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Procedia Computer Science 00 (2018) 000–000 Procedia Computer Science (2018) 000–000 Procedia Computer Science 12500 (2018) 476–482
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6th International Conference on Smart Computing and Communications, ICSCC 2017, 7-8 6th International ConferenceDecember on Smart2017, Computing and Communications, ICSCC 2017, 7-8 Kurukshetra, India December 2017, Kurukshetra, India
Computational Computational Simulations Simulations for for Solving Solving aa Class Class of of Fractional Fractional Models Models via via Caputo-Fabrizio Caputo-Fabrizio Fractional Fractional Derivative Derivative ∗ A.S.V. A.S.V. Ravi Ravi Kanth Kanth∗,, Neetu Neetu Garg Garg
Department of Mathematics, National Institute of Technology Kurukshetra, Kurukshetra-136119, India Department of Mathematics, National Institute of Technology Kurukshetra, Kurukshetra-136119, India
Abstract Abstract The aim of this paper is to investigate computational simulations for a class of fractional models via Caputo-Fabrizio fractional The aim of this paper is to investigate computational simulations for a class of fractional models via Caputo-Fabrizio fractional derivative with non-singular kernel. Analytical solution for a class of fractional models associated with Logistic model, Malthusian derivative with non-singular kernel. Analytical solution for a class of fractional models associated with Logistic model, Malthusian Growth model and Blood Alcohol model is presented graphically for various fractional orders and solution of corresponding Growth model and Blood Alcohol model is presented graphically for various fractional orders and solution of corresponding classical model is recovered as a particular case. An unexpected behaviour of Caputo-Fabrizio fractional derivative is observed. classical model is recovered as a particular case. An unexpected behaviour of Caputo-Fabrizio fractional derivative is observed. We study the impact of order of the fractional derivative on the rate of variation for these fractional models. We study the impact of order of the fractional derivative on the rate of variation for these fractional models. c 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. © c 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility responsibility of of the the scientific scientificcommittee committeeof ofthe the6th 6thInternational InternationalConference Conferenceon onSmart SmartComputing Computingand andCommuPeer-review under responsibility of the scientific committee of the 6th International Conference on Smart Computing and CommuCommunications nications. nications. Keywords: Caputo-Fabrizio fractional derivative, Fractional blood alcohol model, Fractional logistic equation, Non-singular kernel. Keywords: Caputo-Fabrizio fractional derivative, Fractional blood alcohol model, Fractional logistic equation, Non-singular kernel.
1. Introduction 1. Introduction Fractional calculus is the study of derivatives and integrals of non integer order which provides an excellent tool to Fractional calculus is the study of derivatives and integrals of non integer order which provides an excellent tool to explain memory and hereditary properties of complex systems [1–3]. During last decades, it has gained huge interest explain memory and hereditary properties of complex systems [1–3]. During last decades, it has gained huge interest and become a powerful instrument for better modeling of real world problems such as in mathematical biology [4, 5]; and become a powerful instrument for better modeling of real world problems such as in mathematical biology [4, 5]; in electric circuits [6]; in medicine [7] etc. As we know that friction and external forces can perturb the system, thus in electric circuits [6]; in medicine [7] etc. As we know that friction and external forces can perturb the system, thus integer order derivatives may not properly describe the system. Fractional derivatives provide us infinite choices for integer order derivatives may not properly describe the system. Fractional derivatives provide us infinite choices for fractional order that can help us to choose the fractional differential equation that can better describe the dynamics of fractional order that can help us to choose the fractional differential equation that can better describe the dynamics of the model. Fractional calculus modeling has been used to generalize various important models such as logistic model, the model. Fractional calculus modeling has been used to generalize various important models such as logistic model, Malthusian growth model and blood alcohol model etc. [4, 8, 9]. Logistic model and Malthusian growth model are Malthusian growth model and blood alcohol model etc. [4, 8, 9]. Logistic model and Malthusian growth model are used in population biology to describe animal population or the growth of tumor and bacteria. Blood alcohol model used in population biology to describe animal population or the growth of tumor and bacteria. Blood alcohol model [10] predicts alcohol concentration in blood that can be useful in the study of alcohol effects on brain and other body [10] predicts alcohol concentration in blood that can be useful in the study of alcohol effects on brain and other body organs. Fractional version of these models describes the problem more efficiently than the integer order model. organs. Fractional version of these models describes the problem more efficiently than the integer order model. ∗ ∗
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c 2018 The Authors. Published by Elsevier B.V. 1877-0509 c 2018 The Authors. Published by Elsevier B.V. 1877-0509 Peer-review under responsibility of the scientific committee of the 6th International Conference on Smart Computing and Communications. Peer-review under responsibility of the scientific committee of the 6th International Conference on Smart Computing and Communications. 1877-0509 © 2018 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 6th International Conference on Smart Computing and Communications 10.1016/j.procs.2017.12.063
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The well-known definitions of Riemann-Liouville and Caputo fractional derivative have some limitations. Riemann Liouville derivative of a constant is not zero. Definitions of Caputo and Riemann Liouville fractional derivative involve singular kernel and this weakness affects the real world problems. Caputo and Fabrizio [11] proposed a definition of fractional derivative with non singular kernel and Losada [12] described its properties. This definition is capable of describing the heterogeneities and structures with different scales which cannot be well portrayed by classical local derivatives or by fractional models with singular kernel. So the study of Caputo-Fabrizio definition with non-singular kernel becomes more appropriate. The aim of this paper is to study Caputo-Fabrizio fractional derivative without singular kernel for fractional models like logistic equation, Malthusian growth model and blood alcohol model. The article is organized as follows. In the next section, we reviewed some basic definitions of fractional calculus. In section 3, analytical solutions of the fractional models are obtained. The conclusion of our work is presented in section 4. 2. Preliminary Concepts In this section, we present some basic definitions and properties of fractional calculus. Definition 1. Let f (t) ∈ C([a, b]) and a < τ < b, the Riemann Liouville’s definition of left fractional derivative of f (t) of order α (> 0) is defined as [2]:
RL α a Dt f (t)
dn 1 = Γ(n − α) dtn
t
a
f (τ) dτ, n − 1 < α ≤ n. (t − τ)α−n+1
Definition 2. The Riemann Liouville’s definition of right fractional derivative of f (t) of order α (> 0) is defined as:
RL α t Db f (t)
=
dn 1 (− n ) Γ(n − α) dt
t
b
f (τ) dτ, n − 1 < α ≤ n. (τ − t)α−n+1
Definition 3. The Riemann Liouville fractional integral of f (t) of order α (α > 0) is defined as:
RL α a It f (t)
1 = Γ(α)
t
f (τ) dτ, n − 1 < α ≤ n. (t − τ)1−α
a
Definition 4. The Caputo fractional derivative of a function f (t) of order α (> 0) as defined in [2]:
C α a Dt f (t)
=
1 Γ(n − α)
t a
f (n) (τ) dτ, n − 1 < α ≤ n. (t − τ)α−n+1
Definition 5. The Caputo-Fabrizio fractional derivative of order α as defined by [11]:
CF α a Dt f (t)
M(α) = (1 − α)
t a
f (τ)exp
−α(t−τ) 1−α
dτ, 0 < α ≤ 1,
where f ∈ H 1 (a, b), b > a and M(α) is a normalization constant depending on α such that M(0) = M(1) = 1.
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Note that Caputo-Fabrizio fractional derivative of a constant is zero and the kernel does not have singularity for t = τ. Definition 6. The Laplace transform of Caputo-Fabrizio fractional derivative is defined as [11]:
α+n f (t)] = L[ CF 0 Dt
sn+1 LT [ f (t)] − f (0)sn − f (0)sn−1 ... − f n (0) , 0 < α ≤ 1, s + (1 − s)α
(1)
where M(α) = 1. 3. Applications In this section, we have used Caputo-Fabrizio fractional derivative without singular kernel to the fractional models (i) Fractional Logistic equation (ii) Fractional Malthusian Growth model and (iii) Fractional Blood Alcohol model. Recently these models have gained interest by many researchers (see [4, 8–10]). 3.1. Fractional Logistic Equation We consider the fractional logistic equation of the form [4]: CF α 0 Dt x(t)
= k [1 − x(t)] , 0 < α ≤ 1,
(2)
1 and N(t) is the population at time t, k is the growth parameter and r (environmental carrying where x(t) = N(t) capacity) is assumed to be one. By applying Laplace transform on equation (2) and using equation (1), we get
X(s) =
x(0) − 1 (1 + k − αk)(s +
αk 1+k−αk )
1 + . s
(3)
where L(x(t)) = X(s). By applying inverse Laplace transform on both sides of Equation (3), we get: x(t) =
Substituting N(t) =
1 x(t)
and x(0) =
1 N(0)
−αkt x(0) − 1 e 1+k(1−α) + 1. 1 + k(1 − α)
in (4), we get 1
N(t) = 1+
1 N(0) −1
−αkt
.
1+k(1−α) 1+k(1−α) e
Note that
lim N(t) =
α→1
1 1+
1 ( N(0)
− 1)e−kt
,
(4)
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1 0.9 0.8
N(t)
0.7 0.6 0.5 α=1 α=0.9 α=0.8 α=0.7 α=0.6
0.4 0.3 0.2
0
5
10
t
15
Fig. 1. Population growth for different values of α.
i.e. the solution of integer order is a particular case of the fractional solution. Figure 1 represents the population growth for different values of fractional order α, by taking k = 1 , N(0) = 0.2 and carrying capacity r = 1. Also note that 1
lim N(t) = lim
t→∞
t→∞
1+
1 N(0) −1
−αkt
= 1,
1+k(1−α) 1+k(1−α) e
that is, all the values converge to the value of carrying capacity. It is observed from Fig. 1, when α (fractional order) is increasing, the population growth rate is also increasing and reaches to the value of carrying capacity. This behaviour of logistic equation plays an important role in the study of cancer tumor growth [4]. 3.2. Fractional Malthusian Growth Model We consider the fractional Malthusian growth model of the form [8]: CF α 0 Dt P(t)
= kP(t), α > 0,
(5)
where P(t) denote the population at time t and k is a positive constant. Case 1. 0 < α ≤ 1
By applying Laplace transform on (5) and using equation (1), we have
P(s) =
P(0) , s(1 − k + αk) − αk
where L(P(t)) = P(s). By applying the inverse Laplace transform on both sides of (6), we get: P(t) =
αkt P(0) e 1+k(α−1) . 1 + k(α − 1)
(6)
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25
5
40 α=1 α=0.9 α=0.8 α=0.7 α=0.6
20
α=1 α=5/4 α=3/2 α=7/4 α=2
35 30 25
P(t)
P(t)
15
10
20 15 10
5
5 0
0
0.5
1
t
1.5
2
2.5
0
0
0.5
1
1.5
t
2
2.5
3
Fig. 2. (a) Population growth for different values of α lying between 0 and 1; (b) Population growth for different values of α lying between 1 and 2.
Note that lim P(t) = P(0)ekt ,
α→1
i.e. the solution of the integer order is a particular case of the fractional solution. Figure 2(a) represents the population for different values of fractional order α lying between 0 and 1, with P(0) = 1 and k = 1. It is observed from Fig.2(a) that the population growth is decreasing with increasing fractional order, which is not expected. As growth rate of P(t) is supposed to be decreasing, for decreasing fractional order α. So we will consider (5) for order 1 < α ≤ 2. Case 2. 1 < α ≤ 2 Equation (5) can be written as: CF β+1 0 Dt P(t)
= kP(t), α = β + 1, 0 < β ≤ 1.
(7)
By applying Laplace transform on both sides of (7) and using (1), we get
P(s) =
s2
s+1 , − k(s + β(1 − s))
(8)
for P(0) = P (0) = 1. Then the inverse Laplace transform of equation (8) with k = 1 is as follows
P(t) =
2 t α − 2 −(α−1)t . e + e α α
Figure 2(b) represents the population growth for different values of fractional order α lying between 1 and 2. It is observed that the population growth is increasing with decreasing fractional order.
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400 α=0.5 α=0.7 α=0.9 α=1.2
350 300
A(t)
250 200 150 100 50 0
0
20
40
t
60
80
100
Fig. 3. Alcohol Concentration A(t) in stomach for different values of α.
300
300 α=β=1 α=β=1.2 α=β=1.4 α=β=1.6
250
250
200
200
B(t)
B(t)
150 150
100 100
50 α=β=1 α=β=0.9 α=β=0.7 α=β=0.5
50
0
0
20
40
60
80
t
100
120
140
160
180
0 −50
0
20
40
60
80
t
100
120
140
160
180
Fig. 4. (a) Alcohol Concentration in blood for β and α (0 < α, β ≤ 1); (b) Alcohol concentration in blood for β and α (1 ≤ α, β < 2).
3.3. Fractional Blood Alcohol Model We consider the fractional blood alcohol model as given by [9]:
CF α 0 Dt A(t)
= −k1 A(t), A(0) = A0 ,
CF β 0 Dt B(t)
= k1 A(t) − k2 B(t), 0 < α, β < 1, B(0) = 0,
(9)
where A(t) is the concentration of alcohol in stomach, B(t) is the concentration of alcohol in the blood, A0 is the initial alcohol consumed by the subject. k1 and k2 are real constants. By applying Laplace transform on both sides of (9) and using equation (1), we get
A(s) =
A0 αk1 +s(1+k1 (1−α)) ,
B(s) =
k1 F(s)(α+s(1−α)) αk2 +s(1+k2 (1−α)) .
(10)
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Then the inverse Laplace transform of equation (10) ia as follows −αk1 t A0 e 1+k1 (1−α) , 1 + k1 (1 − α) αt −k2 βt β k1 A0 β + k1 (β − α) 1+k−k1(1−α) − B(t) = [ e 1 e 1+k2 (1−β) ]. (k2 β − k1 α) + k1 k2 (β − α) 1 + k1 (1 − α) 1 + k2 (1 − β)
A(t) =
Figures 3 and 4 represent A(t) and B(t) for different values of fractional orders α, β with A0 = 373, k1 = 0.064 and k2 = 0.008. It is observed from Fig. 3, A(t) is decreasing exponentially with time for different fractional orders α. Figure 4(a) depicts decay of B(t) for various fractional orders α and β (0 < α, β ≤ 1). Figure 4(b) represents blood alcohol concentration for different fractional orders α and β (1 ≤ α, β < 2). Here alcohol level in blood decays faster for higher fractional order after a certain time. 4. Conclusion In the present paper, an attempt has been made to describe few fractional biological models with the help of Caputo-Fabrizio fractional derivative with non-singular kernel. Analytical solutions of these models are obtained for various fractional order and solution of corresponding classical equation are recovered as a particular case. Fractional derivatives provide us infinite choices for fractional order to choose the fractional differential equation that can model the problem more efficiently. We observed that growth rate is decreasing with decreasing fractional order derivative for the fractional logistic model. But an opposite behaviour is observed for Malthusian model. In the case of fractional blood alcohol model, we have discussed three different cases. The decay of alcohol concentration in stomach and blood is depicted for different fractional orders α. Thus it is observed that there is not any general relation between the order of the fractional differential equation and the rate of variation. Acknowledgements The second author acknowledges the University Grants Commission of India for providing financial support for the above research (Sr.No. 2061440951, reference no.22/06/14(i)EU-V). Also, the authors would like to thank the anonymous reviewers for their valuable suggestions and comments. References [1] Kilbas A.A., Srivastava H.M., Trujillo J.J. (2006) Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier. [2] Podlubny I. (1999) Fractional Differential equations, San Diego, Academic press. [3] Khalil R., Al Horani M., Yousef A., Sababheh M. (2014) “A new definition of fractional derivative.” Journal of Computational and Applied Mathematics 264: 65–70. [4] Varalta N., Gomes A.V., Camargo R.F. (2014) “A Prelude to the Fractional Calculus Applied to Tumor Dynamic.” Tema 15 (2): 211-221. [5] Yuzbasi S. (2015) “A collocation method for numerical solutions of fractional-order logistic population model.” International Journal of Biomathematics 9 (2): 1650031-45. [6] Aguilar J.F.G. et al. (2016) “Analytical and numerical solutions of electrical circuits described by fractional derivatives.” Applied Mathematical Modelling 40 : 9079-9094. [7] Diethelm K. (2013) “A fractional calculus based model for the simulation of an outbreak of dengue fever.” Nonlinear Dynamics 71: 613-619. [8] Kuroda L.K.B., Gomes A.V., Tavoni R. et al. (2017) “Unexpected behavior of Caputo fractional derivative.” Computational and Applied Mathematics 36: 1173-1183. [9] Almeida R., Nuno Bastos, M.Teresa T.Monteiro. (2016) “A Prelude to the Fractional Calculus Applied to Tumor Dynamic.” Mathematical Methods in the Applied Sciences 39(16): 4846-4855. [10] Ludwin C. (2011) “Blood Alcohol Content.” Undergraduate of Mathematical Modeling 3 (2): 1–8. [11] Caputo M., Fabrizio M. (2015) “A new Definition of Fractional Derivative without Singular Kernel.” Progress in Differentiation and Applications 1 (2): 73-85. [12] Losada J., Nieto J.J. (2015) “Properties of a New Fractional Derivative without Singular Kernel.” Progress in Differentiation and Applications 1(2): 87-92.
