Solutions of Nonlinear Fractional Di erential

Haci Mehmet Baskonus, Fethi Bin Muhammad Belgacem, and Hasan Bulut

Solutions of Nonlinear Fractional Differential Equations Systems through an Implementation of the Variational Iteration Method Abstract: In this book chapter, the Variational Iteration Method is used to solve systems of nonlinear fractional differential equations. Implementations with two separate nonlinear fractional differential equations systems illustrate a high degree of effectiveness. In the absence of explicit analytical methods, the Variational Iterative Method turns out to be a powerful tool to understand the behavior of nonlinear fractional differential equations and systems. Numerical solutions plots are obtained for each system via the Mathematica Symbolic Algebra Software 9. Keywords: Variational Iterational method; Fractional differential equation; Nonlinear fractional differential equation systems.

1 Introduction Fractional differential equations and systems have come to earn a high standing as important mathematical tools and as a means for describing and modeling linear and non-linear phenomena. (See for instance, Goswami & Belgacem, 2012; Gupta, Sharma, & Belgacem, 2011; and Katatbeh & Belgacem, 2011, as well as references therein). Such importance can be measured through the growing attention of scientists to nonlinear studies, thereby affecting nearly every aspect of an intertwined world of nonlinear sciences, both at the physical and theoretical level (Belgacem et al. 2014; Bulut, Belgacem & Baskonus, 2013). The ramifications of this increased focus have deeply influenced engineering applications, the physical sciences and applied mathematics, (Bulut, Baskonus & Belgacem, 2013; Dubey, Goswami & Belgacem, 2014; Tuluce, Bulut &Belgacem, 2014). This is particularly true with Nonlinear Fractional Differential Equations and Systems (NFDEs & NFDES’s).

Haci Mehmet Baskonus: Department of Computer Engineering, University of Tunceli, Tunceli, Turkey, E-mail: [email protected] Fethi Bin Muhammad Belgacem: Department of Mathematics, Arap Open University, Al-Ardhyia, Kuwait, E-mail: [email protected] Hasan Bulut: Department of Mathematics, Firat University, Elazig, Turkey, E-mail: [email protected]

© 2015 Haci Mehmet Baskonus et al. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.

Unauthenticated Download Date | 12/22/15 5:22 PM

334 | Haci Mehmet Baskonus et al. In physics and mathematics new methods are constantly needed to discover exact or approximate solutions for NFDEs and NFDESs. Nonlinear mathematical theory remains scarce and researchers need to specify the type of nonlinearity at hand in order to proceed. NFDEs and NFDES’s may not always be solved exactly or explicitly due to the lack of analytical schemes either due to the difficulty of the problems or the specificity of the nonlinearities. Fortunately, such problems may at least be treated numerically and approximate solutions of equations and systems can most often be obtained. Numerical methods, more than analytical ones, are therefore at the forefront of solving NFDE’s, and NFDES’s (Momani & Abuasad, 2006; He & Wu, 2007; Sweilam, 2007). Most of the nonlinear fractional differential equation systems do not have known exact analytical solutions. There are however some analytical techniques for solving nonlinear fractional differential equation systems. Some of the available methods include iteration techniques (Mousa, Kaltayev & Ragab, 2009). Many applications in this area are based on He’s Variational Iterative Method (VIM), (He, 1997; He, 1998). The application of VIM can be found in many works from the turn of the century. For instance, we refer the esteemed readers’ attention to the treatments in (Abdou and Soliman, 2005; Odibat and Momani, 2006). In 2007, new and important developments and advanced applications of the VIM took place, in particular, in the works by Abbasbandy, 2007; He and Wu, 2007; Sweilam, 2007; Wazwaz, 2007. In this work, we implement the VIM and deliver numerical solutions and plots by means of the Mathematica Symbolic Algebra Package 9.

2 The Variational Iteration Method In order to illustrate the basic concepts of VIM, the following general nonlinear partial differential equation can be considered as a prototype, Lu (x, t) + Ru (x, t) + Nu (x, t) = f (x, t) ,

(1)

where L is a linear time derivative operator, R is a linear operator which has derivatives with respect to x and t, N is a nonlinear differential operator, and f (x, t) is an inhomogeneous source term (also see Bulut, Baskonus, & Tuluce, 2013). To implement the VIM we construct the next iteration for Eq. 1, u n+1 (x, t) = u n (x, t) +

Zt

fn (x, τ) + N u fn (x, τ) − f (x, τ) dτ, (2) λ (τ) Lu n (x, τ) + R u

0

where the parameter λ is a general Lagrange multiplier, which can be identified optimally via variational theory, the subscript n denotes the nth-order approximation,


Solutions of Nonlinear Fractional Differential Equations Systems | 335

fn is considered as a restricted variation which implies that δ u fn = 0, (following and u for instance the treatments in Sweilam, 2007 and Abbasbandy, 2007). Clearly, the main steps of He’s VIM require first the determination of the Lagrange multiplier λ, which needs to be identified optimally. Once λ is determined, the successive approximations, u n+1 , n ≥ 0 of the solution u are obtained by using a suitably selected function u0 , which a priority satisfies the boundary conditions.

3 VIM Application to NFDES Here we implement the VIM for two separate NFDES, both of which were treated in Khan et al. 2011, by alternative means. The fractional derivations are in the sense of Riemann-Liouville fractional derivation throughout paper.

3.1 Example 1 First, we consider and treat the following NFDES, D αt u (t) = k (−1 − ε) u (t) + k (1 − ε) v (t) , D αt v (t) = k (1 − ε) u (t) + k (−1 − ε) v (t) ,

(3)

u (0) = 1, v (0) = 3, 0 < α < 1.

(4)

with the conditions We start by defining a structure of iteration for Eq. 3 in the following form, α Rt ∂ u k (τ) + k (1 + ε) u k (τ) + k (ε − 1) v k (τ) dτ, k = 0, 1, 2, 3 · · · , u k+1 = u k + λ (τ) ∂τ α 0 Rt ∂ α v k (τ) v k+1 = v k + λ (τ) dτ . + k −1 + ε u τ + k ε + 1 v τ ( ) ( ) ( ) ( ) k k ∂τ α 0 (5) Calculating variation with respect to u n , we have the stationary conditions as follows; ( 1 + λ (τ) |τ=x = 0, (6) λ′ (τ) = 0. The Lagrange multiplier can be easily identified as λ (τ) = −1. Substituting Eq. 7 in Eq. 5, we have the iteration formula as follows; Rt ∂ α u k (τ) u k+1 = u k − + k 1 + ε u τ + k ε − 1 v τ dτ, k = 0, 1, 2, 3 · · · ( ) ( ) ( ) ( ) k k ∂τ α 0 Rt ∂ α v k (τ) v k+1 = v k − + k (−1 + ε) u k (τ) + k (ε + 1) v k (τ) dτ . ∂τ α 0


(7)

(8)

336 | Haci Mehmet Baskonus et al. We choose the initial approximate solution in the form u (0) = 1, v (0) = 3 and substitute in Eq. 8 and construct a simple algorithm in any computational software such as Mathematica 9. Then, we obtain the approximate solution as follows, Rt ∂ α u0 + k 1 + ε u + k ε − 1 v dt, k = 0 ⇒ u1 = u0 − ( ) ( ) 0 0 α 0 ∂t Rt ∂ α 1 t1−α (9) dt − k (4ε − 2) t = 1 − − k (4ε − 2) t, =1− α ∂t (1 − α)Γ(1 − α) 0 t1−α = 1 − k(4ε − 2)t − (1−α)Γ(1−α) , Rt ∂ α v0 + k −1 + ε u + k ε + 1 v dt, ( ) ( ) 0 0 ∂t α 0 Rt ∂ α 3 Rt ∂ α 3 + k −1 + ε + k ε + 1 3 dt = 3 − + k 4ε + 2 dt, =3− ( ) ( ) ( ) α ∂t α 0 ∂t 0 Rt ∂ α 3 3t1−α =3− dt − k (4ε + 2) t = 3 − − k (4ε + 2) t, α ∂t (1 − α)Γ(1 − α) 0 1−α 3t = 3 − k (4ε + 2) t − . (1 − α)Γ(1 − α)

⇒ v1 = v0 −

(10) Therefore, we get the following approximate solution, t1−α 3t1−α , 3 − k (4ε + 2) t − . (u1 , v1 ) = 1 − k (4ε − 2) t − (1 − α)Γ(1 − α) (1 − α)Γ(1 − α) (11) k = 1 ⇒ u2 = Rt ∂ α u1 + k (1 + ε) u1 + k (ε − 1) v1 dt, u1 − ∂t α 0 Rt Rt Rt ∂ α u1 t1−α = 1 − k (4ε − 2) t − dt − k 1 + ε u dt − − [ ( ) ] [k (ε − 1) v1 ] dt, 1 (1 − α)Γ(1 − α) 0 ∂t α 0 0 Zt Zt Zt α t1−α ∂ u1 = 1 − k (4ε − 2) t − − dt −k 1 + ε u dt − k ε − 1 ( ) [ 1] ( ) [v1 ] dt, ∂t α (1 − α)Γ(1 − α) 0 0 0 | {z } | {z } | {z } I1

I2

We calculate the terms of I1 , I2 and I3 as follows;


I3


Rt ∂ α u1 Rt ∂ α t1−α dt = 1 − k 4ε − 2 t − dt, ( ) α ∂t α (1 − α)Γ(1 − α) 0 0 ∂t Rt ∂ α t Rt ∂ α 1 Rt ∂ α t1−α 1 dt + 2k 1 − 2ε dt − dt, = ( ) α ∂t α ∂t α (1 − α)Γ(1 − α) 0 0 ∂t 0 t1−α 2k (1 − 2ε) t2−α Γ (2 − α) t2−2α = + − , (1 − α)Γ(1 − α) (2 − α)Γ(2 − α) 2(1 − α)2 Γ(1− α)Γ(2 − 3α) 1−α Rt Rt t thus dt I2 = [u1 ] dt = 1 − k (4ε − 2) t − (1 − α)Γ(1 − α) 0 0 t2−α = t − k (2ε − 1) t2 − , (2 − α)(1 − α)Γ(1 − α) Rt Rt 3t1−α dt 3 − k (4ε + 2) t − I3 = [v1 ] dt = (1 − α)Γ(1 − α) 0 0 3t2−α = 3t − k (2ε + 1) t2 − . (2 − α)(1 − α)Γ(1 − α)

I1 =

2t1−α (1 − α)Γ(1 − α) Γ(2 − α)t2−2α k(1 + ε)t2−α 2k(1 − 2ε)t2−α (12) + + − 2 (2 − α)Γ(2 − α) 2(1 − α) Γ(1 − α)Γ(2 − 3α) (2 − α)(1 − α)Γ(1 − α) 2−α 3k(ε − 1)t −3k(ε − 1)t + (ε − 1)(2ε + 1)k2 t2 + , (2 − α)(1 − α)Γ(1 − α) Rt ∂ α v1 ⇒ v2 = v1 − + k (−1 + ε) u1 + k (ε + 1) v1 dt , ∂t α 0 Rt ∂ α 3t1−α 3t1−α 3 − k 4ε + 2 t − − dt = 3 − k (4ε + 2) t − ( ) (1 − α)Γ(1 − α) 0 ∂t α (1 − α)Γ(1 − α) Rt t1−α k (−1 + ε) 1 − k (4ε − 2) t − dt − (1 − α)Γ(1 − α) 0 Rt 3t1−α k (ε + 1) 3 − k (4ε + 2) t − − dt, (1 − α)Γ(1 − α) 0 Rt ∂ α 3t1−α 3t1−α = 3 − k (4ε + 2) t − − 3 − k 4ε + 2 t − dt ( ) (1 − α)Γ(1 − α) 0 ∂t α (1 − α)Γ(1 − α) k(ε − 1)t2−α −k(−1 + ε)t + (ε − 1)(2ε − 1)k2 t2 + (2 − α)(1 − α)Γ(1 − α) 3k(ε + 1)t2−α 2 2 −3k(ε + 1)t + k(ε + 1)(2ε + 1)k t + , (2 − α)(1 − α)Γ(1 − α) u2 = 1 − k (4ε − 2) t − k(1 + ε)t + k2 (1 + ε)(2ε − 1)t2 −

= 3 − k (4ε + 2) t − k(−1 + ε)t + (ε − 1)(2ε − 1)k2 t2 − 3k(ε + 1)t 6t1−α k(4ε + 2)t2−α 3Γ(2 − α)t2−2α − + + (1 − α)Γ(1 − α) (2 − α)Γ(2 − α) 2(1 − α)2 Γ(1 − α)Γ(2 − 3α) k(ε − 1)t2−α 3k(ε + 1)t2−α +k(ε + 1)(2ε + 1)k2 t2 + + , (2 − α)(1 − α)Γ(1 − α) (2 − α)(1 − α)Γ(1 − α) .. ..

(13)

The remainder components of this iteration formula Eq. 8 are obtained in the same manner using Mathematica 9.


338 | Haci Mehmet Baskonus et al.

(a)

(b) Fig. 1. The surfaces generated from the VIM result of u (t) in Eq. 12 α = k = ε = 0.91 over the interval 0 < t < 15 shown in (a), shown in (b) over the same intervals for approximate solution compound v (t) in Eq. 13.

3.2 Example 2 Our second example was also treated in Khan et al. 2011 by alternative methods. We apply the VIM to the NFDES, D αt u (t) = −1002u (t) + 1000v2 (t) , D αt v (t) = u (t) − v (t) − v2 (t) ,

(14)

with the initial conditions u (0) = 1, v (0) = 1, 0 < α < 1.


(15)


When we apply the VIM, we define the structure of Eq. 14 in the following form, α Rt ∂ u k (τ) 2 + 1002u k (τ) − 1000v k (τ) dτ, k = 0, 1, 2, 3 · · · , u k+1 = u k + λ (λ) ∂τ α 0 α t R ∂ v k (τ) 2 v k+1 = v k + λ (λ) − u k (τ) + v k (τ) + v k (τ) dτ . ∂τ α 0 (16) Calculating variation with respect to u n , we have the stationary conditions as follows; ( 1 + λ (τ) |τ=x = 0, (17) λ′ (τ) = 0. in this case, the Lagrange multiplier can be easily identified as, λ (τ) = −1. Substituting Eq. 18 in Eq. 16, we obtain the following iteration formula, Rt ∂ α u k (τ) 2 + 1002u k (τ) − 1000v k (τ) dτ, k = 0, 1, 2, 3 · · · , u k+1 = u k − ∂τ α 0 Rt ∂ α v k (τ) 2 v k+1 = v k − dτ . − u τ + v τ + v τ ( ) ( ) ( ) k k k ∂τ α 0

(18)

(19)

When we choose the initial approximate solution to be u (0) = 1, v (0) = 1 and substitute in Eq. 19, we get the next approximate solution, Rt ∂ α u0 (t) 2 k = 0 ⇒ u1 = u0 − + 1002u0 − 1000v0 dt, ∂t α 0 Rt ∂ α 1 Rt ∂ α 1 (20) =1− + 2 dt = 1 − 2t − dt, ∂t α ∂t α 0 0 t1−α = 1 − 2t − , (1 − α)Γ(1 − α) Rt ∂ α 1 Rt ∂ α v0 (t) 2 dt = 1 − ⇒ v1 = v0 − − u t + v t + v t + 1 dt, 0( ) 0( ) 0( ) ∂t α ∂t α 0 0 (21) t1−α =1−t− . (1 − α)Γ(1 − α) Therefore, we get an approximate solution as follows; t1−α t1−α , 1−t− , (22) (u1 , v1 ) = 1 − 2t − (1 − α)Γ(1 − α) (1 − α)Γ(1 − α) Rt ∂ α u1 2 k = 1 ⇒ u2 = u1 − + 1002u − 1000v 1 1 dt, ∂t α 0 2t1−α 2000t3−α = 1 − 2t + 1000t − 1000t2 + 1000t3 − + (1 − α)(1 − α)Γ(1 − α) − α)Γ(1 − α) (3 t2−α 1000t3−2α 2 + , −1002 t − t − (2 − α)(1 − α)Γ(1 − α) (3 − 2α)[(1 − α)Γ(1 − α)]2 2000t2−α t2−α Γ(2 − α)t2−2α − + + (2 − α)(1 − α)Γ(1 − α) (2 − α)Γ(1 − α) 2(1 − α)2 Γ(1 − α)Γ(2 − 3α) (23)


340 | Haci Mehmet Baskonus et al. Rt ∂ α v1 2 − u + v + v 1 1 1 dt, ∂t α 0 2t1−α t3−2α t2 t3 − − − = 1 − 2t + 2 3 (1 − α)Γ(1 − α) (3 − 2α)[(1 − α)Γ(1 − α)]2 2t3−α t2−α 2t2−α − + + (2 − α)(1 − α)Γ(1 − α) (3 − α)(1 − α)Γ(1 − α) (2 − α)Γ(2 − α) Γ(2 − α)t2−2α + . 2(1 − α)2 Γ(1 − α)Γ(2 − 3α)

⇒ v2 = v1 −

(24)

The rest components of the iteration formula Eq. 19, can easily be obtained, in the same manner using software such as Mathematica 9.

(a)

(b) Fig. 2. The surfaces generated from the VIM result of u (t) in Eq. 23 α = 0.91 over the intervals 0 < t < 5 shown in (a), shown in (b) over the same intervals for Eq. 24


Solutions of Nonlinear Fractional Differential Equations Systems |

341

4 Conclusions and Future Work In this research, the Variational Iteration Method was successfully applied for solving two systems of nonlinear fractional differential equations systems. Approximate solutions were obtained and plotted by using the Mathematica symbolic algebra package software 9. The VIM set up, despite its simplicity, turns out to be a valuable tool to treat NFDES. Consequently, we recommend the use of VIM for such systems when direct analytical approaches are absent, which are mostly the case. For future work, we plan consider the Hyper Chaotic NFDES treated in (Mohamed & Ghany 2011), by different means, where we hope to enrich the understanding of such four by four chaos engendering systems, thanks to this now celebrated method, the VIM. We look forward to any communications, and remarks, and collaborations from esteemed fellow readers. Acknowledgment: F. B. M. Belgacem wishes to acknowledge the continued support of the Public Authority for Applied Education and Training Research Department, (PAAET RD), Kuwait. The Authors wish to acknowledge the helpful support of Editor Prof. Carlo Cattani, and DGO Managing Editor, Mathematics, Agnieszka Bednarczyk-Drag.

Bibliography Abbasbandy, S. A. (2007), New application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials, J. Comput. Appl. Math., 207(1) 59-63. Abdou M. A. and Soliman, A. A. (2005). Variational iteration method for solving Burger’s and coupled Burger’s equations, J. Comput. Appl. Math., 181(2), 245–51. Belgacem, F.B.M., Bulut H., Baskonus M.H., & Akurk T. (2014). Mathematical Analysis of the Generalized Benjamin Kdv Equations via the Extended Trial Equations Method, Journal of the Associations of Arab Universities for Basic and App. Sciences, Vol.16, 91-100. Bulut H., Baskonus, H. M., & Tuluce S. (2013), The solutions of homogenous and nonhomogeneous linear fractional differential equations by variational iteration method, Acta Universitatis Apulensis. Mathematics-Informatics, 36, 235-243. Bulut H., Baskonus M.H., & Belgacem, F.B.M. (2013); The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu transform Method, Applied and Abstract Analysis, Article ID 203875, 1-6. Bulut H., Belgacem, F.B.M, & Baskonus, H.M. (2013); Fractional Partial Differential Equations Systems Solutions by Adomian Decomposition Method Implementation. ICMCA 2013: 4th International Conference on Mathematical and computational Applications, Manisa, Turkey, June, 11-13, Dubey, R.S., Goswami, P., & Belgacem, F.B.M. (2014). Generalized Time-Fractional Telegraph Equation Analytical Solution by Sumudu and Fourier Transforms. Journal of Fractional Calculus and Applications, 5(2), 52-58. Goswami, P., &, Belgacem, F.B.M. (2012). Fractional Differential Equations Solutions through a Sumudu Rational, Nonlinear Studies Journal, 19(4), 591-598.


342 | Haci Mehmet Baskonus et al.

Gupta, V.G.; Sharma, B.; & Belgacem, F. B. M. (2011). On the Solutions of Generalized Fractional Kinetic Equations. Journal of Applied Math. Sciences, Vol.5, No. 19. 899 – 910. He J.H. (1997). Variational iteration method for delay differential equations, Commun. Nonlinear Sci. Numer. Simul. 2(4), 235–236. He J.H. (1998), Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Eng. 167 (1–2), 69–73. He J.H. & Wu, X.-H. (2007). Variational iteration method: new development and applications, Comput. Math. Appl. 54 (7-8), 881–894. Katatbeh, Q.K. & Belgacem, F.B.M. (2011). Applications of the Sumudu Transform to Fractional Diff. Equations, Nonlinear Studies, Vol. 18, No.1, 99-112. Khan, N.A., Jamil M., Ara, A., & Khan, N.U. (2011), On Efficient Method for System of fractional Differential Equations. Advances in Difference Equations, (2011), 15 pages . Mohamed S. M., & Ghany H.A. (2011), Analytic approximations for fractional-order hyperchaotic system, Journal of Advanced Research in Dynamical and Control Systems, 3(3), 1-12. Momani S., & Abuasad S. (2006). Application of He’s variational iteration method to Helmholtz equation. Chaos Soliton Fractals, 27, 1119–1123. Mousa, M.M. Kaltayev A., & Ragab S.F., (2009). Investigation of a transition from steady convection to chaos in porous media using piecewise variational iteration method, world Academy of Science, Engineering and Technology, 58, 1088–1097. Odibat Z. M. and Momani S., (2006) Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Non-linear Sci. Numer. Simul. 7(1), 27–34. Sweilam, N. H. (2007), Fourth order integro-differential equations using variational iteration method, Comput. Math. Appl. 54, 7-81086–1091. Tuluce D. S., Bulut H., & Belgacem F. B. M., (2014). Sumudu Transform Method for Analytical Solutions of Fractional Type Ordinary Differential Equations, Mathematical Problems in Engineering, in a Special Issue on: Partial Fractional Equations and their Applications. Wazwaz, A.-M. (2007), The variational iteration method for exact solutions of Laplace equation, Phys. Lett. A, 363(4), 260–262.
