the model becomes evident when it is compared the Figures 11 and 12, and note that a small variation in the parameter κ returns in a wide change in dynamics, from divergent behavior for convergent. 4 Conclusion The spacial equation was solved using the differential transform method. It also has been shown that the Euler-Bernoulli beam subjected to a parametric excitation can be approximately modeled by Landau solution. In case where there is dissipation, a generalization of Landau solution was presented. It was observed that the generalized Landau approximation works well for large range of parameters. Also it was observed that there is a combination of parameters that allow the system to behave as it was periodic, we call it Almost Stable Attractor. In this regime, the dynamics are dominated by those almost stable orbit. It was also observed that the system is strongly sensible to changes in

218


0.8

0.8 ϑ1

ϑ1

ϑ2

0.7

ϑ2

0.7

ϑ3

ϑ3

0.6

ϑ

ϑ

0.6

0.5

0.5

0.4

0.4

0.3

0.3 0

0.02

0.04

0.06

0.08

0.1

0

0.5

1

t (a)

t (b)

Fig. 16 Temporal mean of ϑk as function of time. The used parameters are ω = 48.824, ϕ = 1, ξ = −4.195 × 10−4, ε = 0.1 and κ = 0.099, φ = 0.668. In red ϑ1 , blue ϑ2 , in green ϑ3 . The x axis is t in units of ω −1 .

Fig. 17 Temporal mean of ϑk as function of time. The used parameters are ω = 48.824, ϕ = 1, ξ = −4.195 × 10−4, ε = 0.1 and κ = 0.099, φ = 0.668. In red ϑ1 , blue ϑ2 , in green ϑ3 . The x axis is t in units of ω −1 .

parameters. The parameters are determined by the set of natural frequencies of the beam, and these in turn depend on the geometry of the beam as well as its mechanical properties. Then, the dynamics can range from divergent to convergent just because of a small change in mechanical or geometrical beam’s parameters. This result indicates that a detailed dynamic analysis should be important for the dimensioning of beams, especially of mobile equipments. A Appendix - Differential transform method (DTM) The differential transformation method (DTM) is an interactive method based on the Taylor series expansion that transform the governing differential equations and boundary conditions into a set of algebraic equations using a transformation rule. The first study of the DTM in vibration analysis of beams was made by [22] and was employed for deriving frequency equations and mode functions. In this paper the DTM is implemented to find the dimensionless natural frequencies of of a beam that the cross section varies linear in direction of its

1.5


219

Table 1 Basic theorems of DTM (centered in x = x0 )for equations of motion, initial conditions and boundary conditions. Original Function

Transformed Function

f (x) = g(x) + h(x)

F[k] = G(k) + H(k)

f (x) = λ g(x)

F[k] = λ G(k)

f (x) = g(x)h(x)

F[k] = ∑ G(k − i)H[i]

k

i=0 (k+n)! k! G[k + n]

dn dxn g(x) f (x) = xn

f (x) =

F[k] =

F[k] = δ (k − n) = ( 0 if k 6= n, 1 if k=n)

Boundary conditions

Transformed

f (0) = 0

F[0] = 0

d dx f (0) = 0 d2 f (0) = 0 dx2 d3 f (0) = 0 dx3

F[1] = 0 F[2] = 0 F[3] = 0

∞

∑ F[k] = 0

f (1) = 0 d dx 2

d dx2 3

d dx3

k=0 ∞

∑ kF[k] = 0

f (1) = 0 ∞

k=0

∑ k(k − 1)F[k]) = 0

f (1) = 0 ∞

k=0

∑ k(k − 1)(k − 2)F [k] = 0

f (1) = 0

k=0

length. The basic definitions and operations of differential transformation are introduced as follows. Consider a function f (x) which is analytic in a domain D containing x0 . The differential transform F[k] of the function f (x) centered in x = x0 is given by: F[k] = The inverse transformation is defined as

1 dk ( f (x))x=x0 . k! dxk

(31)

∞

∑ (x − x0 )k F[k].

(32)

N (x − x0 )k dk ( f (x)) = x=x ∑ k! dxk ∑ (x − x0 )k F[k], 0 k=0 k=0

(33)

f (x) =

k=0

In practical problems, f (x) is presented as: N

f (x) ≈ where N is selected such that

∞

∑

(x − x0 )k F[k] < ε ,

(34)

k=N+1

and ε is a small predefined number (as small as we want). Theorems that are frequently used to transform (centered in x0 = 0) equations of motion and the boundary conditions are listed in Table 1. 1.1

Finding the dimensionless natural frequencies by DTM

As summary, the methodology for calculating the dimensionless natural frequencies is presented in the following. In order to find the natural frequencies of a beam through the DTM, it is necessary to know the boundary conditions of the problem. For simplicity, it will make the DTM centred in x0 = 0.

220


Having the boundary conditions of the problem under study and using the definition of the DTM, are found four finite sums equal to zero, two related to conditions on the edge x = 0 beam and two for x = L, where L the longitudinal length of the beam. It is from these sums with the transform of the equation of motion that are obtained the values of the dimensionless natural frequencies. For the calculation of the coefficients F[k], it is used the recursive relation given by the transformed equation of motion. It is noteworthy that since our problem is an equation of the fourth order, the coefficient values are identified since F[k + 4] for the recurrence relation, being necessary to calculate the first four coefficients through boundary conditions. Through the boundary conditions of the problem, it is possible find the values of two of the first four coefficients F[0], F[1], F[2] and F[3] to calculate the frequencies. The other two unknown coefficients will be called A and B. Once the recurrence relation utilized to calculate the coefficients, they will be written depending on variables adopted above, A and B, as a function of Ω described in the general equation of the beam. Consequently, to replace the values of the coefficients in the boundary conditions on η = 0 and η = 1 ( respectively to x = 0 and x = L in dimensional terms) are obtained two linear equations equalled zero in terms of A, B and Ω and they can be written as: P1A (Ω) P1B (Ω) A 0 = , P2A (Ω) P2B (Ω) B 0 where P1A (Ω), P1B (Ω), P2A (Ω) and P2B (Ω) are polynomials in Ω, independent of A e B. Once the trivial solution is not interesting, the determinant of the matrix of the linear system above should be zero, and therefore the real roots of the polynomial Ω coming from determinant above give the dimensionless natural frequencies of the beam. Acknowledgements The authors acknowledge FAPEMIG for financial support. ACO and ACLA acknowledge the support of the Funda¸cão de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG) through grant No. APQ-01366-16. References [1] Marion, J.B. and Thornton, S.T. (1995), Classical Dynamics of Particles and Systems (Fort Worth: Saunders College Pub.). [2] de Magalhaes, A.B. and Oliveira, A.C. (2016), Physics Letters A, 380(4), 554, [3] Oyarzabal, R., Jr., J.S., Batista, A.,de Souza, S.,Caldas, I.,Viana, R., and Sanjuan, M. (2016), Physics Letters A, 380(18-19), 1621, [4] Bonetti, R.C., de Souza, S.L.T., Batista, A.M., Jr, J.D.S., Caldas, I.L., Viana, R.L., Lopes, S.R., and Baptista, M.S. (2014), Journal of Physics A: Mathematical and Theoretical, 47(40), 405101. [5] Seoane, Aguirre, J.M., Sanjuan, J., M.A., and Lai, Y.C. (2006), Chaos, 16(2), 023101, [6] Barrio, R., Blesa, F., Sanju´ an, M.A.F., and Seoane, J.M. (2012), International Journal of Bifurcation and Chaos, 22(6), 1230010. [7] Landau, L. and Lifshits, E. Mechanics (Pergamon, 1976) [8] de Barros, V.P. (2007), Revista Brasileira de Ensino de F´ısica, 29(4), 549. [9] Dias, C. and Alves, M. (2013), Journal of Sound and Vibration, 332(5), 1372, [10] Hou, C. and Lu, Y. (2016), Journal of Sound and Vibration, 385, 104. [11] Shali, S., Nagaraja, S.R., and Jafarali, P. (2016), IOP Conference Series: Materials Science and Engineering, 149(1), 012158. [12] Abdelghany, S., Ewis, K., Mahmoud, A., and Nassar, M.M. (2015), Beni-Suef University Journal of Basic and Applied Sciences, 4(3), 185.


221

[13] Mao, X.Y.,Ding, H.,Lim, C., and Chen, L.Q. (2016), Journal of Sound and Vibration, 385, 267. [14] Mignolet, M.P.,Przekop, A., Rizzi, S.A., and Spottswood, S.M. (2013), Journal of Sound and Vibration, 332(10), 2437. [15] Abrate, S. (1995), Journal of Sound and Vibration, 185(4), 703. [16] Balachandran, B. and Magrab, E.B. (2004), Vibrations (Cengage Learning). [17] Brito, R.Y., Fabrino, D.L., and Oliveira, A. (2016), in Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, 10020. [18] Jiang, H. and Zhang, J. (2008), Journal of Applied Mechanics, Transactions ASME, 75(6), 610191. [19] Garc´ıa, D.A., Sampaio, R., and Rosales, M.B. (2016), Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38(8), 2661. URL http://dx.doi.org/10.1007/s40430-015-0418-1. [20] Biazar, J. and Mohammadi, F. (2010), Appl. Appl. Math. 5(10), 1726. [21] Ebrahimi, F. and Mokhtari, M. (2015), Journal of the Brazilian Society of Mechanical Sciences and Engineering, 37(4), 1435. URL http://dx.doi.org/10.1007/s40430-014-0255-7. [22] Malik, M. and Dang, H.H. (1998), Applied Mathematics and Computation, 96(1), 17. .

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Xilin Fu School of Mathematical Science Shandong Normal University Jinan 250014, China Email: [email protected]

C. Nataraj Department of Mechanical Engineering Villanova University, Villanova PA 19085, USA Fax: +1 610 519 7312 Email: [email protected]

Juan J. Trujillo Department de Análisis Matemático Universidad de La Laguna C/Astr. Fco. Sánchez s/n 38271 La Laguna, Tenerife, Spain Fax: +34 922318195 Email: [email protected]

Hamid R. Hamidzadeh Department of Mechanical and Manufacturing Engineering Tennessee State University Nashville, TN37209-1561, USA Email: [email protected]

Raoul R. Nigmatullin Department of Theoretical Physics Kremlevskaiya str.18 Kazan State University, 420008 KAZAN, Tatarstan Russia Email: [email protected]

Yuefang Wang Department of Engineering Mechanics Dalian University of Technology Dalian, Liaoning, 116024, China Email: [email protected]

Reza N. Jazar School of Aerospace, Mechanical and Manufacturing RMIT University Bundoora VIC 3083, Australia Fax: +61 3 9925 6108 Email: [email protected]

Lev Ostrovsky Zel Technology/NOAA ETL Boulder CO, USA 80305 Fax:+1 303 497 7384 Email: [email protected]

Mikhail V. Zakrzhevsky Institute of Mechanics Riga Technical University 1 Kalku Str., Riga LV-1658 Latvia Email: [email protected]

Stefano Lenci Dipartimento di Ingegneria Civile Edile e Architettura, Universita' Politecnica delle Marche via Brecce Bianche, 60131 ANCONA, Italy Fax: +39 071 2204576 Email: [email protected]

Richard H. Rand Department of Mathematics Cornell University Ithaca, NY 14853-1503, USA Fax:+1 607 255 7149 Email: [email protected]

Jiazhong Zhang School of Energy and Power Engineering Xi’an Jiaotong University Shaanxi Province 710049, China Email: [email protected]

Edson Denis Leonel Departamento de Física UNESP - Univ Estadual Paulista Av. 24A, 1515 - Rio Claro – SP Brazil - 13.506-900 Fax: 55 19 3534 0009 Email: [email protected]

S.C. Sinha Department of Mechanical Engineering Auburn University Auburn, Alabama 36849, USA Fax: +1 334 844 3307 Email: [email protected]

Yufeng Zhang College of Sciences China University of Mining and Technology Xuzhou 221116, China Email: [email protected]

Gennady A. Leonov Department of Mathematics and Mechanics St-Petersburg State University St-Petersburg, 198504, Russia Email: [email protected]

Jian-Qiao Sun School of Engineering University of California Merced, 5200 N. Lake Road Merced, CA 95344, USA Fax: +1 209 228 4047 Email: [email protected]

Shijun Liao Department of Naval Architecture and Ocean Engineering Shanghai Jiaotong University Shanghai 200030, China Fax: +86 21 6485 0629 Email: [email protected]

József K. Tar Institute of Intelligent Engineering Systems Óbuda University Bécsi út 96/B, H-1034 Budapest, Hungary Fax: +36 1 219 6495 Email: [email protected]

Indexed by Scopus and zbMATH

Journal of Applied Nonlinear Dynamics Volume 7, Issue 2

June 2018

Contents Trajectory Controllability of Fractional-order   (1, 2] Systems with Delay V. Srinivasan, N. Sukavanam….……………………….………....……..……......

111－122

On the Localization of Invariant Tori in a Family of Generalized Standard Mappings and its Applications to Scaling in a Chaotic Sea Diogo Ricardo da Costa, Iberê L. Caldas, Denis G. Ladeira, Edson D. Leonel….

123－129

Runge-Kutta Method of Order Four for Solving Fuzzy Delay Differential Equations under Generalized Differentiability S. Indrakumar, K. Kanagarajan ..............................................................................

131－146

Existence of Positive Solutions for System of Second Order Integro-differential Equations with Multi-point Boundary Conditions on Time Scales V. Krishnaveni, K. Sathiyanathan...………...…........................………………..…

147－163

Spatiotemporal Patterns of a Pursuit-evasion Generalist Predator-prey Model With Prey Harvesting Lakshmi Narayan Guin, Benukar Mondal, Santabrata Chakravarty................…

165－177

Dynamics and Stability Results of Fractional Pantograph Equations with Complex Order D. Vivek, K. Kanagarajan, S. Harikrishnan……...................………...….…..…...

179－187

Variational Iteration Method in the Fractional Burgers Equation A. R. Gómez Plata, E. Capelas de Oliveira………………………………..…......

189－196

Influence of Round-off Errors on the Reliability of Numerical Simulations of Chaotic Dynamic Systems Shijie Qin, Shijun Liao……………….……….............................…………...…...

197－204

Dynamics of a Non-uniform Euler-Bernoulli Beam: Sensitivity Study in the Parameter Space Lilian M. Ribeiro, Gilson V. Soares, Alexandre C. L. Almeida, Adélcio C. Oliveira……………………………………………………………………………………… 205－221

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Printed in USA

Journal of Applied Nonlinear Dynamics

Journal of Applied Nonlinear Dynamics

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