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Applied Mathematics and Computation 200 (2008) 87–95 www.elsevier.com/locate/amc
Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives Yong Chen
a,b,c,*
, Hong-Li An
b,c
a
b c
Institute of Theoretical Computing, East China Normal University, Shanghai 200062, China Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo 315211, China Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080, China
Abstract In this paper, by introducing the fractional derivative in the sense of Caputo, the Adomian decomposition method is directly extended to study the coupled Burgers equations with time- and space-fractional derivatives. As a result, the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The figures show the effectiveness and good accuracy of the proposed method. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Adomian decomposition method; Fractional coupled nonlinear equations; Fractional calculus; Numerical solution
1. Introduction Since Adomian firstly proposed the decomposition method [1] at the beginning of 1980s, the algorithm has been widely and effectively used for solving the analytic solutions of physically significant equations arranging from linear to nonlinear, from ordinary differential to partial differential, from integer to fractional, etc., [1–8]. With this method, we don’t need to take any special technique and can easily obtain the realistic solution in the form of a rapidly convergent infinite series with each term computed conveniently. As we all know, for the nonlinear equations of integer order, there exist many methods used to derive the explicit solutions [1,9–15]. However, for the fractional differential equations, there are only limited approaches, such as Laplace transform method [16], the Fourier transform method [17], the iteration method [18] and the operational method [19]. In recent ten years, the fractional differential equations have been attracted great attention and widely been used in the areas of physics and engineering [20]. Particularly in some interdisciplinary fields, the fractional derivatives are considered to be a very powerful and useful tool [16–18,20,21]. With the help of fractional derivatives, phenomena in electromagnetics, acoustics electrochem-
*
Corresponding author. Address: Institute of Theoretical Computing, East China Normal University, Shanghai 200062, China. E-mail address: [email protected] (Y. Chen).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.10.050
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Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
istry and material science can be elegantly described [16–18,20,21]. With the help of fractional derivatives, the nonlinear oscillation of earthquake can be well modelled [21]; with fractional derivatives, the fluid dynamic traffic model can eliminate the deficiency arising from the assumption of continuum traffic flow [21]. The study to coupled Burgers equations is very significant for that the system is a simple model of sedimentation or evolution of scaled volume concentrations of two kinds of particles in fluid suspensions or colloids, under the effect of gravity [22]. It has been studied by many authors by different methods [23–25]. Especially recently, Dehghan et al. have obtained a good numerical results by using Adomian–Pade technique [25]. However, as we know, the study for the coupled Burgers equations with time- and space-fractional derivatives of this form oa1 u o2 u oa2 u oðuvÞ ; ¼ þ 2u ota1 ox2 oxa2 ox
ð1Þ
ob1 v o2 v ob2 v oðuvÞ ; ¼ þ 2v otb1 ox2 oxb2 ox
by the Adomian method (ADM) has not been investigated. Here ai and bi (i = 1, 2) are the parameters standing for the order of the fractional time and space derivatives, respectively, and they satisfy 0 < ai, bi 6 1 (i = 1, 2) and t > 0. In fact, different response systems can be obtained when at less one of the parameters varies. When ai = bi = 1, the fractional equations reduce to the classical coupled Burgers equation. We introduce Caputo fractional derivative and extend the ADM to derive explicit and numerical solutions of the coupled Burgers equations with time- and space-fractional derivatives. The solutions obtained by us are calculated in the form of convergent series with easily computable components. The paper is organized as follows. In Section 2, some necessary details on the fractional calculus are provided. In Section 3, the coupled Burgers equations with time- and space- fractional derivatives are studied with the ADM and figures are used to show the efficiency as well as the accuracy of the approximate results achieved. Finally, conclusions are followed. 2. Description of the fractional calculus There are several mathematical definitions about fractional derivative [16,18]. Here, we adopt the two usually used definitions: the Caputo and its reverse operator Riemann–Liouville. That is because Caputo fractional derivative allows traditional initial condition assumption and boundary conditions. More details one can consults Ref. [16]. In the following, we will give the necessary notation and basic definition. Definition 1. The Riemann–Liouville fractional integral operator of order a P 0, for a function f 2 Cl (l P 1) is defined as J a f ðxÞ ¼
1 CðaÞ
Z
x
ðx tÞ
a1
f ðtÞdt;
a > 0; x > 0;
0
ð2Þ
J 0 f ðxÞ ¼ f ðxÞ: For the convenience of establishing the results for the fractional coupled Burgers equations, we give two basic properties J a J b f ðxÞ ¼ J aþb f ðxÞ; J a J b f ðxÞ ¼ J b J a f ðxÞ:
ð3Þ
For expression (2), when f(x) = xb we get another expression that will be used later J a xb ¼
Cðb þ 1Þ aþb x : Cða þ b þ 1Þ
ð4Þ
Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
Definition 2. The fractional derivative of f 2 C n1 in the Caputo sense is defined as Z t 1 na1 ðnÞ Da f ðtÞ ¼ ðt sÞ f ðsÞds; ðn 1 < ReðaÞ 6 n; n 2 N Þ: Cðn aÞ 0 According to Caputo’s derivative, we can easily obtain the following expression: Da K ¼ 0; K is a constant; ( 0; b 6 a 1; a b Dt ¼ Cðbþ1Þ ba t ; b > a 1: Cðbaþ1Þ
89
ð5Þ
ð6Þ
Details on Caputo’s derivative can be found in Ref. [16]. Here we just give the expressions used later: the linear relationship and the so-called Leibnitz rule, that’s 1 X a Da ðf ðtÞgðtÞÞ ¼ ðk Þf ðkÞ ðtÞDak f ðtÞ; ð7Þ k¼0 Da ðkf ðtÞ þ lgðtÞÞ ¼ kDa f ðtÞ þ lDa gðtÞ; where k, l are constants, f(t) is continuous in [a, t] and g(t) has n + 1 continuous derivatives in [a, t]. In addition, we also need the following two relations: 1 X xk x > 0; Da J a f ðxÞ ¼ f ðxÞ; J a Da f ðxÞ ¼ f ðxÞ f k ð0þ Þ ; k! k¼0 where f 2 C ml ; l P 1; a P 0 and b P 0. Remark. In this paper, we need to discuss the coupled Burgers equations with time- and space-fractional derivatives. When a 2 R+ we just copy (5), when a = n 2 N fractional derivative reduces to the commonly used derivative. That’s to say ( 1 Rt n ðt sÞna1 o uðx;sÞ ds; n 1 < a < n; oa uðx; tÞ CðnaÞ 0 osn a ¼ Dt uðx; tÞ ¼ ð8Þ n a o uðx;tÞ ot ; a ¼ n 2 N: n ot
The form of the space-fractional derivative is similar to the above and we just omit it here. 3. Applications of the ADM Consider the coupled Burgers equations with time- and space-fractional derivatives Eq. (1). For convenience, we just discuss the following case a1 = a2 = a and b1 = b2 = b. The other cases are similar. In order to solve numerical solutions for Eq. (1) by using ADM, we write it in the operator form Dat u ¼ L2x u þ 2uDax u Lx ðuvÞ;
0 < a 6 1;
Dbt v
0 < b 6 1;
¼ L2x v þ
2vDbx v
Lx ðuvÞ;
ð9Þ
n
where Lnx ¼ oxo n ; the operators Dat ; Dbt ; Dax and Dbx stand for the fractional derivative and are defined as in (8). Take the initial condition as uðx; 0Þ ¼ f ðxÞ;
vðx; 0Þ ¼ gðxÞ: a
b
ð10Þ a
b
Applying the operator J and J , the inverse of D and D , respectively, on corresponding sub-equation of Eq. (9), using the initial condition (10), yields uðx; tÞ ¼ f ðxÞ þ J a L2x u þ 2J a U1 ðuÞ J a Wðu; vÞ; vðx; tÞ ¼ gðxÞ þ J b L2x v þ 2J b U2 ðvÞ J b Wðu; vÞ;
ð11Þ
where U1 ðuÞ ¼ uDax u; U2 ðvÞ ¼ vDbx v and Wðu; vÞ ¼ Lx ðuvÞ. Following Adomian decomposition method [1], the solutions are represented as infinite series like
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Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
uðx; tÞ ¼
X
1un ðx; tÞ; vðx; tÞ ¼
n¼0
X
1vn ðx; tÞ:
ð12Þ
n¼0
The nonlinear operators U1(u), U2(v) and W(u, v) are decomposed in these forms X X X U1 ðuÞ ¼ 1An ; U2 ðvÞ ¼ 1Bn ; Wðu; vÞ ¼ 1C n ; n¼0
n¼0
ð13Þ
n¼0
where An, Bn and Cn are the so-called Adomian polynomials and have the form " !# " ! !!# 1 1 1 X X X 1 dn 1 dn k k k a An ¼ U1 k uk ¼ k uk k uk ; Dx n! dkn n! dkn k¼0 k¼0 k¼0 " !# k¼0 " ! !!# k¼0 1 1 1 n X X X 1 dn 1 d Bn ¼ U2 kk v k ¼ kk v k kk v k ; Dbx n n! dkn n! dk k¼0 k¼0 k¼0 k¼0 k¼0 " !# " ! !!# 1 1 1 1 X X X X 1 dn 1 dn k k k k Cn ¼ W k uk ; k vk ¼ Lx k uk k vk n! dkn n! dkn k¼0 k¼0 k¼0 k¼0 k¼0
ð14Þ : k¼0
In fact, with the aid of Maple, these Adomian polynomials (14) can be easily calculated. Here we give the expressions n n n X X X o ðuk vnk Þ: uk Dax unk ; Bn ¼ vk Dbx vnk ; C n ¼ ð15Þ An ¼ ox k¼0 k¼0 k¼0 Then substituting the decomposition series (12) and (13) into Eq. (11), yields the following recursive formulas: u0 ¼ f ðxÞ; unþ1 ¼ J a L2x un þ 2J a An J a C n ; b
b
b
v0 ¼ gðxÞ; vnþ1 ¼ J L2x vn þ 2J Bn J C n ;
n P 0; n P 0:
ð16Þ
In the following, according to the above steps, we will derive the numerical solutions for the coupled Burgers equations with time- and space-fractional derivatives in details. 3.1. Numerical solutions of the time-fractional coupled Burgers equations Consider the following form of time-fractional coupled Burgers equations: a Dt u ¼ L2x u þ 2uLx u Lx ðuvÞ ð0 < a 6 1Þ; Dbt v ¼ L2x v þ 2vLx v Lx ðuvÞ
ð0 < b 6 1Þ;
with the initial condition uðx; 0Þ ¼ f ðxÞ ¼ sin x; vðx; 0Þ ¼ gðxÞ ¼ sin x: The exact solutions of (17) for the special case a = b = 1 is uðx; tÞ ¼ et sin x; vðx; tÞ ¼ et sin x:
ð17Þ
ð18Þ
ð19Þ
In order to obtain the numerical solutions of Eq. (17), substituting the initial condition (18) and using the Adomian polynomials (15) into the expression (16), we can calculate the results. For simplify, we only give the first few terms of series: u0 ¼ f ðxÞ; v0 ¼ gðxÞ; u1 ¼ J a Lxx u0 þ 2J a A0 J a C 0 ¼ J a ½u0xx þ 2J a ½u0 u0x J a ½u0 v0x þ u0x v0
Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
¼ f1 ðxÞ
91
ta ; Cða þ 1Þ
v1 ¼ J b Lxx v0 þ 2J b B0 J b C 0 ¼ J b ½v0xx þ 2J b ½v0 v0x J b ½u0 v0x þ u0x v0 tb ; Cðb þ 1Þ u2 ¼ J a Lxx u1 þ 2J a A1 J a C 1 ¼ g1 ðxÞ
¼ J a ½u1xx þ 2J a ½u0 u1x þ u0x u1 J a ½u0 v1x þ u1x v0 ¼ f2 ðxÞ
t2a taþb þ f3 ðxÞ ; Cð2a þ 1Þ Cða þ b þ 1Þ
v2 ¼ J b Lxx v1 þ 2J b B1 J b C 1 ¼ J b ½v1xx þ 2J b ½v0 v1x þ v0x v1 J b ½u0 v1x þ u1x v0 ¼ g2 ðxÞ
t2b taþb þ g3 ðxÞ ; Cð2b þ 1Þ Cða þ b þ 1Þ
where f ðxÞ ¼ sin x;
f 1 ðxÞ ¼ fxx þ 2ffx fgx fx g;
f 2 ðxÞ ¼ f1xx þ 2ff1x þ 2f 1 fx f1x g; f3 ðxÞ ¼ fx g1 ;
gðxÞ ¼ sin x;
g1 ðxÞ ¼ gxx þ 2ggx fgx fx g;
g2 ðxÞ ¼ g1xx þ 2gg1x þ 2g1 gx fx g1 ;
g3 ðxÞ ¼ f1x g:
Then we can have the numerical solutions of time-fractional coupled Burgers equation (17) in series form ta t2a taþb þ f2 ðxÞ þ f3 ðxÞ þ ; Cða þ 1Þ Cð2a þ 1Þ Cða þ b þ 1Þ tb t2b taþb vðx; tÞ ¼ g þ g1 ðxÞ þ g2 ðxÞ þ g3 ðxÞ þ : Cðb þ 1Þ Cð2b þ 1Þ Cða þ b þ 1Þ uðx; tÞ ¼ f þ f1 ðxÞ
ð20Þ ð21Þ
In order to verify the efficiency and accuracy of the proposed ADM for the time-fractional coupled Burgers equations, we draw figures for the numerical solutions with a ¼ 12 ; b ¼ 14 as well as the exact solutions (19) when a = b = 1. Fig. 1 stands for the numerical solutions of (20) and (21). Fig. 2 shows the exact solutions (19). From the figs, we can know the series solutions converge rapidly and we nearly cannot tell the difference between solutions obtained by different methods. That’s to say a good approximation is achieved by using N-term approximation of the ADM solutions. Remark. The accuracy of the numerical solutions obtained depends on how many terms we choose. The more terms we calculate, the smaller the error becomes. That’s to say: in order to reduce the overall errors, what we need is to add new terms to the decomposition series.
0.05
–1
0.04
u 0 1 10
0.03 0.02 0.01
5
x
0
–5
–10
0
t
1
v
0.05 0.04 0.03
0 –1 –10
0.02 –5
x
0
0.01 5
10
0
Fig. 1. Explicit numerical solutions for Eq. (17): (a) (20) u(x, t), (b) (21) v(x, t), with a ¼ 12 and b ¼ 14.
t
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Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
0.05
–1
0.04
u 0
0.03 0.02
1 10
t
0.01
5
x
0
–5
–10
1
0.05 0.04 0.03
v 0 –1 –10
–5
0
x
0
0.02 0.01 5
10
t
0
Fig. 2. Exact solutions (18) for Eq. (17): (a) u(x, t) = et sin x, (b) v(x, t) = et sin x, with a = 1 and b = 1.
3.2. Numerical solutions for the space-fractional coupled Burgers equations In this section, we will take the space-fractional coupled equations as another example to illustrate the efficiency of the method. As the main method is the same as the above, we will omit the heavy calculation and only give some necessary expressions. Considering the operator form of the space-fractional coupled Burgers equations Dt u ¼ L2x u þ 2uDax u Lx ðuvÞ; ð0 < a 6 1Þ; ð22Þ Dt v ¼ L2x v þ 2vDbx v Lx ðuvÞ; ð0 < b 6 1Þ: Assuming the initial condition as uðx; 0Þ ¼ f ðxÞ ¼ x2 ; vðx; 0Þ ¼ gðxÞ ¼ x3 :
ð23Þ
Remark. In fact, arbitrary function in the initial condition (23) can be chosen. In order to avoid the difficult of fractional differentiation computation, we just set it as the simple form xn. In order to estimate the numerical solutions of Eq. (22), substituting (12), (13) and the initial condition (23) into (16), we can get the Adomian solutions. Here we only give the first few terms of series solutions: u0 ¼ x 2 ;
v 0 ¼ x3 ;
u1 ¼ JLxx u0 þ 2JA0 JC 0 ¼ J ½u0xx þ 2J ½u0 Dax u0x J ½u0 v0x þ u0x v0 ¼ ð2 5x4 þ f1 x4a Þt; v1 ¼ JLxx v0 þ 2JB0 JC 0 ¼ J ½v0xx þ 2J ½v0 Dbx v0x J ½u0 v0x þ u0x v0 ¼ ð6x 5x4 þ g1 x6b Þt; u2 ¼ JLxx u1 þ 2JA1 JC 1 ¼ J ½u1xx þ 2J ½u0 Dax u1x þ Dax u0x u1 J ½u0 v1x þ u1x v0 t2 ðf2 x62a þ f3 x6a þ f4 x2a þ f5 x7b þ 20x6 þ 10x5 72x2 Þ; 2 v2 ¼ JLxx v1 þ 2JB1 JC 1 ¼
¼ J ½v1xx þ 2J ½v0 Dbx v1x þ Dbx v0x v1 J ½u0 v1x þ u1x v0 ¼
t2 ðg x92b þ g3 x7b þ g4 x4b þ g5 x6a þ 20x6 þ 10x5 72x2 Þ; 2 2
Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
u
93
–2e+08
v –1e+08
–1e+07 –5e+06
1
0 1 0.8
0.8 0.6
t
0.6
t 0.4 0.2 0
10
8
4
6
0.4
0
2
0.2
x
0
10
0
2
4
6
8
x
Fig. 3. Approximate numerical solutions for Eq. (22): (a) (24) u(x, t), (b) (25) v(x, t), with a ¼ 14 and b ¼ 13.
–1.5e+08
v –1e+08 –5e+07
1e+06 800000 600000 400000 200000 0
u
1 0.8 0 0.2 0.4
10
8
0.6
6
x
4
2
0.8 0
t
0.6
t 0.4 0.2 0
1
10
8
4
6
2
0
x
Fig. 4. Approximate numerical solutions u(x, t) and v(x, t) for Eq. (22) with a = b = 1. (a) u(x, t), (b) v(x, t).
where
4 4 2Cð5 aÞ ; f 2 ðxÞ ¼ þ f ðxÞ ¼ x ; f 1 ðxÞ ¼ f1 ; Cð3 aÞ Cð3 aÞ Cð5 2aÞ 20 240 8 ; f 4 ðxÞ ¼ ð4 aÞð3 aÞf1 þ ; f 5 ðxÞ ¼ 2g1 ; f 3 ðxÞ ¼ ða 4Þf1 Cð3 aÞ Cð5 aÞ Cð3 aÞ 12 12 2Cð7 bÞ ; g2 ðxÞ ¼ gðxÞ ¼ x3 ; g1 ðxÞ ¼ g; Cð4 bÞ Cð4 bÞ Cð7 2bÞ 1 60 240 12 72 ; g4 ðxÞ ¼ ð6 bÞð5 bÞg1 þ þ ; g3 ðxÞ ¼ 2g1 Cð4 bÞ Cð5 bÞ Cð2 bÞ Cð4 bÞ g5 ðxÞ ¼ ða 4Þf1 : 2
Then we obtain the numerical solutions of space-fractional equation (22) in series form t2 uðx; tÞ ¼ x2 þ ð2 5x4 þ f1 x4a Þt þ ðf2 x62a þ f3 x6a þ f4 x2a þ f5 x7b þ 20x6 þ 10x5 72x2 Þ þ ; 2 ð24Þ 2
t ðg x92b þ g3 x7b þ g4 x4b þ g5 x6a þ 20x6 þ 10x5 72x2 Þ þ : 2 2 ð25Þ Fig. 3a and b shows the numerical solutions (24) and (25) for the space-fractional equation (22) with a ¼ 14 and b ¼ 13. Fig. 4a and b is the figure for Eq. (22) with a = b = 1. From figures we can see that the Adomian solutions converge rapidly, which indicate that good results are achieved. vðx; tÞ ¼ x3 þ ð6x 5x4 þ g1 x6b Þt þ
4. Conclusion In this paper, combining the Caputo fractional derivative, the ADM has been successfully extended to derive the explicit numerical solutions for the time- and space-fractional coupled Burgers equations with initial condi-
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Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
tion. The above procedure shows that: (1) the ADM is an efficient and powerful method in solving a wide class of equations, in particular, coupled fractional order equations; (2) the method is straightforward without any restrictive assumptions and special techniques; (3) the continuity of the solution depends on the time- and space-fractional derivatives and the convergent speed is related with terms. Whether we can introduce other new feasible derivative operator or algorithms to solve differential equations and whether existing other techniques that can accelerate the convergent speed for the ADM solution, we hope these questions will be further studied. Acknowledgements The author would like to thank the referees very much for their careful reading of the manuscript and many valuable suggestions. The work is supported by the National Natural Science Foundation of China (Grant No. 10735030), Shanghai Leading Academic Discipline Project (No. B412), Zhejiang Provincial Natural Science Foundations of China (Grant No. Y604056) and Doctoral Foundation of Ningbo City (No. 2005A61030). References [1] G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic Publishers, Dordrecht, 1989; G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994; G. Adomian, R. Rach, On the solution of algebraic equations by the decomposition method, Math. Anal. Appl. 105 (1) (1985) 141– 166; G. Adomian, Explicit solutions of nonlinear partial differential equations, 88 (1997) 117–126. [2] A.M. Wazwaz, A new approach to the nonlinear advection problem: an application of the decomposition technique, Appl. Math. Comput. 72 (1995) 175–181; A.M. Wazwaz, A reliable modification of Adomian’s decomposition method, Appl. Math. Comput. 102 (1999) 77–86; A.M. Wazwaz, A new algothrim for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput. 111 (2000) 53– 69; A.M. Wazwaz, A computational approach to soliton solutions of the Kadomtsev–Petviashvili equation, 123 (2001) 205–217; A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. [3] D. Kaya, S.M. El-Sayed, A numerical implementation of the decomposition method for the Lienard equation, Appl. Math. Comput. 171 (2005) 1095–1103. [4] Y.Z. Yan, Numerical doubly periodic solution of the KdV equation with the initial condition via the decomposition method, Appl. Math. Comput. 168 (2005) 1065–1078. [5] Serdal Pamuk, An application for linear and nonlinear heat equations by Adomian’s decomposition method, Appl. Math. Comput. 163 (2005) 89–96. [6] E. Babolian, A. Davari, Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind, Appl. Math. Comput. 165 (2005) 223–227. [7] Q. Wang, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl. Math. Comput. 182 (2006) 1048–1055. [8] H.F. Gu, Z.B. Li, A modified Adomian method for system of nonlinear differential equations, Appl. Math. Comput. 187 (2007) 748– 755. [9] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scatting, Cambridge University Press, NewYork, 1991. [10] S.Y. Lou, J.Z. Lu, Special solutions from variable separation approach: Davey–Stewartson equation, J. Phys. A: Math. Gen. 29 (1996) 4209–4215. [11] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218. [12] Z.Y. Yan, New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A 292 (2001) 100–106. [13] Y. Chen, Q. Wang, A unified rational expansion method to construct a series of explicit exact solutions to nonlinear evolution equations, 177 (2006) 396–409. [14] B. Li, Exact soliton solutions for the higher-order nonlinear Schrodinger equation, Int. J. Mod. Phys. C 16 (2005) 1225–1237; B. Li, Y. Chen, H.Q. Zhang, Exact travelling wave solutions for a generalized Zakharov–Kuznetsov equation, 146 (2003) 653–666. [15] Y. Chen, Y.Z. Yan, Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota– Satsuma coupled KdV system, 177 (2006) 85–91. [16] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [17] S. Kemple, H. Beyer, Global and causal solutions of fractional differential equations, in: Transform Methods and Special Functions: Varna96, Proceedings of Second International Workshop (SCTP), Singapore, 1997.
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Applied Mathematics and Computation 200 (2008) 87–95 www.elsevier.com/locate/amc
Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives Yong Chen
a,b,c,*
, Hong-Li An
b,c
a
b c
Institute of Theoretical Computing, East China Normal University, Shanghai 200062, China Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo 315211, China Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080, China
Abstract In this paper, by introducing the fractional derivative in the sense of Caputo, the Adomian decomposition method is directly extended to study the coupled Burgers equations with time- and space-fractional derivatives. As a result, the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The figures show the effectiveness and good accuracy of the proposed method. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Adomian decomposition method; Fractional coupled nonlinear equations; Fractional calculus; Numerical solution
1. Introduction Since Adomian firstly proposed the decomposition method [1] at the beginning of 1980s, the algorithm has been widely and effectively used for solving the analytic solutions of physically significant equations arranging from linear to nonlinear, from ordinary differential to partial differential, from integer to fractional, etc., [1–8]. With this method, we don’t need to take any special technique and can easily obtain the realistic solution in the form of a rapidly convergent infinite series with each term computed conveniently. As we all know, for the nonlinear equations of integer order, there exist many methods used to derive the explicit solutions [1,9–15]. However, for the fractional differential equations, there are only limited approaches, such as Laplace transform method [16], the Fourier transform method [17], the iteration method [18] and the operational method [19]. In recent ten years, the fractional differential equations have been attracted great attention and widely been used in the areas of physics and engineering [20]. Particularly in some interdisciplinary fields, the fractional derivatives are considered to be a very powerful and useful tool [16–18,20,21]. With the help of fractional derivatives, phenomena in electromagnetics, acoustics electrochem-
*
Corresponding author. Address: Institute of Theoretical Computing, East China Normal University, Shanghai 200062, China. E-mail address: [email protected] (Y. Chen).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.10.050
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Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
istry and material science can be elegantly described [16–18,20,21]. With the help of fractional derivatives, the nonlinear oscillation of earthquake can be well modelled [21]; with fractional derivatives, the fluid dynamic traffic model can eliminate the deficiency arising from the assumption of continuum traffic flow [21]. The study to coupled Burgers equations is very significant for that the system is a simple model of sedimentation or evolution of scaled volume concentrations of two kinds of particles in fluid suspensions or colloids, under the effect of gravity [22]. It has been studied by many authors by different methods [23–25]. Especially recently, Dehghan et al. have obtained a good numerical results by using Adomian–Pade technique [25]. However, as we know, the study for the coupled Burgers equations with time- and space-fractional derivatives of this form oa1 u o2 u oa2 u oðuvÞ ; ¼ þ 2u ota1 ox2 oxa2 ox
ð1Þ
ob1 v o2 v ob2 v oðuvÞ ; ¼ þ 2v otb1 ox2 oxb2 ox
by the Adomian method (ADM) has not been investigated. Here ai and bi (i = 1, 2) are the parameters standing for the order of the fractional time and space derivatives, respectively, and they satisfy 0 < ai, bi 6 1 (i = 1, 2) and t > 0. In fact, different response systems can be obtained when at less one of the parameters varies. When ai = bi = 1, the fractional equations reduce to the classical coupled Burgers equation. We introduce Caputo fractional derivative and extend the ADM to derive explicit and numerical solutions of the coupled Burgers equations with time- and space-fractional derivatives. The solutions obtained by us are calculated in the form of convergent series with easily computable components. The paper is organized as follows. In Section 2, some necessary details on the fractional calculus are provided. In Section 3, the coupled Burgers equations with time- and space- fractional derivatives are studied with the ADM and figures are used to show the efficiency as well as the accuracy of the approximate results achieved. Finally, conclusions are followed. 2. Description of the fractional calculus There are several mathematical definitions about fractional derivative [16,18]. Here, we adopt the two usually used definitions: the Caputo and its reverse operator Riemann–Liouville. That is because Caputo fractional derivative allows traditional initial condition assumption and boundary conditions. More details one can consults Ref. [16]. In the following, we will give the necessary notation and basic definition. Definition 1. The Riemann–Liouville fractional integral operator of order a P 0, for a function f 2 Cl (l P 1) is defined as J a f ðxÞ ¼
1 CðaÞ
Z
x
ðx tÞ
a1
f ðtÞdt;
a > 0; x > 0;
0
ð2Þ
J 0 f ðxÞ ¼ f ðxÞ: For the convenience of establishing the results for the fractional coupled Burgers equations, we give two basic properties J a J b f ðxÞ ¼ J aþb f ðxÞ; J a J b f ðxÞ ¼ J b J a f ðxÞ:
ð3Þ
For expression (2), when f(x) = xb we get another expression that will be used later J a xb ¼
Cðb þ 1Þ aþb x : Cða þ b þ 1Þ
ð4Þ
Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
Definition 2. The fractional derivative of f 2 C n1 in the Caputo sense is defined as Z t 1 na1 ðnÞ Da f ðtÞ ¼ ðt sÞ f ðsÞds; ðn 1 < ReðaÞ 6 n; n 2 N Þ: Cðn aÞ 0 According to Caputo’s derivative, we can easily obtain the following expression: Da K ¼ 0; K is a constant; ( 0; b 6 a 1; a b Dt ¼ Cðbþ1Þ ba t ; b > a 1: Cðbaþ1Þ
89
ð5Þ
ð6Þ
Details on Caputo’s derivative can be found in Ref. [16]. Here we just give the expressions used later: the linear relationship and the so-called Leibnitz rule, that’s 1 X a Da ðf ðtÞgðtÞÞ ¼ ðk Þf ðkÞ ðtÞDak f ðtÞ; ð7Þ k¼0 Da ðkf ðtÞ þ lgðtÞÞ ¼ kDa f ðtÞ þ lDa gðtÞ; where k, l are constants, f(t) is continuous in [a, t] and g(t) has n + 1 continuous derivatives in [a, t]. In addition, we also need the following two relations: 1 X xk x > 0; Da J a f ðxÞ ¼ f ðxÞ; J a Da f ðxÞ ¼ f ðxÞ f k ð0þ Þ ; k! k¼0 where f 2 C ml ; l P 1; a P 0 and b P 0. Remark. In this paper, we need to discuss the coupled Burgers equations with time- and space-fractional derivatives. When a 2 R+ we just copy (5), when a = n 2 N fractional derivative reduces to the commonly used derivative. That’s to say ( 1 Rt n ðt sÞna1 o uðx;sÞ ds; n 1 < a < n; oa uðx; tÞ CðnaÞ 0 osn a ¼ Dt uðx; tÞ ¼ ð8Þ n a o uðx;tÞ ot ; a ¼ n 2 N: n ot
The form of the space-fractional derivative is similar to the above and we just omit it here. 3. Applications of the ADM Consider the coupled Burgers equations with time- and space-fractional derivatives Eq. (1). For convenience, we just discuss the following case a1 = a2 = a and b1 = b2 = b. The other cases are similar. In order to solve numerical solutions for Eq. (1) by using ADM, we write it in the operator form Dat u ¼ L2x u þ 2uDax u Lx ðuvÞ;
0 < a 6 1;
Dbt v
0 < b 6 1;
¼ L2x v þ
2vDbx v
Lx ðuvÞ;
ð9Þ
n
where Lnx ¼ oxo n ; the operators Dat ; Dbt ; Dax and Dbx stand for the fractional derivative and are defined as in (8). Take the initial condition as uðx; 0Þ ¼ f ðxÞ;
vðx; 0Þ ¼ gðxÞ: a
b
ð10Þ a
b
Applying the operator J and J , the inverse of D and D , respectively, on corresponding sub-equation of Eq. (9), using the initial condition (10), yields uðx; tÞ ¼ f ðxÞ þ J a L2x u þ 2J a U1 ðuÞ J a Wðu; vÞ; vðx; tÞ ¼ gðxÞ þ J b L2x v þ 2J b U2 ðvÞ J b Wðu; vÞ;
ð11Þ
where U1 ðuÞ ¼ uDax u; U2 ðvÞ ¼ vDbx v and Wðu; vÞ ¼ Lx ðuvÞ. Following Adomian decomposition method [1], the solutions are represented as infinite series like
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uðx; tÞ ¼
X
1un ðx; tÞ; vðx; tÞ ¼
n¼0
X
1vn ðx; tÞ:
ð12Þ
n¼0
The nonlinear operators U1(u), U2(v) and W(u, v) are decomposed in these forms X X X U1 ðuÞ ¼ 1An ; U2 ðvÞ ¼ 1Bn ; Wðu; vÞ ¼ 1C n ; n¼0
n¼0
ð13Þ
n¼0
where An, Bn and Cn are the so-called Adomian polynomials and have the form " !# " ! !!# 1 1 1 X X X 1 dn 1 dn k k k a An ¼ U1 k uk ¼ k uk k uk ; Dx n! dkn n! dkn k¼0 k¼0 k¼0 " !# k¼0 " ! !!# k¼0 1 1 1 n X X X 1 dn 1 d Bn ¼ U2 kk v k ¼ kk v k kk v k ; Dbx n n! dkn n! dk k¼0 k¼0 k¼0 k¼0 k¼0 " !# " ! !!# 1 1 1 1 X X X X 1 dn 1 dn k k k k Cn ¼ W k uk ; k vk ¼ Lx k uk k vk n! dkn n! dkn k¼0 k¼0 k¼0 k¼0 k¼0
ð14Þ : k¼0
In fact, with the aid of Maple, these Adomian polynomials (14) can be easily calculated. Here we give the expressions n n n X X X o ðuk vnk Þ: uk Dax unk ; Bn ¼ vk Dbx vnk ; C n ¼ ð15Þ An ¼ ox k¼0 k¼0 k¼0 Then substituting the decomposition series (12) and (13) into Eq. (11), yields the following recursive formulas: u0 ¼ f ðxÞ; unþ1 ¼ J a L2x un þ 2J a An J a C n ; b
b
b
v0 ¼ gðxÞ; vnþ1 ¼ J L2x vn þ 2J Bn J C n ;
n P 0; n P 0:
ð16Þ
In the following, according to the above steps, we will derive the numerical solutions for the coupled Burgers equations with time- and space-fractional derivatives in details. 3.1. Numerical solutions of the time-fractional coupled Burgers equations Consider the following form of time-fractional coupled Burgers equations: a Dt u ¼ L2x u þ 2uLx u Lx ðuvÞ ð0 < a 6 1Þ; Dbt v ¼ L2x v þ 2vLx v Lx ðuvÞ
ð0 < b 6 1Þ;
with the initial condition uðx; 0Þ ¼ f ðxÞ ¼ sin x; vðx; 0Þ ¼ gðxÞ ¼ sin x: The exact solutions of (17) for the special case a = b = 1 is uðx; tÞ ¼ et sin x; vðx; tÞ ¼ et sin x:
ð17Þ
ð18Þ
ð19Þ
In order to obtain the numerical solutions of Eq. (17), substituting the initial condition (18) and using the Adomian polynomials (15) into the expression (16), we can calculate the results. For simplify, we only give the first few terms of series: u0 ¼ f ðxÞ; v0 ¼ gðxÞ; u1 ¼ J a Lxx u0 þ 2J a A0 J a C 0 ¼ J a ½u0xx þ 2J a ½u0 u0x J a ½u0 v0x þ u0x v0
Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
¼ f1 ðxÞ
91
ta ; Cða þ 1Þ
v1 ¼ J b Lxx v0 þ 2J b B0 J b C 0 ¼ J b ½v0xx þ 2J b ½v0 v0x J b ½u0 v0x þ u0x v0 tb ; Cðb þ 1Þ u2 ¼ J a Lxx u1 þ 2J a A1 J a C 1 ¼ g1 ðxÞ
¼ J a ½u1xx þ 2J a ½u0 u1x þ u0x u1 J a ½u0 v1x þ u1x v0 ¼ f2 ðxÞ
t2a taþb þ f3 ðxÞ ; Cð2a þ 1Þ Cða þ b þ 1Þ
v2 ¼ J b Lxx v1 þ 2J b B1 J b C 1 ¼ J b ½v1xx þ 2J b ½v0 v1x þ v0x v1 J b ½u0 v1x þ u1x v0 ¼ g2 ðxÞ
t2b taþb þ g3 ðxÞ ; Cð2b þ 1Þ Cða þ b þ 1Þ
where f ðxÞ ¼ sin x;
f 1 ðxÞ ¼ fxx þ 2ffx fgx fx g;
f 2 ðxÞ ¼ f1xx þ 2ff1x þ 2f 1 fx f1x g; f3 ðxÞ ¼ fx g1 ;
gðxÞ ¼ sin x;
g1 ðxÞ ¼ gxx þ 2ggx fgx fx g;
g2 ðxÞ ¼ g1xx þ 2gg1x þ 2g1 gx fx g1 ;
g3 ðxÞ ¼ f1x g:
Then we can have the numerical solutions of time-fractional coupled Burgers equation (17) in series form ta t2a taþb þ f2 ðxÞ þ f3 ðxÞ þ ; Cða þ 1Þ Cð2a þ 1Þ Cða þ b þ 1Þ tb t2b taþb vðx; tÞ ¼ g þ g1 ðxÞ þ g2 ðxÞ þ g3 ðxÞ þ : Cðb þ 1Þ Cð2b þ 1Þ Cða þ b þ 1Þ uðx; tÞ ¼ f þ f1 ðxÞ
ð20Þ ð21Þ
In order to verify the efficiency and accuracy of the proposed ADM for the time-fractional coupled Burgers equations, we draw figures for the numerical solutions with a ¼ 12 ; b ¼ 14 as well as the exact solutions (19) when a = b = 1. Fig. 1 stands for the numerical solutions of (20) and (21). Fig. 2 shows the exact solutions (19). From the figs, we can know the series solutions converge rapidly and we nearly cannot tell the difference between solutions obtained by different methods. That’s to say a good approximation is achieved by using N-term approximation of the ADM solutions. Remark. The accuracy of the numerical solutions obtained depends on how many terms we choose. The more terms we calculate, the smaller the error becomes. That’s to say: in order to reduce the overall errors, what we need is to add new terms to the decomposition series.
0.05
–1
0.04
u 0 1 10
0.03 0.02 0.01
5
x
0
–5
–10
0
t
1
v
0.05 0.04 0.03
0 –1 –10
0.02 –5
x
0
0.01 5
10
0
Fig. 1. Explicit numerical solutions for Eq. (17): (a) (20) u(x, t), (b) (21) v(x, t), with a ¼ 12 and b ¼ 14.
t
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Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
0.05
–1
0.04
u 0
0.03 0.02
1 10
t
0.01
5
x
0
–5
–10
1
0.05 0.04 0.03
v 0 –1 –10
–5
0
x
0
0.02 0.01 5
10
t
0
Fig. 2. Exact solutions (18) for Eq. (17): (a) u(x, t) = et sin x, (b) v(x, t) = et sin x, with a = 1 and b = 1.
3.2. Numerical solutions for the space-fractional coupled Burgers equations In this section, we will take the space-fractional coupled equations as another example to illustrate the efficiency of the method. As the main method is the same as the above, we will omit the heavy calculation and only give some necessary expressions. Considering the operator form of the space-fractional coupled Burgers equations Dt u ¼ L2x u þ 2uDax u Lx ðuvÞ; ð0 < a 6 1Þ; ð22Þ Dt v ¼ L2x v þ 2vDbx v Lx ðuvÞ; ð0 < b 6 1Þ: Assuming the initial condition as uðx; 0Þ ¼ f ðxÞ ¼ x2 ; vðx; 0Þ ¼ gðxÞ ¼ x3 :
ð23Þ
Remark. In fact, arbitrary function in the initial condition (23) can be chosen. In order to avoid the difficult of fractional differentiation computation, we just set it as the simple form xn. In order to estimate the numerical solutions of Eq. (22), substituting (12), (13) and the initial condition (23) into (16), we can get the Adomian solutions. Here we only give the first few terms of series solutions: u0 ¼ x 2 ;
v 0 ¼ x3 ;
u1 ¼ JLxx u0 þ 2JA0 JC 0 ¼ J ½u0xx þ 2J ½u0 Dax u0x J ½u0 v0x þ u0x v0 ¼ ð2 5x4 þ f1 x4a Þt; v1 ¼ JLxx v0 þ 2JB0 JC 0 ¼ J ½v0xx þ 2J ½v0 Dbx v0x J ½u0 v0x þ u0x v0 ¼ ð6x 5x4 þ g1 x6b Þt; u2 ¼ JLxx u1 þ 2JA1 JC 1 ¼ J ½u1xx þ 2J ½u0 Dax u1x þ Dax u0x u1 J ½u0 v1x þ u1x v0 t2 ðf2 x62a þ f3 x6a þ f4 x2a þ f5 x7b þ 20x6 þ 10x5 72x2 Þ; 2 v2 ¼ JLxx v1 þ 2JB1 JC 1 ¼
¼ J ½v1xx þ 2J ½v0 Dbx v1x þ Dbx v0x v1 J ½u0 v1x þ u1x v0 ¼
t2 ðg x92b þ g3 x7b þ g4 x4b þ g5 x6a þ 20x6 þ 10x5 72x2 Þ; 2 2
Y. Chen, H.-L. An / Applied Mathematics and Computation 200 (2008) 87–95
u
93
–2e+08
v –1e+08
–1e+07 –5e+06
1
0 1 0.8
0.8 0.6
t
0.6
t 0.4 0.2 0
10
8
4
6
0.4
0
2
0.2
x
0
10
0
2
4
6
8
x
Fig. 3. Approximate numerical solutions for Eq. (22): (a) (24) u(x, t), (b) (25) v(x, t), with a ¼ 14 and b ¼ 13.
–1.5e+08
v –1e+08 –5e+07
1e+06 800000 600000 400000 200000 0
u
1 0.8 0 0.2 0.4
10
8
0.6
6
x
4
2
0.8 0
t
0.6
t 0.4 0.2 0
1
10
8
4
6
2
0
x
Fig. 4. Approximate numerical solutions u(x, t) and v(x, t) for Eq. (22) with a = b = 1. (a) u(x, t), (b) v(x, t).
where
4 4 2Cð5 aÞ ; f 2 ðxÞ ¼ þ f ðxÞ ¼ x ; f 1 ðxÞ ¼ f1 ; Cð3 aÞ Cð3 aÞ Cð5 2aÞ 20 240 8 ; f 4 ðxÞ ¼ ð4 aÞð3 aÞf1 þ ; f 5 ðxÞ ¼ 2g1 ; f 3 ðxÞ ¼ ða 4Þf1 Cð3 aÞ Cð5 aÞ Cð3 aÞ 12 12 2Cð7 bÞ ; g2 ðxÞ ¼ gðxÞ ¼ x3 ; g1 ðxÞ ¼ g; Cð4 bÞ Cð4 bÞ Cð7 2bÞ 1 60 240 12 72 ; g4 ðxÞ ¼ ð6 bÞð5 bÞg1 þ þ ; g3 ðxÞ ¼ 2g1 Cð4 bÞ Cð5 bÞ Cð2 bÞ Cð4 bÞ g5 ðxÞ ¼ ða 4Þf1 : 2
Then we obtain the numerical solutions of space-fractional equation (22) in series form t2 uðx; tÞ ¼ x2 þ ð2 5x4 þ f1 x4a Þt þ ðf2 x62a þ f3 x6a þ f4 x2a þ f5 x7b þ 20x6 þ 10x5 72x2 Þ þ ; 2 ð24Þ 2
t ðg x92b þ g3 x7b þ g4 x4b þ g5 x6a þ 20x6 þ 10x5 72x2 Þ þ : 2 2 ð25Þ Fig. 3a and b shows the numerical solutions (24) and (25) for the space-fractional equation (22) with a ¼ 14 and b ¼ 13. Fig. 4a and b is the figure for Eq. (22) with a = b = 1. From figures we can see that the Adomian solutions converge rapidly, which indicate that good results are achieved. vðx; tÞ ¼ x3 þ ð6x 5x4 þ g1 x6b Þt þ
4. Conclusion In this paper, combining the Caputo fractional derivative, the ADM has been successfully extended to derive the explicit numerical solutions for the time- and space-fractional coupled Burgers equations with initial condi-
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tion. The above procedure shows that: (1) the ADM is an efficient and powerful method in solving a wide class of equations, in particular, coupled fractional order equations; (2) the method is straightforward without any restrictive assumptions and special techniques; (3) the continuity of the solution depends on the time- and space-fractional derivatives and the convergent speed is related with terms. Whether we can introduce other new feasible derivative operator or algorithms to solve differential equations and whether existing other techniques that can accelerate the convergent speed for the ADM solution, we hope these questions will be further studied. Acknowledgements The author would like to thank the referees very much for their careful reading of the manuscript and many valuable suggestions. The work is supported by the National Natural Science Foundation of China (Grant No. 10735030), Shanghai Leading Academic Discipline Project (No. B412), Zhejiang Provincial Natural Science Foundations of China (Grant No. Y604056) and Doctoral Foundation of Ningbo City (No. 2005A61030). References [1] G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic Publishers, Dordrecht, 1989; G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994; G. Adomian, R. Rach, On the solution of algebraic equations by the decomposition method, Math. Anal. Appl. 105 (1) (1985) 141– 166; G. Adomian, Explicit solutions of nonlinear partial differential equations, 88 (1997) 117–126. [2] A.M. Wazwaz, A new approach to the nonlinear advection problem: an application of the decomposition technique, Appl. Math. Comput. 72 (1995) 175–181; A.M. Wazwaz, A reliable modification of Adomian’s decomposition method, Appl. Math. Comput. 102 (1999) 77–86; A.M. Wazwaz, A new algothrim for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput. 111 (2000) 53– 69; A.M. Wazwaz, A computational approach to soliton solutions of the Kadomtsev–Petviashvili equation, 123 (2001) 205–217; A.M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, The Netherlands, 2002. [3] D. Kaya, S.M. El-Sayed, A numerical implementation of the decomposition method for the Lienard equation, Appl. Math. Comput. 171 (2005) 1095–1103. [4] Y.Z. Yan, Numerical doubly periodic solution of the KdV equation with the initial condition via the decomposition method, Appl. Math. Comput. 168 (2005) 1065–1078. [5] Serdal Pamuk, An application for linear and nonlinear heat equations by Adomian’s decomposition method, Appl. Math. Comput. 163 (2005) 89–96. [6] E. Babolian, A. Davari, Numerical implementation of Adomian decomposition method for linear Volterra integral equations of the second kind, Appl. Math. Comput. 165 (2005) 223–227. [7] Q. Wang, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl. Math. Comput. 182 (2006) 1048–1055. [8] H.F. Gu, Z.B. Li, A modified Adomian method for system of nonlinear differential equations, Appl. Math. Comput. 187 (2007) 748– 755. [9] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scatting, Cambridge University Press, NewYork, 1991. [10] S.Y. Lou, J.Z. Lu, Special solutions from variable separation approach: Davey–Stewartson equation, J. Phys. A: Math. Gen. 29 (1996) 4209–4215. [11] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218. [12] Z.Y. Yan, New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A 292 (2001) 100–106. [13] Y. Chen, Q. Wang, A unified rational expansion method to construct a series of explicit exact solutions to nonlinear evolution equations, 177 (2006) 396–409. [14] B. Li, Exact soliton solutions for the higher-order nonlinear Schrodinger equation, Int. J. Mod. Phys. C 16 (2005) 1225–1237; B. Li, Y. Chen, H.Q. Zhang, Exact travelling wave solutions for a generalized Zakharov–Kuznetsov equation, 146 (2003) 653–666. [15] Y. Chen, Y.Z. Yan, Weierstrass semi-rational expansion method and new doubly periodic solutions of the generalized Hirota– Satsuma coupled KdV system, 177 (2006) 85–91. [16] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [17] S. Kemple, H. Beyer, Global and causal solutions of fractional differential equations, in: Transform Methods and Special Functions: Varna96, Proceedings of Second International Workshop (SCTP), Singapore, 1997.
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