Numerical comparison of methods for solving linear differential equations of fractional order

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1 Chaos, Solitons and Fractals 31 (27) Numerical comparison of methods for solving linear differential equations of fractional order Shaher Momani a, *, Zaid Odibat b a Department of Mathematics, Mutah University, P.O. Box 7, Al-Karak, Jordan b Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa Applied University, Salt, Jordan Accepted 2 October 25 Communicated by Prof. Ji-Huan He Abstract In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving linear differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper will present a numerical comparison between the two methods and a conventional method such as the fractional difference method for solving linear differential equations of fractional order. The numerical results demonstrates that the new methods are quite accurate and readily implemented. Ó 25 Elsevier Ltd. All rights reserved. 1. Introduction In this paper, we consider the numerical solution of linear fractional differential equation of the form d m u dt a da u bu ¼ f ðtþ; m dta t > ; m 1 < a 6 m; ð1:1þ subject to the initial conditions u ðjþ ðþ ¼c j ; j ¼ 1; ;...; m 1; ð1:2þ where c j, j =,1,...,m 1, are arbitrary constants and u(t) is assumed to be a causal function of time, i.e., vanishing for t <. The fractional derivatives is considered in the Caputo sense. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. We refer to Eq. (1.1) as to the composite fractional relaxation and to the composite fractional oscillation equation in the cases { < a 6 1,m = 1} and {1 < a 6 2,m = 2}, respectively. * Corresponding author. Tel.: ; fax: addresses: shahermm@yahoo.com (S. Momani), odibat@bau.edu.jo (Z. Odibat) /$ - see front matter Ó 25 Elsevier Ltd. All rights reserved. doi:1.116/j.chaos

2 S. Momani, Z. Odibat / Chaos, Solitons and Fractals 31 (27) The composite fractional relaxation equation corresponds to the Basset Problem [1]. It represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the gravity force. The composite fractional oscillation equation describes the motion of a rigid plate immersed in a Newtonian fluid [2]. For m = 2 and a = 1.5, Eq. (1.1) is the Bagley-Torvik equation and the initial value problem has been solved analytically using the decomposition method [3], and numerically [4] using the fractional difference method. Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics and engineering. Consequently, considerable attention has been given to the solutions of fractional ordinary differential equations, integral equations and fractional partial differential equations of physical interest. Recently, a large amount of literatures developed concerning the application of fractional differential equations in nonlinear dynamics [4 7]. Most fractional differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used. The variational iteration method [8 13] and the Adomian decomposition method [14,15] are relatively new approaches to provide an analytical approximation to linear and nonlinear problems, and they are particularly valuable as tools for scientists and applied mathematicians, because they provide immediate and visible symbolic terms of analytic solutions, as well as numerical approximate solutions to both linear and nonlinear differential equations without linearization or discretization. In this paper, the two methods are implemented to derive analytic approximate solutions to linear differential equations of fractional order, and then a numerical comparison with the fractional difference is demonstrated through some examples. In the literature, the decomposition method has been used to obtain approximate solutions of a large class of linear or nonlinear differential equations (see Ref. [16] and the references therein). Recently, the application of the method is extended for fractional differential equations [3,17,18]. The variational iteration method, which proposed by He [8 13], was successfully applied to autonomous ordinary and partial differential equations [19 23] and other fields. He [9] was the first to apply the variational iteration method to fractional differential equations. Recently Odibat and Momani [24] implemented the variational iteration method to solve nonlinear differential equations of fractional order. 2. Definitions For the concept of fractional derivative we will adopt CaputoÕs definition which is a modification of the Riemann Liouville definition and has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their integer order which is the case in most physical processes. Definition 2.1. A real function f(x), x >, is said to be in the space C l, l 2 R if there exists a real number p(>l), such that f(x)=x p f 1 (x), where f 1 (x) 2 C[,1), and it is said to be in the space C m l iff f(m) 2 C l, m 2 N. Definition 2.2. The Riemann Liouville fractional integral operator of order a P, of a function f 2 C l, l P 1, is defined as J a f ðxþ ¼ 1 Z x ðx tþ a 1 f ðtþdt; a > ; x > ; CðaÞ J f ðxþ ¼f ðxþ. Properties of the operator J a can be found in [25 27], we mention only the following: For f 2 C l, l P 1, a,b P and c > 1: 1. J a J b f(x)=j a+b f(x), 2. J a J b f(x)=j b J a f(x), 3. J a x c ¼ Cðcþ1Þ Cðaþcþ1Þ xaþc. The Riemann Liouville derivative has certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we shall introduce a modified fractional differential operator D a proposed by Caputo in his work on the theory of viscoelasticity [28]. Definition 2.3. The fractional derivative of f(x) in the Caputo sense is defined as Z D a f ðxþ ¼J m a D m 1 x f ðxþ ¼ ðx tþ m a 1 f ðmþ ðtþdt; ð2:1þ Cðm aþ

3 125 S. Momani, Z. Odibat / Chaos, Solitons and Fractals 31 (27) for m 1<a 6 m, m 2 IN, x >,f 2 C m 1. Also, we need here two of its basic properties. Lemma 2.1. If m 1 < a 6 m, m 2 N and f 2 C m l, l P 1, then and D a J a f ðxþ ¼f ðxþ; J a D a f ðxþ ¼f ðxþ Xm 1 k¼ f ðkþ ð þ Þ xk k! ; x >. 3. Fractional difference method In this section we introduce another definition of the fractional derivative. With regard to [4,27] we define the fractional derivative in the Grünwald Letnikov sense as D a X½t=hŠ a uðtþ ¼lim h ð 1Þ j a uðt jhþ; ð3:1þ h! j j¼ where [t] means the integer part of t and h is the step size. The definition of operator in the Grünwald Letnikov sense (3.1) is equivalent to the definition of operator in the Riemann Liouville sense [4]. Nevertheless the Grünwald Letnikov operator is more flexible and most straightforward in numerical calculations. Approximating the fractional derivative in (1.1) by (3.1), we obtain the following approximation for Eq. (1.1): X m j¼ ½tm=hŠ c m j u m j a X c a j u m j bu m ¼ f m ðm ¼ 1; 2; 3;...Þ; ð3:2þ j¼ where t m = mh, u m = u(t m ) and c a j are Grünwald Letnikov coefficients defined as c a j ¼ h a ð 1Þ j a ðj ¼ ; 1; 2;...Þ. ð3:3þ j Using the recurrence relationship [4] c a ¼ h a ; c a j ¼ 1 1 þ a c a j 1 ðj ¼ 1; 2; 3;...Þ; ð3:4þ j we can compute the coefficient in a simple way. For j = 1 we have c a 1 ¼ ah a. The next two equations describe the numerical solution algorithms for the composite fractional relaxation and composite fractional oscillation equations, respectively, u ¼ c ; u m ¼ u m 1 þ ah P m j¼1 ca j u m j þ hf m 1 hðb þ ah a ðm ¼ 1; 2; 3...Þ; ð3:5þ Þ u 1 u u ¼ c ; ¼ c 1 ; h u m ¼ 2u m 1 u m 2 þ ah 2P m j¼1 ca j u m j þ h 2 f m ðm ¼ 2; 3; 4...Þ. ð3:6þ 1 h 2 ðb þ ah a Þ These two algorithms are simple for computational performance for all values of a and h. For more details about the fractional difference method and its applications for solving fractional differential equations, we refer the reader to [4]. 4. Decomposition method as The decomposition method requires that the fractional differential Eq. (1.1) be expressed in terms of operator from

4 S. Momani, Z. Odibat / Chaos, Solitons and Fractals 31 (27) D m t u ada tu bu ¼ f ðtþ; ð4:1þ where D m t ¼ dm and the fractional differential operator D a dt m t is defined as in Eq. (2.1) denoted by D a t ¼ da dt. a Applying the operator J m, the inverse of the operator D m t, to both sides of Eq. (4.1) and using the initial conditions lead to uðtþ ¼/ 1 ðtþþa/ 2 ðtþþj m f ðtþþ½aj m a þ bj m ŠuðtÞ; ð4:2þ where / 1 ðtþ ¼ Xm 1 i¼ / 2 ðtþ ¼ Xm 1 i¼ c i t i i! ; c i t m aþi Cðm a þ i þ 1Þ. The AdomianÕs decomposition method [14,15] suggests the solution u(t) be decomposed by the infinite series of components uðtþ ¼ X1 u n ðtþ; ð4:3þ n¼ Substitution the decomposition series (4.3) into both sides of (4.2) gives X 1 n¼ u n ðtþ ¼/ 1 ðtþþa/ 2 ðtþþj m f ðtþþ½aj m a þ bj m Š X1 n¼ u n ðtþ. From this equation, the iterates are determined by the following recursive way: u ¼ / 1 ðtþþa/ 2 ðtþþj m f ðtþ; u 1 ¼ðaJ m a þ bj m Þu ðtþ ¼ðaJ m a þ bj m Þ½/ 1 ðtþþa/ 2 ðtþþj m f ðtþš; u 2 ¼ðaJ m a þ bj m Þu 1 ðtþ ¼ðaJ m a þ bj m Þ 2 ½/ 1 ðtþþa/ 2 ðtþþj m f ðtþš;. u k ¼ðaJ m a þ bj m Þu k 1 ðtþ ¼ðaJ m a þ bj m Þ k ½/ 1 ðtþþa/ 2 ðtþþj m f ðtþš. The components of u(t) are then defined as uðtþ ¼ X1 ðaj m a þ bj m Þ k ½/ 1 ðtþþa/ 2 ðtþþj m f ðtþš. k¼ Expanding the operator in (4.5) using the binomial formula we obtain the solution of Eq. (1.1) in a series form uðtþ ¼ X1 X k k a j b k j J km ja ½/ 1 ðtþþa/ 2 ðtþþj m f ðtþš. ð4:6þ j k¼ j¼ ð4:4þ ð4:5þ 5. Variational iteration method The principles of the variational iteration method and its applicability for various kinds of differential equations are given in [8 13]. We consider the following fractional differential equation D m t u ada tu bu ¼ f ðtþ; ð5:1þ where D m t ¼ dm and the fractional differential operator D a dt m t is defined as in Eq. (2.1), subject to the initial conditions (1.2). The correction functional for Eq. (5.1) can be constructed as u nþ1 ðtþ ¼u n ðtþþ kðd m t u nðsþ ad a t ~u nðsþ b~u n ðsþ f ðsþþds; ð5:2þ

5 1252 S. Momani, Z. Odibat / Chaos, Solitons and Fractals 31 (27) where k is a general Lagrange multiplier [29], which can be identified optimally via the variational theory [29 35], here ~u n and D a t ~u n are considered as restricted variations. We begin with the initial approximation u ¼ c þ c 1 t þ c 2 t 2 þþc m 1 t m 1. ð5:3þ Making the above functional stationary, noticing that d~u n ¼, du nþ1 ðtþ ¼du n ðtþþd kðd m t u nðsþ f ðsþþds; ð5:4þ yields the following Lagrange multipliers: k ¼ 1; for m ¼ 1; k ¼ s t; for m ¼ 2. Therefore, for m = 1, we obtain the following iteration formula: u nþ1 ðtþ ¼u n ðtþ ðd 1 t u nðsþ ad a t u nðsþ bu n ðsþ f ðsþþds. For m = 2, we obtain the following iteration formula: u nþ1 ðtþ ¼u n ðtþþ ðs tþðd 2 t u nðsþ ad a t u nðsþ bu n ðsþ f ðsþþds. ð5:5þ ð5:6þ 6. Numerical experiments To give a clear overview of the methodology as a numerical tool, we consider two examples in this section. We apply the fractional difference method, Adomian decomposition method and the variational iteration method on these examples so that the comparisons are made numerically. For numerical approximate solution we truncate the series solution (4.6) up to 1 terms and we choose the step size h to be.1 for Eqs. (3.5) and (3.6). All the results are calculated by using the symbolic calculus software Mathematica. Example 6.1. Consider the following composite fractional relaxation equation: du dt a da u bu ¼ ; t > ; < a 6 1; ð6:1þ dta subject to the initial condition uðþ ¼1. ð6:2þ According to the formula (4.6), using the Adomian decomposition method, we obtain the following series solution: uðtþ ¼ X1 X k k a j b k j j Cðk ja þ 1Þ tk ja. ð6:3þ k¼ j¼ In view of (5.5), the iteration formula for Eq. (6.1) is given by u nþ1 ðtþ ¼u n ðtþ ðd 1 u n ðsþ ad a t u nðsþ bu n ðsþþds. By the above variational iteration formula (6.4), we can obtain the following results: u ðtþ ¼1; u 1 ðtþ ¼1 þ bt; u 2 ðtþ ¼1 þ bt þ 1 2 b2 t 2 ab þ Cð3 aþ t2 a ; u 3 ðtþ ¼1 þ bt þ 1 2 b2 t 2 ab þ Cð3 aþ t2 a þ b3 6 t3 þ 2ab2 a 2 b Cð4 aþ t3 a þ Cð4 2aÞ t3 2a ;. ð6:4þ

6 S. Momani, Z. Odibat / Chaos, Solitons and Fractals 31 (27) Table 1 Numerical values when a =.25,.5,.75 and a = b = 1 for Eq. (6.1) t a =.25 a =.5 a =.75 u FDM u ADM u VIM u FDM u ADM u VIM u FDM u ADM u VIM and so on, in the same manner the rest of components of the iteration formula (6.4) can be obtained using the Mathematica package. The nth approximation converges to the exact series solution (6.3) obtained using Adomian decomposition method. Table 1 shows the approximate solutions for Eq. (6.1) obtained for different values of a using the decomposition method, variational iteration method and fractional difference method. It is clear that the approximations obtained using the decomposition method and the variational iteration method are in high agreement with those obtained using the fractional difference method. Example 6.2. Consider the composite fractional oscillation equation d 2 u dt a da u bu ¼ 8; t > ; 1 < a 6 2; ð6:5þ 2 dta subject to the initial conditions uðþ ¼; u ðþ ¼. ð6:6þ According to the formula (4.6), using the Adomian decomposition method, we obtain the following series solution: uðtþ ¼ X1 X k k 8a j b k j j Cð2k ja þ 3Þ t2k jaþ2. ð6:7þ k¼ j¼ In view of (5.6), the iteration formula for Eq. (6.5) is given by u nþ1 ðtþ ¼u n ðtþþ ðs tþðd 2 u n ðsþ ad a t u nðsþ bu n ðsþ 8Þds. Consequently, we find the following approximations: u ðtþ ¼; u 1 ðtþ ¼4t 2 ; u 2 ðtþ ¼4t 2 þ 1 8a 3 bt4 þ Cð5 aþ t4 a ; u 3 ðtþ ¼4t 2 þ 1 8a 3 bt4 þ Cð5 aþ t4 a þ b2 9 t6 þ 16ab 8a 2 Cð7 aþ t6 a þ Cð7 2aÞ t6 2a ;. and so on, in the same manner the rest of components of the iteration formula (6.8) can be obtained using the Mathematica package. The nth approximation converges to the exact series solution (6.7) obtained using Adomian decomposition method. Table 2 shows the approximate solutions for Eq. (6.5) obtained for different values of a using the decomposition method, variational iteration method and fractional difference method. Clear conclusion can be drawn ð6:8þ

7 1254 S. Momani, Z. Odibat / Chaos, Solitons and Fractals 31 (27) Table 2 Numerical values when a =.25,.5,.75 and a = b = 1 for Eq. (6.5) t a =.5 a = 1. a = 1.5 u FDM u ADM u VIM u FDM u ADM u VIM u FDM u ADM u VIM from the numerical results in Tables 1 and 2 that the decomposition method and the variational iteration method provide highly accurate solution without spatial discretization of the problem. 7. Concluding remarks This present analysis exhibits the applicability of the variational iteration method and the Adomian decomposition method to solve linear differential equations of fractional order. It may be concluded that the two methods are very powerful and efficient in finding analytical as well as numerical solutions for wide classes of fractional differential equations. They provide more realistic series solutions that converge very rapidly in real physical problems. It is also worth noting that the advantage of the two methods is that they do not require linearization, discretization, or perturbation, and they do not need closure approximation, or smallness assumptions, or physically unrealistic white noise assumption in the nonlinear stochastic case [8 15]. References [1] Basset AB. On the descent of a sphere in a viscous liquid. Quart J Math 191;42: [2] Bagley RL, Torvik PJ. On the appearance of the fractional derivative in the behavior of real materials. J Appl Mech 1994;51: [3] Shawagfeh NT. Analytical approximate solutions for nonlinear fractional differential equations. Appl Math Comput 22;131: [4] Podlubny I. Fractional differential equations. New York: Academic Press; [5] Gao X, Yu J. Synchronization of two coupled fractional-order chaotic oscillators. Chaos, Solitons & Fractals 25;26(1): [6] Lu JG. Chaotic dynamics and synchronization of fractional-order ArneodoÕs systems. Chaos, Solitons & Fractals 25;26(4): [7] Lu JG, Chen G. A note on the fractional-order Chen system. Chaos, Solitons & Fractals 26;27(3): [8] He JH. Variational iteration method for delay differential equations. Commun Nonlinear Sci Numer Simulat 1997;2(4): [9] He JH. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Meth Appl Mech Eng 1998;167: [1] He JH. Approximate solution of non linear differential equations with convolution product nonlinearities. Comput Meth Appl Mech Eng 1998;167: [11] He JH. Variational iteration method a kind of non-linear analytical technique: some examples. Int J Nonlinear Mech 1999;34: [12] He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 2;114: [13] He JH, Wan YQ, Guo Q. An iteration formulation for normalized diode characteristics. Int J Circ Theory Appl 24;32(6): [14] Adomian G. A review of the decomposition method in applied mathematics. J Math Anal Appl 1988;135: [15] Adomian G. Solving Frontier problems of physics: the decomposition method. Boston: Kluwer Academic Publishers; [16] Shawagfeh N, Kaya D. Comparing numerical methods for the solutions of systems of ordinary differential equations. Appl Math Lett 24;17:323 8.

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