Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method

Abstract

The fractional Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, space-time fractional Korteweg-de Vries equation, regularized long-wave equation, Boussinesq equation, and Klein-Gordon equation are considered. As a result, abundant types of exact analytical solutions are obtained. These solutions include generalized trigonometric and hyperbolic functions solutions which may be useful for further understanding of the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The periodic and kink solutions are founded as special case.

1. Introduction

During recent years, fractional differential equations (FDEs) have attracted much attention due to their numerous applications in areas of physics, biology, and engineering [1–3]. Many important phenomena in non-Brownian motion, signal processing, systems identification, control problem, viscoelastic materials, polymers, and other areas of science are well described by fractional differential equation [4–7]. The most important advantage of using FDEs is their nonlocal property, which means that the next state of a system depends not only upon its current state but also upon all of its historical states [8, 9]. Recently, the fractional functional analysis has been investigated by many researchers [10, 11]. For example, the properties and theorems of Yang-Laplace transforms and Yang-Fourier transforms [12] and their applications to the fractional ordinary differential equations, fractional ordinary differential systems, and fractional partial differential equations have been discussed. Many powerful methods have been established and developed to obtain numerical and analytical solutions of FDEs, such as finite difference method [13], finite element method [14], Adomian decomposition method [15, 16], differential transform method [17], variational iteration method [18–20], homotopy perturbation method [21, 22], the fractional sub-equation method [23], and generalized fractional subequation method [24]. How to extend the existing methods to solve other FDEs is still an interesting and important research problem. Thanks to the efforts of many researchers, several FDEs have been investigated and solved, such as the impulsive fractional differential equations [25], space- and time-fractional advection-dispersion equation [26–28], fractional generalized Burgers’ fluid [29], and fractional heat- and wave-like equations [30], and so forth. The finding of a new mathematical algorithm to construct exact solutions of nonlinear FDEs is important and might have significant impact on future research. In this research paper, we introduce the fractional Riccati expansion method to construct many exact traveling wave solutions of nonlinear FDEs with the modified Riemann-Liouville derivative defined by Jumarie. We use the fractional Riccati expansion method for solving the space-time fractional Korteweg-de Vries (KdV) equation, space-time fractional regularized long-wave (RLW) equation, space-time fractional Boussinesq equation, and space-time fractional Klein-Gordon equation.

The structure of this paper is as follows: in Section 2, we introduce some basic definitions and mathematical preliminaries of the fractional calculus theory. Section 3 describes the fractional Riccati expansion method for solving nonlinear FDEs. In Section 4, we applied the proposed method to the space-time fractional KdV, RLW, Boussinesq, and Klein-Gordon equations. Finally, we give some conclusions and discussions.

2. Mathematical Preliminaries

Fractional calculus is one of the generalizations of ordinary calculus. Generally speaking, there are two kinds of fractional derivatives. One of them is nonlocal fractional derivative, that is, Caputo derivative and Riemann-Liouville derivative which have been used successfully in various fields of science and engineering. However, the Caputo derivative requires the function to be smooth and differentiable. Obviously, the nonlocal derivatives are not suitable for the investigation of the local behavior of fractional differentiable equations. The other one is the local fractional derivative, that is, Kolwankar-Gangal (K-G) derivative [31], Chen’s fractal derivative [32] and Cresson’s derivative [33]. One of the famous examples is the devi-stair curve, which can be described by a continuous but nowhere differentiable function. Recently, there is new development of continuous but nowhere differentiable functions [34]. This study is motivated by the need to propose a fractional Riccati expansion method to construct exact analytical solutions of nonlinear FDEs with the following modified Riemann-Liouville derivative defined by Jumarie [35]:𝐷^𝛼_𝑥𝑓(𝑥)=1Γ(1−𝛼)^𝑥₀(𝑥−𝜉)^−𝛼−1𝑓(𝜉)(01−𝑓)𝑑𝜉,𝛼<0,Γ()𝑑1−𝛼𝑑𝑥^𝑥₀(𝑥−𝜉)^−𝛼𝑓(𝜉)(0𝑓−𝑓)𝑑𝜉,0<𝛼<1,_{(𝛼−𝑛)}(𝑥)^(𝑛),𝑛≤𝛼<𝑛+1,𝑛≥1, (1)which has merits over the original one; for example, the 𝛼-order derivative of a constant is zero. Owing to these merits, Jumarie’s modified Riemann-Liouville derivative was successfully applied to the probability calculus [36], fractional Laplace problems [37], and fractional variational calculus [38]. Some useful formulas and results of Jumarie’s modified Riemann-Liouville derivative were summarized in [35], three of them are 𝐷^𝛼_𝑥𝑥^𝛾=Γ𝛾+1Γ𝑥𝛾+1−𝛼^𝛾−𝛼𝐷,𝛾>0,_𝛼𝑥𝑓(𝑥)𝑔(𝑥)=𝑔(𝑥)𝐷^𝛼_𝑥𝑓(𝑥)+𝑓(𝑥)𝐷^𝛼_𝑥𝑔(𝑥),𝐷^𝛼_𝑥𝑓𝑔(𝑥)=𝑓^′_𝑔𝑔(𝑥)𝐷^𝛼_𝑥𝑔(𝑥)=𝐷^𝛼_𝑔𝑓𝑔(𝑥)𝑔^′_𝑥^𝛼, (2) which will be used in the following sections.

3. Fractional Riccati Expansion Method

In this section, we outline the main steps of the fractional Riccati expansion method for solving nonlinear FDEs. For a given nonlinear FDE, say, in two variables 𝑥 and 𝑡𝑃𝑢,𝐷^𝛼_𝑡𝑢,𝐷^𝛼_𝑥𝑢,𝐷_𝑡^2𝛼𝑢,𝐷_𝑥^2𝛼𝑢,…=0, (3) where 𝐷^𝛼_𝑡𝑢 and 𝐷^𝛼_𝑥𝑢 are Jumarie’s modified Riemann-Liouville derivatives of 𝑢, 𝑢=𝑢(𝑥,𝑡) is an unknown function, and 𝑃 is a polynomial in 𝑢 and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.

Step 1. By using the traveling wave transformation: 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑥+𝜔𝑡, (4) where 𝜔 is a constant to be determined later, the nonlinear FDE (3) is reduced to the following nonlinear fractional ordinary differential equation (FODE) for 𝑢=𝑢(𝜉): 𝑃𝑢,𝜔^𝛼𝐷^𝛼_𝜉𝑢,𝐷^𝛼_𝜉𝑢,𝜔^2𝛼𝐷_𝜉^2𝛼𝑢,𝐷_𝜉^2𝛼𝑢,…=0. (5)

Step 2. We suppose that 𝑢(𝜉) can be expressed by a finite power series of 𝐹(𝜉) as follows: 𝑢(𝜉)=𝑎₀+^𝑛_𝑖=1𝑎_𝑖𝐹^𝑖,𝑎_𝑛≠0, (6) where 𝑎_𝑖 (𝑖=0,1,2,…,𝑛) are constants to be determined later, 𝑛 is a positive integer determined by balancing the linear term of the highest order with the nonlinear term in (5), and 𝐹=𝐹(𝜉) satisfies the following fractional Riccati equation: 𝐷^𝛼_𝜉𝐹=𝐴+𝐵𝐹²,0<𝛼≤1, (7) where 𝐴 and 𝐵 are constants. Using the Mittag-Leffler function in one parameter 𝐸_𝛼(𝑥)=∑^∞_𝑘=0(𝑥^𝑘/Γ(1+𝑘𝛼))(𝛼>0), we obtain the following solution of (7).

Case 1. If 𝐴=1 and 𝐵=1, then (7) has the solution 𝐹=tan(𝜉,𝛼).

Case 2. If 𝐴=−1 and 𝐵=−1, then (7) has the solution 𝐹=cot(𝜉,𝛼).

Case 3. If 𝐴=1 and 𝐵=−1, then (7) has the solutions 𝐹=tanh(𝜉,𝛼),𝐹=coth(𝜉,𝛼). (8)

Case 4. If 𝐴=1/2 and 𝐵=−1/2, then (7) has the solutions ()𝐹=tanh𝜉,𝛼()()().1±sech𝜉,𝛼,𝐹=coth𝜉,𝛼±csch𝜉,𝛼 (9)

Case 5. If 𝐴=1/2 and 𝐵=1/2, then (7) has the solutions ()𝐹=tan𝜉,𝛼()()(),()().1±sec𝜉,𝛼,𝐹=csc𝜉,𝛼−cot𝜉,𝛼𝐹=tan𝜉,𝛼±sec𝜉,𝛼 (10)

Case 6. If A=−1/2 and 𝐵=−1/2, then (7) has the solutions ()𝐹=cot𝜉,𝛼(),()(),()().1±csc𝜉,𝛼𝐹=sec𝜉,𝛼−tan𝜉,𝛼𝐹=cot𝜉,𝛼±csc𝜉,𝛼 (11)

Case 7. If 𝐴=1 and 𝐵=−4, then (7) has the solution 𝐹=(tanh(𝜉,𝛼))/(1+tanh²(𝜉,𝛼)).

Case 8. If 𝐴=1 and 𝐵=4, then (7) has the solution 𝐹=(tan(𝜉,𝛼))/(1−tan²(𝜉,𝛼)).

Case 9. If 𝐴=−1 and 𝐵=−4, then (7) has the solution 𝐹=(cot(𝜉,𝛼))/(1−cot²(𝜉,𝛼)), where the generalized hyperbolic and trigonometric functions are defined as ()=𝐸cosh𝜉,𝛼^𝛼𝜉^𝛼+𝐸_𝛼−𝜉^𝛼2,()=𝐸sinh𝜉,𝛼^𝛼𝜉^𝛼−𝐸_𝛼−𝜉^𝛼2,()=𝐸cos𝜉,𝛼^𝛼𝑖𝜉^𝛼+𝐸_𝛼−𝑖𝜉^𝛼2,()=𝐸sin𝜉,𝛼^𝛼𝑖𝜉^𝛼−𝐸_𝛼−𝑖𝜉^𝛼,()=()2𝑖tanh𝜉,𝛼sinh𝜉,𝛼()()=()cosh𝜉,𝛼,tan𝜉,𝛼sin𝜉,𝛼(),()=1cos𝜉,𝛼coth𝜉,𝛼()()=1tanh𝜉,𝛼,sech𝜉,𝛼(),()=1cosh𝜉,𝛼cot𝜉,𝛼()()=1tan𝜉,𝛼,sec𝜉,𝛼(),()=1cos𝜉,𝛼csch𝜉,𝛼()()=1sinh𝜉,𝛼,csc𝜉,𝛼().sin𝜉,𝛼 (12)

Step 3. After substituting the fractional Riccati expansion method (6) into the FODE (5) the left-hand side of (5) can be converted into a polynomial in 𝐹(𝜉). Setting each coefficient of the polynomial to zero yields system of algebraic equations for 𝑎₀,𝑎₁,…,𝑎_𝑛, and 𝜔.

Step 4. By solving the system obtained in Step 3, the constants 𝑎₀,𝑎₁,…,𝑎_𝑛, and 𝜔 can be expressed by the parameters 𝐴 and 𝐵. Depending on the chosen values of 𝐴 and𝐵 the function 𝐹(𝜉) possesses the traveling wave solutions as given above; then the fractional Riccati expansion method (6) has the traveling wave solution of the nonlinear FDE (3).

Remark 1. If we take 𝐴=𝜎 and 𝐵=1, this agrees with the results obtained by S. Zhang and H. Q. Zhang [23].

Remark 2. In [39], Professor He introduced the simple fractional complex transform to convert nonlinear FDEs into its nonlinear differential equations.

4. Application

4.1. Space-Time Fractional KdV Equation

The KdV equation is the earliest soliton equation that was firstly derived by Korteweg and de Vries to model the evolution of shallow water wave in 1895. The KdV-type equations have applications in shallow-water waves [40], optical solitons in the two cycle regime [41], density waves in traffic flow of two kinds of vehicles [42], short waves in nonlinear dispersive models [43], surface acoustic soliton in a system supporting long waves [44], quantum field theory, plasma physics, and solid-state physics. The space-time fractional KdV equation is 𝐷^𝛼_𝑡𝑢+𝜇𝑢𝐷^𝛼_𝑥𝑢+𝜏𝐷_𝑥^3𝛼𝑢=0,0<𝛼≤1, (13) where 𝜇 and 𝜏 are constants. In order to solve (13) by the fractional Riccati expansion method, we use the travelling wave transformation 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥+𝜔𝑡, where 𝜔 is the dimensionless velocity of the wave. Then, (13) is reduced to the following nonlinear FODE: 𝜔^𝛼𝐷^𝛼_𝜉𝑢+𝜇𝑢𝐷^𝛼_𝜉𝑢+𝜏𝐷_𝜉^3𝛼𝑢=0. (14) Balancing 𝐷_𝜉^3𝛼𝑢 with 𝑢𝐷^𝛼_𝜉𝑢 gives 𝑛=2. Therefore, the solution of (14) can be expressed as 𝑢=𝑎₀+𝑎₁𝐹+𝑎₂𝐹². (15) By substituting (15) into (14) using (7) and setting the coefficients of 𝐹^𝑖 (𝑖=0,1,2,3,4,5) to zero, we get 𝑎₀𝜔=−^𝛼+8𝜏𝐴𝐵𝜇,𝑎₁=0,𝑎₂=−12𝜏𝐵²𝜇. (16) The general formula for the traveling wave solution of the space-time fractional KdV equation (13) 𝑢()1𝑥,𝑡=−𝜇𝜔^𝛼+8𝜏𝐴𝐵+12𝜏𝐵²𝐹²(𝑥+𝜔𝑡). (17) By selecting the special value of the 𝐴, 𝐵, and the corresponding function 𝐹(𝜉), we get the following solutions of (13): 𝑢₁1=−𝜇𝜔^𝛼+8𝜏+12𝜏tan²(,𝑢𝑥+𝜔𝑡,𝛼)₂1=−𝜇𝜔^𝛼−8𝜏+12𝜏tanh²(𝑥+𝜔𝑡,𝛼). (18) The remaining solutions can be obtained in a similar manner. When 𝛼=1, we obtain the classical KdV equation 𝑢_𝑡+𝜇𝑢𝑢_𝑥+𝜏𝑢_𝑥𝑥𝑥=0, (19) as special case of (13). Solutions given in (18) reduced to the well-known periodic and kink solutions of the KdV equation 𝑢_1KdV1=−𝜇𝜔+8𝜏+12𝜏tan²(,𝑢𝑥+𝜔𝑡)_2KdV1=−𝜇𝜔−8𝜏+12𝜏tanh²(𝑥+𝜔𝑡). (20)

4.2. Space-Time Fractional Regularized Long-Wave Equation

The RLW equation that describes approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems has been proposed by Benjamin et al. in 1972. The RLW equation is considered as an alternative to the KdV equation, which is modeled to govern a large number of physical phenomena such as shallow waters and plasma waves [45]. The space-time fractional RLW equation has the form 𝐷^𝛼_𝑡𝑢+𝜈𝐷^𝛼_𝑥𝑢+𝜇𝑢𝐷^𝛼_𝑥𝑢−𝜏𝐷^𝛼_𝑡𝐷_𝑥^2𝛼𝑢=0,0<𝛼≤1, (21) where 𝜈, 𝜇, and 𝜏 are constants. By using the travelling wave transformation 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥+𝜔𝑡, (21) is reduced to the following nonlinear FODE: 𝜔^𝛼+𝜈𝐷^𝛼_𝜉𝑢+𝜇𝑢𝐷^𝛼_𝜉𝑢−𝜏𝜔^𝛼𝐷_𝜉^3𝛼𝑢=0. (22) Thus, the solution of (22) has the form 𝑢=𝑎₀+𝑎₁𝐹+𝑎₂𝐹². (23) By substituting (23) into (22) using (7) and setting the coefficients of 𝐹^𝑖 (𝑖=0,1,2,3,4,5) to zero, we have 𝑎₀𝜔=−^𝛼−8𝜏𝐴𝐵𝜔^𝛼+𝜈𝜇,𝑎₁=0,𝑎₂=12𝜏𝐵²𝜔^𝛼𝜇. (24) The general formula of the travelling wave solution of the space-time fractional RLW equation (21) is 𝑢()1𝑥,𝑡=−𝜇𝜔^𝛼−8𝜏𝐴𝐵𝜔^𝛼+𝜈−12𝜏𝐵²𝜔^𝛼𝐹²(𝑥+𝜔𝑡). (25) By selecting the special value of the 𝐴, 𝐵, and the corresponding function 𝐹(𝜉), we get the following solutions of (21): 𝑢₁1=−𝜇𝜔^𝛼−8𝜏𝜔^𝛼+𝜈−12𝜏𝜔^𝛼tan²(,𝑢𝑥+𝜔𝑡,𝛼)₂1=−𝜇𝜔^𝛼+8𝜏𝜔^𝛼+𝜈−12𝜏𝜔^𝛼tanh²(𝑥+𝜔𝑡,𝛼). (26) The remaining solutions can be obtained in a similar manner. As a special case, when 𝛼=1 (21) reduced to the classical RLW equation 𝑢_𝑡+𝜈𝑢_𝑥+𝜇𝑢𝑢_𝑥−𝜏𝑢_𝑥𝑥𝑡=0. (27) The solutions (26) take the form of the well-known periodic and kink solutions of the RLW equation 𝑢_1RLW1=−𝜇𝜔−8𝜏𝜔+𝜈−12𝜏𝜔tan²(,𝑢𝑥+𝜔𝑡)_2RLW1=−𝜇𝜔+8𝜏𝜔+𝜈−12𝜏𝜔tanh²(𝑥+𝜔𝑡). (28)

4.3. Space-Time Fractional Boussinesq Equation

The Boussinesq equation was first derived to describe the propagation of long waves in shallow water [46]. It also arises in many other applications of physical interest including one-dimensional nonlinear lattice waves, vibrations in a nonlinear string, and ion sound waves in a plasma [47, 48]. Space-time fractional Boussinesq equation has the form 𝐷_𝑡^2𝛼𝑢−𝜈𝐷_𝑥^2𝛼𝑢−𝜇𝐷_𝑥^2𝛼𝑢²−𝜏𝐷_𝑥^4𝛼𝑢=0,0<𝛼≤1, (29) where 𝜈, 𝜇, and 𝜏 are constants. By using the travelling wave transformation 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥+𝜔𝑡, (29) is reduced to the following nonlinear FODE: 𝜔^2𝛼−𝜈𝐷_𝜉^2𝛼𝑢−2𝜇𝐷^𝛼_𝜉𝑢²+𝑢𝐷_𝜉^2𝛼𝑢−𝜏𝐷_𝜉^4𝛼𝑢=0. (30) Thus, the solution of (30) has the form 𝑢=𝑎₀+𝑎₁𝐹+𝑎₂𝐹². (31) By substituting (31) into (30) using (7) and setting the coefficients of 𝐹^𝑖 (𝑖=0,1,…,6) to zero, we have 𝑎₀𝜔=−^2𝛼−8𝜏𝐴𝐵−𝜈2𝜇,𝑎₁=0,𝑎₂=−6𝜏𝐵²𝜇. (32) The general formula of the travelling wave solution of the fractional space-time Boussinesq equation (29) is 𝑢()1𝑥,𝑡=−𝜔2𝜇^2𝛼−8𝜏𝐴𝐵−𝜈+12𝜏𝐵²𝐹²(𝑥+𝜔𝑡). (33) By selecting the special value of the 𝐴, 𝐵, and the corresponding function 𝐹(𝜉), we get the following solutions of (29): 𝑢₁1=−𝜔2𝜇^2𝛼−8𝜏−𝜈+12𝜏tan²(,𝑢𝑥+𝜔𝑡,𝛼)₂1=−𝜔2𝜇^2𝛼+8𝜏−𝜈+12𝜏tanh²(𝑥+𝜔𝑡,𝛼). (34) The remaining solutions can be obtained in a similar manner. When 𝛼=1, we obtain the Boussinesq equation 𝑢_𝑡𝑡−𝜈𝑢_𝑥𝑥+𝜇𝑢²_𝑥𝑥−𝜏𝑢_{𝑥𝑥𝑥𝑥}=0. (35) The solutions (34) reduced to the famous periodic and kink solutions of the Boussinesq equation 𝑢_1Boussinesq1=−𝜔2𝜇²−8𝜏−𝜈+12𝜏tan²(,𝑢𝑥+𝜔𝑡)_2Boussinesq1=−𝜔2𝜇²+8𝜏−𝜈+12𝜏tanh²(𝑥+𝜔𝑡). (36)

4.4. Space-Time Fractional Klein-Gordon Equation

The nonlinear Klein-Gordon equation is a good physical equation in the sense that it appears in many fields of applications [49]. For example, in relativistic quantum mechanics, it describes the processes involving particles of spin zero. The nonlinear space-time fractional Klein-Gordon equation is 𝐷_𝑡^2𝛼𝑢−𝜈𝐷_𝑥^2𝛼𝑢+𝜇𝑢−𝜏𝑢³=0,0<𝛼≤1, (37) where 𝜈, 𝜇, and 𝜏 are constants. Let 𝑢(𝑥,𝑡)=𝑢(𝜉), 𝜉=𝑥+𝜔𝑡, (37) transforms to the reduced nonlinear FODE 𝜔^2𝛼−𝜈𝐷_𝜉^2𝛼𝑢+𝜇𝑢−𝜏𝑢³=0. (38) Balancing 𝐷_𝜉^2𝛼𝑢 with 𝑢³ yields 𝑛=1, so we may choose 𝑢=𝑎₀+𝑎₁𝐹. (39) By substituting (39) into (38) using (7) and setting the coefficients of 𝐹^𝑖 (𝑖=0,1,2,3) to zero, we have 𝑎²₁=−𝐵𝜇𝐴𝜏,𝑎₀=0,𝜔^2𝛼𝜇=𝜈−.2𝐴𝐵 (40) The general formula of the travelling wave solution of the fractional space-time Klein-Gordon equation (37) is 𝑢(𝑥,𝑡)=±−𝐵𝜇𝐹()𝐴𝜏𝑥+𝜔𝑡,𝜔^2𝛼𝜇=𝜈−.2𝐴𝐵 (41) By selecting the special value of the 𝐴, 𝐵, and the corresponding function 𝐹(𝜉), we get the following solutions of (37): 𝑢(𝑥,𝑡)=±−𝜇𝜏tan(𝑥+𝜔𝑡,𝛼),𝜔^2𝛼𝜇=𝜈−2𝑢(),𝜇<0.𝑥,𝑡=±𝜇𝜏tanh(𝑥+𝜔𝑡,𝛼),𝜔^2𝛼𝜇=𝜈−2,𝜇>0. (42) The remaining solutions can be obtained in a similar manner. When 𝛼=1 (37) reduced to the well-known Klein-Gordon equation 𝑢_𝑡𝑡−𝜈𝑢_𝑥𝑥+𝜇𝑢−𝜏𝑢³=0. (43) The solutions (42) take the form of periodic and kink solutions of the Klein-Gordon equation 𝑢_1KG=±−𝜇𝜏tan(𝑥+𝜔𝑡),𝜔²𝜇=𝜈−2𝑢,𝜇<0,_2KG=±𝜇𝜏tanh(𝑥+𝜔𝑡),𝜔²𝜇=𝜈−2,𝜇>0. (44)

5. Conclusions and Discussions

We have proposed the fractional Riccati expansion method to solve nonlinear FDEs. The space-time fractional KdV equation, RLW equation, Boussinesq equation, and Klein-Gordon equation are selected to test the effectiveness of the proposed method. As a result, some exact analytical solutions are obtained including the generalized hyperbolic function and generalized trigonometric function solutions. To the best of our knowledge, some of the solutions obtained in this research paper have not been reported in the literature. The fractional Riccati expansion method is more effective and simpler than other methods, and a number of solutions can be obtained simultaneously. Mathematical packages can be used to perform more complicated and tedious algebraic calculations. The fractional Riccati expansion method can be applied to other nonlinear FDEs. How to extend other methods used for solving differential equations, such as F-expansion method, Fan sub-equation method, auxiliary sub-equation method, and the projective Riccati equation method, to handle FDEs is worthy of study. This is our task in the future. We hope that the present solutions may be useful in further numerical analysis and may help one to explain some physical phenomena. This paper is merely an initial work; more applications to the other nonlinear physical systems could be conducted and deserve further investigation.

Acknowledgment

The authors would like to express their sincere thanks to the referees for their valuable suggestions and comments.