Approximate Analytic Solutions of Time-Fractional Hirota-Satsuma Coupled KdV Equation and Coupled MKdV Equation

Abstract

By introducing the fractional derivative in the sense of Caputo and combining the pretreatment technique to deal with long nonlinear items, the generalized two-dimensional differential transform method is proposed for solving the time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation. The presented method is a numerical method based on the generalized Taylor series expansion which constructs an analytical solution in the form of a polynomial. The numerical results show that the generalized two-dimensional differential transform method is very effective for the fractional coupled equations.

1. Introduction

In the last past decade, the fractional differential equations have been widely used in various fields of physics and engineering. The analytical approximation of such problems has attracted great attention and became a considbased onerable interest in mathematical physics. Some powerful methods including the homotopy perturbation method [1], Adomian decomposition method [2, 3], variational iteration method [4, 5], homotopy analysis method [6, 7], fractional complex transform method [8], and generalized differential transform method [9] have been developed to obtain exact and approximate analytic solutions. These solution techniques are more clear and realistic methods for fractional differential equations, because they give the approximate solutions of the considered problems without any linearization or discretization.

The variational iteration method and the homotopy perturbation method were first proposed by Professor He in [10, 11], respectively. The idea of the variational iteration method is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. Recently, Wu and Lee proposed a fractional variational iteration method for fractional differential equation based on the modified Riemann CLiouville derivative, which is more effective to solve fractional differential equation [12]. This method has been developed by many authors, see [13–16] and the references cited therein. The homotopy perturbation method, which does not require a small parameter in an equation, has a significant advantage that it provides an analytical approximate solution to a wide range of nonlinear problems in applied sciences. Recently, the fractional complex transform is developed to convert the fractional differential equation to its differential partner and gave a geometrical explanation [8]. These methods are more effective for solving the linear and nonlinear fractional differential equations.

The differential transform method was used firstly by Zhou in 1986 to study electric circuits [17]. The differential transform is an iterative procedure based on the Taylor series expansion which constructs an analytic solution in the form of a polynomial. The method is well addressed in [18–23]. Recently, the generalized differential transform method is developed for obtaining approximate analytic solutions for some linear and nonlinear differential equations of fractional order [24, 25].

The aim of this paper is to directly extend the generalized two-dimensional differential transform method to obtain the approximate analytic solutions of a time-fractional Hirota-Satsuma coupled KdV equation, 𝐷^𝛼_𝑡1𝑢=2𝑢_𝑥𝑥𝑥−3𝑢𝑢_𝑥+3𝑣𝑤_𝑥+3𝑣_𝑥𝐷𝑤,_𝛼𝑡𝑣=−𝑣_𝑥𝑥𝑥+3𝑢𝑣_𝑥,𝐷^𝛼_𝑡𝑤=−𝑤_𝑥𝑥𝑥+3𝑢𝑤_𝑥,𝑡>0,0<𝛼≤1, (1) and a time-fractional coupled MKdV equation, 𝐷^𝛽_𝑡1𝑢=2𝑢_𝑥𝑥𝑥−3𝑢²𝑢_𝑥+32𝑣𝑣_𝑥+3𝑢𝑣_𝑥+3𝑢_𝑥𝑣−3𝜆𝑢_𝑥,𝐷^𝛽_𝑡𝑣=−𝑣_𝑥𝑥𝑥−3𝑣𝑣_𝑥−3𝑢_𝑥𝑣_𝑥+3𝑢²𝑣_𝑥+3𝜆𝑣_𝑥,𝑡>0,0<𝛽≤1, (2) where 𝜆 is a constant and 𝛼 and 𝛽 are parameters describing the order of the time-fractional derivatives of 𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡), and 𝑤(𝑥,𝑡), respectively. The fractional derivatives are considered in the Caputo sense. In the case of 𝛼=1 and 𝛽=1, (1) and (2) reduce to the classical Hirota-Satsuma coupled KdV equation and coupled MKdV equation [26], respectively.

The Caputo fractional derivative is considered here because it allows traditional initial and boundary conditions to be included in the formulation of the problem.

Definition 1 (see [27]). The fractional derivative of 𝑓(𝑥) in the Caputo sense is defined as 𝐷^𝜈𝑓(𝑥)=𝐽₀^𝑚−𝜈𝐷^𝑚𝑓(𝑥)=1Γ(𝑚−𝜈)^𝑥₀(𝑥−𝑡)^{𝑚−𝜈−1}𝑓^(𝑚)(𝑡)𝑑𝑡, (3) for 𝑚−1<𝜈<𝑚 and 𝑚∈𝑁,𝑥>0. Here, 𝐽^𝜇₀ is the Riemann-Liouville integral operator of order 𝜇>0, defined by 𝐽^𝜇₀𝑓(𝑥)=1Γ𝜇^𝑥₀(𝑥−𝑡)^𝜇−1𝑓(𝑡)𝑑𝑡,𝜇>0. (4)

Definition 2 (see [28]). For 𝑚 to be the smallest integer that exceeds 𝛼, the Caputo time-fractional derivative of order 𝛼>0 is defined as 𝐷^𝛼_𝑡𝑢()=𝜕𝑥,𝑡^𝛼𝑢𝜕𝑡^𝛼=1Γ(𝑚−𝛼)×∫^𝑡₀(𝑡−𝜉)^{𝑚−𝛼−1}×𝜕^𝑚𝑢(𝑥,𝜉)𝜕𝜉^𝑚𝜕𝑑𝜉,for𝑚−1<𝛼<𝑚,_𝑚𝑢(𝑥,𝑡)𝜕𝑡^𝑚,for𝛼=𝑚∈𝑁, (5) and the space-fractional derivatives of Caputo type can be defined analogously.

Lemma 3 (see [29]). The Caputo fractional derivative of the power function satisfies 𝐷^𝛼_𝑡𝑡^𝑝=Γ𝑝+1Γ𝑡𝑝−𝛼+1^𝑝−𝛼,𝑛−1<𝛼<𝑛,𝑝>𝑛−1,𝑝∈ℝ,0,𝑛−1<𝛼<𝑛,𝑝≤𝑛−1,𝑝∈ℕ. (6)

2. Generalized Two-Dimensional Differential Transform Method (GDTM)

Consider a function of two variables 𝑢(𝑥,𝑡) and suppose that it can be represented as a product of two single variable functions, that is, 𝑢(𝑥,𝑡)=𝑓(𝑥)𝑔(𝑡). Based on the properties of generalized two-dimensional differential transform, the function 𝑢(𝑥,𝑡) can be represented as 𝑢(𝑥,𝑡)=^∞_𝑘=0𝐹_𝛼(𝑘)𝑥−𝑥₀^∞𝑘𝛼_ℎ=0𝐺_𝛽(ℎ)𝑡−𝑡₀^ℎ𝛽=^∞∞_𝑘=0ℎ=0𝑈_𝛼,𝛽(𝑘,ℎ)𝑥−𝑥₀^𝑘𝛼𝑡−𝑡₀^ℎ𝛽, (7) where 0<𝛼,𝛽≤1, 𝑈_𝛼,𝛽(𝑘,ℎ)=𝐹_𝛼(𝑘)𝐺_𝛽(ℎ) is called the spectrum of 𝑢(𝑥,𝑡). The generalized two-dimensional differential transform of the function 𝑢(𝑥,𝑡) is given by 𝑈_𝛼,𝛽()=1𝑘,ℎΓ()Γ×𝐷𝛼𝑘+1𝛽ℎ+1_𝛼𝑥𝑘𝐷^𝛽_𝑡^ℎ𝑢(𝑥,𝑡)_{(𝑥0,𝑡0)}, (8) where (𝐷^𝛼_𝑥)^𝑘=𝐷^𝛼_𝑥𝐷^𝛼_𝑥⋯𝐷^𝛼_𝑥, 𝑘 is times. In the case of 𝛼=1 and 𝛽=1, the generalized two-dimensional differential transform (8) reduces to two-dimensional differential transform.

The fundamental theorems of the generalized two-dimensional differential transform are as follows.

Theorem 4. Suppose that 𝑈_𝛼,𝛽(𝑘,ℎ),𝑉_𝛼,𝛽(𝑘,ℎ), and 𝑊_𝛼,𝛽(𝑘,ℎ) are the differential transformations of the functions 𝑢(𝑥,𝑡),𝑣(𝑥,𝑡), and 𝑤(𝑥,𝑡), respectively:(a)if 𝑢(𝑥,𝑡)=𝑣(𝑥,𝑡)±𝑤(𝑥,𝑡), then 𝑈_𝛼,𝛽(𝑘,ℎ)=𝑉_𝛼,𝛽(𝑘,ℎ)±𝑊_𝛼,𝛽(𝑘,ℎ).(b)if 𝑢(𝑥,𝑡)=𝑎𝑣(𝑥,𝑡),𝑎∈ℝ, then 𝑈_𝛼,𝛽(𝑘,ℎ)=𝑎𝑉_𝛼,𝛽(𝑘,ℎ).(c)if 𝑢(𝑥,𝑡)=𝑣(𝑥,𝑡)𝑤(𝑥,𝑡), then 𝑈_𝛼,𝛽(𝑘,ℎ)=∑^𝑘_𝑟=0∑^ℎ_𝑠=0𝑉_𝛼,𝛽(𝑟,ℎ−𝑠)𝑊_𝛼,𝛽(𝑘−𝑟,𝑠).(d)if 𝑢(𝑥,𝑡)=(𝑥−𝑥₀)^𝑛𝛼(𝑡−𝑡₀)^𝑚𝛽, then 𝑈_𝛼,𝛽(𝑘,ℎ)=𝛿(𝑘−𝑛)𝛿(ℎ−𝑚).(e)if 𝑢(𝑥,𝑡)=𝐷^𝛼_𝑥𝑣(𝑥,𝑡),0<𝛼≤1, then 𝑈_𝛼,𝛽(𝑘,ℎ)=(Γ(𝛼(𝑘+1)+1)/Γ(𝛼𝑘+1))𝑉_𝛼,𝛽(𝑘+1,ℎ).

Theorem 5. If 𝑢(𝑥,𝑡)=𝑓(𝑥)𝑔(𝑡), function 𝑓(𝑥)=𝑥^𝜆ℎ(𝑥), where 𝜆>−1 and ℎ(𝑥) has the generalized Taylor series expansion ℎ(𝑥)=∑^∞_𝑛=0𝑎_𝑛(𝑥−𝑥₀)^𝛼𝑛, (i)𝛽<𝜆+1 and 𝛼 arbitrary, or(ii)𝛽≥𝜆+1, 𝛼 arbitrary, and 𝑎_𝑛=0for 𝑛=0,1,…,𝑚−1, where 𝑚−1<𝛽≤𝑚,then the generalized differential transform (8) becomes 𝑈_𝛼,𝛽()=1𝑘,ℎΓ()Γ𝐷𝛼𝑘+1𝛽ℎ+1^{𝑥𝛼𝑘0}𝐷^𝛽_𝑡0^ℎ𝑢(𝑥,𝑡)_{(𝑥0,𝑡0)}. (9) If 𝑢(𝑥,𝑡)=𝐷^𝛾_𝑥0𝑣(𝑥,𝑡),𝑚−1<𝛾≤𝑚, and 𝑣(𝑥,𝑡)=𝑓(𝑥)𝑔(𝑡), then 𝑈_𝛼,𝛽()=Γ𝑘,ℎ𝛼𝑘+𝛾+1Γ()𝑉𝛼𝑘+1^𝛼,𝛽𝛾𝑘+𝛼,ℎ. (10)

Some details of the aformentioned theorems can be found in [30].

3. Applications of GDTM

3.1. Fractional Hirota-Satsuma Coupled KdV Equation

Consider the following time-fractional Hirota-Satsuma coupled KdV equation: 𝐷^𝛼_𝑡1𝑢=2𝑢_𝑥𝑥𝑥−3𝑢𝑢_𝑥+3(𝑣𝑤)_𝑥,𝐷^𝛼_𝑡𝑣=−𝑣_𝑥𝑥𝑥+3𝑢𝑣_𝑥,𝐷^𝛼_𝑡𝑤=−𝑤_𝑥𝑥𝑥+3𝑢𝑤_𝑥,𝑡>0,0<𝛼≤1, (11) subject to the initial conditions 𝑢()=1𝑥,03𝛽−8𝛾²+4𝛾²tanh²,𝑣()=𝛾𝑥𝑥,0−43𝛾⁴𝑐₀−2𝛽𝛾²𝑐₂+4𝛾⁴𝑐₂3𝑐²₂+4𝛾²𝑐₂tanh²,𝑤()𝛾𝑥𝑥,0=𝑐₀+𝑐₂tanh²𝛾𝑥, (12) where 𝑐₀, 𝑐₂, 𝛽, and 𝛾 are arbitrary constants.

The exact solutions of (11) and (12), for the special case of 𝛼=1, given in [26], are 𝑢()=1𝑥,𝑡3𝛽−8𝛾²+4𝛾²tanh²𝛾,𝑣()=𝑥+𝛽𝑡𝑥,𝑡−43𝛾⁴𝑐₀−2𝛽𝛾²𝑐₂+4𝛾⁴𝑐₂3𝑐²₂+4𝛾²𝑐₂tanh²𝛾,𝑤()𝑥+𝛽𝑡𝑥,𝑡=𝑐₀+𝑐₂tanh²𝛾𝑥+𝛽𝑡. (13)

Suppose that the solutions 𝑢(𝑥,𝑡), 𝑣(𝑥,𝑡), and 𝑤(𝑥,𝑡) can be represented as the products of single-valued functions, respectively. Applying the generalized two-dimensional differential transform to both sides of (11) and using the related theorems, we have Γ(𝛼(ℎ+1)+1)Γ()𝑈𝛼ℎ+1^1,𝛼()=1𝑘,ℎ+12(𝑘+1)(𝑘+2)(𝑘+3)𝑈_1,𝛼(𝑘+3,ℎ)−3^𝑘ℎ_𝑟=0𝑠=0(𝑘+1−𝑟)𝑈_1,𝛼(𝑟,ℎ−𝑠)𝑈_1,𝛼()()𝑘+1−𝑟,𝑠+3𝑘+1^𝑘+1ℎ_𝑟=0𝑠=0𝑉_1,𝛼(𝑟,ℎ−𝑠)𝑊_1,𝛼(),Γ(𝛼())𝑘+1−𝑟,𝑠ℎ+1+1Γ()𝑉𝛼ℎ+1^1,𝛼()()𝑉𝑘,ℎ+1=−𝑘+1)(𝑘+2)(𝑘+3_1,𝛼(𝑘+3,ℎ)−3^𝑘ℎ_𝑟=0𝑠=0(𝑘+1−𝑟)𝑈_1,𝛼(𝑟,ℎ−𝑠)𝑉_1,𝛼(),Γ(𝛼())𝑘+1−𝑟,𝑠ℎ+1+1Γ()𝑊𝛼ℎ+1^1,𝛼()()𝑊𝑘,ℎ+1=−𝑘+1)(𝑘+2)(𝑘+3_1,𝛼(𝑘+3,ℎ)−3^𝑘ℎ_𝑟=0𝑠=0(𝑘+1−𝑟)𝑈_1,𝛼(𝑟,ℎ−𝑠)𝑊_1,𝛼(𝑘+1−𝑟,𝑠). (14)

The generalized two-dimensional differential transforms of the initial conditions can be obtained as follows: 𝑈_1,𝛼(𝑘,0)=𝑉_1,𝛼(𝑘,0)=𝑊_1,𝛼()𝑈𝑘,0=0if𝑘=1,3,5,…,_1,𝛼()=10,03𝛽−8𝛾²,𝑈_1,𝛼(2,0)=4𝛾⁴,𝑈_1,𝛼()84,0=−3𝛾⁶,𝑈_1,𝛼(6,0)=68𝛾45⁸𝑉,…,_1,𝛼()40,0=−3𝛾⁴𝑐₀−2𝛽𝛾²𝑐₂+4𝛾⁴𝑐₂3𝑐²₂,𝑉_1,𝛼(2,0)=4𝛾⁴𝑐₂,𝑉_1,𝛼(4,0)=−8𝛾⁶3𝑐₂,𝑉_1,𝛼(6,0)=68𝛾⁸45𝑐₂𝑊,…,_1,𝛼(0,0)=𝑐₀,𝑊_1,𝛼(2,0)=𝑐₂𝛾²,𝑊_1,𝛼(4,0)=−2𝑐₂3𝛾⁴,𝑊_1,𝛼(6,0)=17𝑐₂𝛾45⁶,…. (15)

Utilizing the recurrence relations (14) and the transformed initial conditions, we can obtain all the 𝑈_1,𝛼(𝑘,ℎ), 𝑉_1,𝛼(𝑘,ℎ), and 𝑊_1,𝛼(𝑘,ℎ) with the help of Mathematica. Moreover, substituting all 𝑈_1,𝛼(𝑘,ℎ) into (7), we obtain the series form solution 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+⋯, (16) where 𝑢₀()=1𝑥,𝑡3𝛽−8𝛾²+4𝛾⁴𝑥²−8𝛾⁶3𝑥⁴+68𝛾⁸𝑥45⁶−248𝛾¹⁰𝑥315⁸𝑢+⋯,₁()=𝑥,𝑡8𝛾⁴𝑐₂𝛽−3𝑐₀−1+𝛾²𝑐₂𝑥+16𝛾⁶𝑐₂9−2𝛽−9𝛾²+6𝑐₀−1+𝛾²3𝑐₂𝑥³+8𝛾⁸−51𝑐₀−1+𝛾²+𝑐₂17𝛽+180−1+𝛾²15𝑐₂𝑥⁵𝑡+⋯^𝛼Γ(),𝑢1+𝛼₂()=𝑥,𝑡8𝛾⁴−12𝑐₀𝛾²−1+𝛾²+𝑐₂𝛽²−18𝛾²−1+𝛾²𝑐₂+16𝛾⁶51𝑐₀𝛾²−1+𝛾²−2𝑐₂𝛽²−54𝛾²−1+𝛾²𝑐₂𝑥²−8𝛾⁸744𝑐₀𝛾²−1+𝛾²+𝑐₂−17𝛽²+2016𝛾²−1+𝛾²3𝑐₂𝑥⁴𝑡+⋯^2𝛼Γ(1+2𝛼),…. (17)

The closed forms of 𝑢₀(𝑥,𝑡), 𝑢₁(𝑥,𝑡), and 𝑢₂(𝑥,𝑡) are 𝑢₀()=1𝑥,𝑡3𝛽−8𝛾²+4𝛾²tanh²,𝑢𝛾𝑥₁()=𝛾𝑥,𝑡³sec²𝛾𝑥𝑐₂×24𝑐₀+8𝑐₂𝛽−24𝑐₀𝛾²+48𝑐₂𝛾²−1tanh²×𝑡𝛾𝑥tanh𝛾𝑥^𝛼Γ(),𝑢1+𝛼₂()=𝛾𝑥,𝑡⁶1−𝛾²𝑐₂×96𝑐₀+2𝑐₂sec²+𝛾𝑥8𝑐₂𝛽²𝛾²1−𝛾²2−cosh2𝛾𝑥sec⁴𝛾𝑥+615𝑐₀+94𝑐₂−88𝑐₂−cosh2𝛾𝑥15𝑐₀+14𝑐₂cosh4𝛾𝑥sec⁸×𝑡𝛾𝑥_2𝛼Γ().1+2𝛼 (18)

Similarly, substituting all 𝑉_1,𝛼(𝑘,ℎ) and all 𝑊_1,𝛼(𝑘,ℎ) into (7), respectively, we obtain the series form solutions 𝑣(𝑥,𝑡)=𝑣₀(𝑥,𝑡)+𝑣₁(𝑥,𝑡)+𝑣₂()𝑤()𝑥,𝑡+⋯,𝑥,𝑡=𝑤₀(𝑥,𝑡)+𝑤₁(𝑥,𝑡)+𝑤₂(𝑥,𝑡)+⋯. (19)

The closed forms of 𝑣₀(𝑥,𝑡), 𝑣₁(𝑥,𝑡) and 𝑣₂(𝑥,𝑡) are 𝑣₀(𝑥,𝑡)=−43𝛾⁴𝑐₀−2𝛽𝛾²𝑐₂+4𝛾⁴𝑐₂3𝑐²₂+4𝛾²𝑐₂tanh²,𝑣𝛾𝑥₁(𝑥,𝑡)=8𝑐₀𝛽𝛾³sec²𝛾𝑥tanh𝛾𝑥𝑐₂𝑡^𝛼Γ(),𝑣1+𝛼₂()=𝑥,𝑡8𝛽²𝛾⁴3sec²𝛾𝑥−2𝑐₂cosh²+𝛾𝑥576𝛾⁶1−𝛾²𝑐₀+2𝑐₂tanh²𝛾𝑥tanh²𝛾𝑥𝑐²₂cosh⁴×𝑡𝛾𝑥_2𝛼Γ().1+2𝛼 (20)

The closed forms of 𝑤₀(𝑥,𝑡), 𝑤₁(𝑥,𝑡), and 𝑤₂(𝑥,𝑡) are 𝑤₀(𝑥,𝑡)=𝑐₀+𝑐₂tanh²,𝑤𝛾𝑥₁(𝑥,𝑡)=2𝑐₂𝛽𝛾sec²𝛾𝑥tanh𝛾𝑥𝑐₂𝑡^𝛼Γ(),𝑤1+𝛼₂(𝑥,𝑡)=2𝑐₂𝛽²𝛾²sec²𝛾𝑥3sec²𝑐𝛾𝑥−2+144₀+2𝑐₂𝛾⁴1−𝛾²×sec⁴𝛾𝑥tanh²𝛾𝑥−288𝑐₂𝛾⁴1−𝛾²×sec⁶𝛾𝑥tanh²𝑡𝛾𝑥^2𝛼Γ().1+2𝛼 (21)

The approximate solutions of (11) in finite series forms are given by 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(),𝑣()𝑥,𝑡𝑥,𝑡=𝑣₀(𝑥,𝑡)+𝑣₁(𝑥,𝑡)+𝑣₂(),𝑤()𝑥,𝑡𝑥,𝑡=𝑤₀(𝑥,𝑡)+𝑤₁(𝑥,𝑡)+𝑤₂(𝑥,𝑡). (22)(23)(24)

In order to verify whether the approximate solutions of (22)–(24) lead to higher accuracy, we draw the figures of the approximate solutions of (22)–(24) with 𝛼=1, as well as the exact solutions (13) when 𝑐₀=𝑐₂=1, 𝛾=0.1, and 𝛽=0.08. It can be seen from Figures 1(a) to 3(b) that the solutions obtained by the presented method is nearly identical with the exact solutions. So, we conclude that a good approximation is achieved by using the GDTM.

(a)

(b)

Figure 1 

The surface shows the solution 𝑢(𝑥,𝑡) of (11): (a) exact solution; (b) approximate solution of (22) when 𝛼=𝑐0=𝑐2=1, 𝛾=0.1, and 𝛽=0.08.

In the following, we will construct an approximate solution of (11) with the new initial conditions, 𝑢(𝑥,0)=𝛽−2𝛾²3+2𝛾²tanh²,𝑣()𝛾𝑥𝑥,0=−4𝛾²𝑐₀𝛽+𝛾²3𝑐²₁+4𝛾²𝛽+𝛾²3𝑐₁,𝑤()tanh𝛾𝑥𝑥,0=𝑐₀+𝑐₁tanh𝛾𝑥, (25) where 𝑐₀,𝑐₁,𝛾, and 𝛽 are arbitrary constants.

The generalized two-dimensional differential transforms of the initial conditions of (25) are given by 𝑈_1,𝛼()𝑉𝑘,0=0,if𝑘=1,3,5,…,_1,𝛼(𝑘,0)=𝑊_1,𝛼()𝑈𝑘,0=0if𝑘=2,4,6,…,_1,𝛼()=10,03𝛽−2𝛾²,𝑈_1,𝛼(2,0)=2𝛾⁴,𝑈_1,𝛼()44,0=−3𝛾⁶,𝑈_1,𝛼(6,0)=34𝛾45⁸𝑉,…,_1,𝛼(0,0)=−4𝑐₀𝛾²𝛽+𝛾²3𝑐²₁,𝑉_1,𝛼(1,0)=4𝛾³𝛽+𝛾²3𝑐₁,𝑉_1,𝛼(3,0)=−4𝛾⁵𝛽+𝛾²9𝑐₁𝑊,…,_1,𝛼(0,0)=𝑐₀,𝑊_1,𝛼(1,0)=𝑐₁𝑊𝛾,_1,𝛼()𝑐3,0=−¹3𝛾³,𝑊_1,𝛼(5,0)=2𝑐₁𝛾15⁵,…. (26)

Utilizing the recurrence relations in (14) and the transformed initial conditions, we can obtain the following approximate solutions: 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃(),𝑣()𝑥,𝑡𝑥,𝑡=𝑣₀(𝑥,𝑡)+𝑣₁(𝑥,𝑡)+𝑣₂(𝑥,𝑡)+𝑣₃(),𝑤()𝑥,𝑡𝑥,𝑡=𝑤₀(𝑥,𝑡)+𝑤₁(𝑥,𝑡)+𝑤₂(𝑥,𝑡)+𝑤₃(𝑥,𝑡), (27) where the closed forms of 𝑢_𝑖(𝑥,𝑡),𝑣_𝑖(𝑥,𝑡), and 𝑤_𝑖(𝑥,𝑡)(𝑖=0,1,2,3) are 𝑢₀(𝑥,𝑡)=𝛽−2𝛾²3+2𝛾²tanh²,𝑢𝛾𝑥₁(𝑥,𝑡)=4𝛽𝛾³sech²𝑡𝛾𝑥tanh𝛾𝑥^𝛼Γ(),𝑢1+𝛼₂(𝑥,𝑡)=−4𝛽²𝛾⁴cosh2𝛾𝑥−2sech⁴𝑡𝛾𝑥^2𝛼Γ(),𝑢1+2𝛼₃(𝑥,𝑡)=2𝛾²sech²×𝛾𝑥tanh𝛾𝑥64𝛽²𝛾⁵sech²𝛾𝑥+144𝛽²𝛾⁵tanh⁴𝛾𝑥−2𝛾⁵144𝛽²𝛾²−8𝛽³sech²𝛾𝑥tanh³−1𝛾𝑥𝛽88²𝛾⁵878𝛽+3626𝛾²+179𝛽+2921𝛾²+cosh2𝛾𝑥386𝛽−922𝛾²cosh4𝛾𝑥+7−5𝛽+𝛾²cosh6𝛾𝑥×sech⁸×Γ()𝛾𝑥tanh𝛾𝑥1+2𝛼Γ²()𝑡1+𝛼Γ(^3𝛼),𝑣1+3𝛼₀(𝑥,𝑡)=−4𝛾²𝑐₀𝛽+𝛾²3𝑐²₁+4𝛾²𝛽+𝛾²3𝑐₁,𝑣tanh𝛾𝑥₁(𝑥,𝑡)=4𝛽𝛾³𝛽+𝛾²sech²𝛾𝑥3𝑐₁𝑡^𝛼Γ(),𝑣1+𝛼₂(𝑥,𝑡)=−8𝛽²𝛾⁴𝛽+𝛾²sech²𝛾𝑥tanh𝛾𝑥3𝑐₁𝑡^2𝛼Γ(),𝑣1+2𝛼₃()=𝛽𝑥,𝑡²𝛾⁵30𝑐₁𝛽+𝛾²×408𝛾²−𝛽−153𝛽+56𝛾²cosh2𝛾𝑥+576𝛾²+cosh4𝛾𝑥5𝛽−56𝛾²cosh6𝛾𝑥sech⁸−𝛾𝑥16𝛽²𝛾⁶𝛽+𝛾²×5+cosh2𝛾𝑥sech⁴𝛾𝑥tanh³×𝛾𝑥3𝑐₁⁻¹Γ(1+2𝛼)Γ²()𝑡1+𝛼Γ(^3𝛼),𝑤1+3𝛼₀(𝑥,𝑡)=𝑐₀+𝑐₁,𝑤tanh𝛾𝑥₁(𝑥,𝑡)=𝑐₁𝛽𝛾sech²𝑡𝛾𝑥^𝛼Γ(),𝑤1+𝛼₂(𝑥,𝑡)=−2𝑐₁𝛽²𝛾²sech²𝑡𝛾𝑥tanh𝛾𝑥^2𝛼Γ(),𝑤1+2𝛼₃()=𝑐𝑥,𝑡¹𝛽²𝛾³40408𝛾²−𝛽−153𝛽+56𝛾²cosh2𝛾𝑥+576𝛾²+cosh4𝛾𝑥5𝛽−56𝛾²cosh6𝛾𝑥×sech⁸𝛾𝑥+2𝑐₁𝛽²5𝛾⁵×7cosh4𝛾𝑥−9−58cosh2𝛾𝑥×sech⁶𝛾𝑥tanh²Γ()𝛾𝑥1+2𝛼Γ²()×𝑡1+𝛼_3𝛼Γ().1+3𝛼 (28)

3.2. Fractional Coupled MKdV Equation

Consider the following time-fractional coupled MKdV equation: 𝐷^𝛽_𝑡1𝑢=2𝑢_𝑥𝑥𝑥−3𝑢²𝑢_𝑥+32𝑣_𝑥𝑥+3𝑢𝑣_𝑥+3𝑢_𝑥𝑣−3𝜆𝑢_𝑥,𝐷^𝛽_𝑡𝑣=−𝑣_𝑥𝑥𝑥−3𝑣𝑣_𝑥−3𝑢_𝑥𝑣_𝑥+3𝑢²𝑣_𝑥+3𝜆𝑣_𝑥,𝑡>0,0<𝛽≤1, (29) subject to the initial conditions 𝑢(),𝑣()=1𝑥,0=𝛾tanh𝛾𝑥𝑥,024𝛾²+𝜆−2𝛾²tanh²𝛾𝑥, (30) where 𝛾 is an arbitrary constant.

The exact solutions of (29) and (30), for the special case of 𝛽=1, given in [26], are 𝑢(𝑥,𝑡)=𝛾tanh𝛾𝜉,𝜉=𝑥−𝛾²+32𝜆𝑣()=1𝑡,𝑥,𝑡24𝛾²+𝜆−2𝛾²tanh²𝛾𝜉. (31)

Equation system of (29) is more complex. In order to obtain an explicit iteration scheme, we follow the pretreatment technique introduced recently by Chang [20, 21] to deal with long nonlinear items. Firstly, we suppose that 𝑓=𝑣_𝑥,𝑔=𝑢_𝑥,𝑝=𝑔+3𝑣,𝑞=𝑢²−𝑣+𝜆. (32)

According to (32), (29) can be equivalently written as the following form: 𝑓=𝑣_𝑥,𝑔=𝑢_𝑥,𝑝=𝑔+3𝑣,𝑞=𝑢²𝐷−𝑣+𝜆,_𝛽𝑡1𝑢=2𝑝_𝑥𝑥𝐷−3𝑞𝑔+3𝑢𝑓,_𝛽𝑡𝑣=−𝑔𝑝_𝑥+3𝑞𝑓−𝑓_𝑥𝑥+𝑔𝑔_𝑥. (33)

Suppose that the solutions 𝑢(𝑥,𝑡) and 𝑣(𝑥,𝑡) can be represented as the products of single-valued functions, respectively. Applying the generalized two-dimensional differential transform to both sides of (33) and using the related theorems, we have 𝐹_1,𝛽(𝑘,ℎ)=(𝑘+1)𝑉_1,𝛽(),𝐺𝑘+1,ℎ_1,𝛽(𝑘,ℎ)=(𝑘+1)𝑈_1,𝛽(),𝑃𝑘+1,ℎ_1,𝛽(𝑘,ℎ)=𝐺_1,𝛽(𝑘,ℎ)+3𝑉_1,𝛽(),𝑄𝑘,ℎ_1,𝛽(0,0)=𝑈²_1,𝛽(0,0)−𝑉_1,𝛽()𝑄0,0+𝜆,_1,𝛽(𝑘,ℎ)=^𝑘ℎ_𝑟=0𝑠=0𝑈_1,𝛽(𝑟,ℎ−𝑠)𝑈_1,𝛽(𝑘−𝑟,𝑠)−𝑉_1,𝛽(𝑘,ℎ),𝑘²+ℎ²Γ𝛽()≠0,ℎ+1+1Γ𝑈𝛽ℎ+1^1,𝛽()=1𝑘,ℎ+12(𝑘+1)(𝑘+2)𝑃_1,𝛽(𝑘+2,ℎ)−3^𝑘ℎ_𝑟=0𝑠=0𝑄_1,𝛽(𝑟,ℎ−𝑠)𝐺_1,𝛽(𝑘−𝑟,𝑠)+3^𝑘+1ℎ_𝑟=0𝑠=0𝑈_1,𝛽(𝑟,ℎ−𝑠)𝐹_1,𝛽(),Γ𝛽()𝑘−𝑟,𝑠ℎ+1+1Γ𝑉𝛽ℎ+1^1,𝛽(𝑘,ℎ+1)=−^𝑘ℎ_𝑟=0𝑠=0(𝑘+1−𝑟)𝐺_1,𝛽(𝑟,ℎ−𝑠)𝑃_1,𝛽×(𝑘+1−𝑟,𝑠)−(𝑘+1)(𝑘+2)𝐹_1,𝛽(𝑘+2,ℎ)+3^𝑘ℎ_𝑟=0𝑠=0𝑄_1,𝛽(𝑟,ℎ−𝑠)𝐹_1,𝛽()+𝑘−𝑟,𝑠_{𝑘ℎ𝑟=0𝑠=0}(𝑘+1−𝑟)𝐺_1,𝛽(𝑟,ℎ−𝑠)𝐺_1,𝛽×(𝑘+1−𝑟,𝑠). (34) Herein 𝐹_1,𝛽(𝑘,ℎ),𝐺_1,𝛽(𝑘,ℎ),𝑃_1,𝛽(𝑘,ℎ), and 𝑄_1,𝛽(𝑘,ℎ) denote the differential transformations of the functions 𝑓(𝑥,𝑡),𝑔(𝑥,𝑡),𝑝(𝑥,𝑡), and 𝑞(𝑥,𝑡), respectively.

The generalized two-dimensional differential transforms of the initial conditions of (30) can be obtained as follows: 𝑈_1,𝛽()𝑉𝑘,0=0,if𝑘=0,2,4,…,_1,𝛽()𝑈𝑘,0=0,if𝑘=1,3,5,…,_1,𝛽(1,0)=𝛾²,𝑈_1,𝛽()𝛾3,0=−⁴3,𝑈_1,𝛽(5,0)=2𝛾⁶15,𝑈_1,𝛽(7,0)=−17𝛾⁸𝑉315,…,_1,𝛽()=10,024𝛾²+𝜆,𝑉_1,𝛽(2,0)=−2𝛾⁴,𝑉_1,𝛽(4,0)=4𝛾⁶3,𝑉_1,𝛽(6,0)=−34𝛾⁸45,…. (35)

Utilizing the recurrence relations of (34) and the transformed initial conditions, we can obtain all the 𝑈_1,𝛽(𝑘,ℎ) and 𝑉_1,𝛽(𝑘,ℎ) with the help of Mathematica. Through the complex calculation which is similar to the solving process in Section 3.1, we have the approximate solutions of (29) in finite series, 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃(),𝑣()𝑥,𝑡𝑥,𝑡=𝑣₀(𝑥,𝑡)+𝑣₁(𝑥,𝑡)+𝑣₂(𝑥,𝑡)+𝑣₃(𝑥,𝑡), (36)(37) where the closed forms of 𝑢_𝑖(𝑥,𝑡) and 𝑣_𝑖(𝑥,𝑡)(𝑖=0,1,2,3) are given by 𝑢₀(),𝑢𝑥,𝑡=𝛾tanh𝛾𝑥₁()1𝑥,𝑡=−2𝛾²2𝛾²+3𝜆sec²𝛾𝑥𝑡^𝛼Γ(),𝑢1+𝛼₂()1𝑥,𝑡=−2𝛾³sec²×𝛾𝑥2𝛾²+3𝜆²−144𝛾²𝜆sec²×𝑡𝛾𝑥tanh𝛾𝑥^2𝛼Γ(),𝑢1+2𝛼₃()=1𝑥,𝑡4𝛾⁴sec²×𝛾𝑥−22𝛾²+3𝜆³+3sec²×𝛾𝑥−152𝛾⁶+1092𝛾⁴𝜆+270𝛾²𝜆²+27𝜆³+200𝛾⁶−4776𝛾⁴𝜆−270𝛾²𝜆²×sec²𝛾𝑥+4032𝛾⁴𝜆sec⁴𝑡𝛾𝑥^3𝛼Γ()+31+3𝛼4𝛾⁶10𝛾²−9𝜆2𝛾²××+3𝜆2cosh2𝛾𝑥−3sec⁶𝛾𝑥Γ(1+2𝛼)𝑡^3𝛼Γ(1+𝛼)²Γ(),𝑣1+3𝛼₀()=1𝑥,𝑡24𝛾²+𝜆−2𝛾²tanh²,𝑣𝛾𝑥₁(𝑥,𝑡)=2𝛾³2𝛾²−3𝜆sec²𝑡𝛾𝑥tanh𝛾𝑥^𝛼Γ(),𝑣1+𝛼₂(𝑥,𝑡)=22𝛾²−3𝜆²−3sec²×𝛾𝑥4𝛾⁴−60𝛾²𝜆+9𝜆²+48𝛾²𝜆sec²×𝛾𝛾𝑥₄sec²𝛾𝑥𝑡^2𝛼Γ(),𝑣1+2𝛼₃(𝑥,𝑡)=2𝛾⁵sec⁸×𝛾𝑥92𝛾⁶+3420𝛾⁴𝜆+351𝛾²𝜆²+54𝜆³−912𝛾⁶+1456𝛾⁴𝜆+63𝛾²𝜆²−9𝜆³×sinh²𝛾𝑥−96𝛾²2𝛾⁴−48𝛾²𝜆+9𝜆²sinh⁴+𝛾𝑥2𝛾²−3𝜆³sinh𝛾𝑥⁶×tanh𝛾𝑥𝑡^3𝛼Γ()−1+3𝛼27𝜆−2𝛾²²sinh𝛾𝑥−96𝛾²2𝛾²−3𝜆sinh³×𝛾𝛾𝑥₇sec⁷𝛾𝑥Γ(1+2𝛼)𝑡^3𝛼Γ(1+𝛼)²Γ().1+3𝛼 (38)

The effectiveness and accuracy of the approximate solutions can be seen from the comparison figures.

Figures 4 and 5 show the approximate solutions of (36) and (37) and the exact ones of (31) with 𝛼=𝜆=1 and 𝛾=0.1, respectively. Comparing Figures 1(a) and 1(b) with Figures 2(a) and 2(b), we can see that the solutions obtained by different methods are nearly identical. From these figures, we can know that the series solutions converge rapidly, so a good approximation has been achieved.

(a)

(b)

Figure 2 

The surface shows the solution 𝑣(𝑥,𝑡) of (11): (a) exact solution; (b) approximate solution of (23) when 𝛼=𝑐0=𝑐2=1, 𝛾=0.1, and 𝛽=0.08.

(a)

(b)

Figure 3 

The surface shows the solution 𝑤(𝑥,𝑡) of (11): (a) exact solution; (b) approximate solution (24) when 𝛼=𝑐0=𝑐2=1, 𝛾=0.1 and 𝛽=0.08.

(a)

(b)

Figure 4 

The surface shows the exact solutions (31): (a) 𝑢(𝑥,𝑡); (b) 𝑣(𝑥,𝑡) when 𝜆=1 and 𝛾=0.1.

(a)

(b)

Figure 5 

The surface shows the approximate solutions of (29): (a) 𝑢(𝑥,𝑡) of (36); (b) 𝑣(𝑥,𝑡) of (37) when 𝛼=𝜆=1 and 𝛾=0.1.

4. Summary and Discussion

In this paper, combining the Caputo fractional derivative, the GDTM was applied to derive approximate analytical solutions of the time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation with initial conditions. The numerical solutions obtained from the GDTM are shown graphically. The obtained results demonstrate the reliability of the algorithm and its wider applicability to nonlinear fractional coupled partial differential equations.

In [8] and the references cited therein, the so-called fractional complex transform (FCT) is suggested to convert a fractional differential equation with Jumarie's modification of Riemann-Liouville derivative into its classical differential partner. According to the idea of FCT, for some fractional differential equations with Caputo time-fractional derivative 𝐷^𝛽_𝑡, we assume that 𝑢(𝑥,𝑡)=^+∞_𝑚=0𝑢_𝑚(𝑥)𝑡^𝑚𝛽Γ,𝑣()=1+𝑚𝛽𝑥,𝑡^+∞_𝑚=0𝑣_𝑚(𝑥)𝑡^𝑚𝛽Γ,1+𝑚𝛽 (39) consequently, from Lemma 3, 𝐷^𝛽_𝑡𝑢(𝑥,𝑡)=^+∞_𝑚=0𝑢_𝑚+1(𝑥)𝑡^𝑚𝛽Γ,1+𝑚𝛽 (40) and it can be seen easily that 𝑢(𝑥,𝑡)𝑣(𝑥,𝑡)=^+∞_𝑚=0^𝑚_𝑟=0𝑢_𝑟(𝑥)𝑣_𝑚−𝑟(𝑥)×Γ1+𝑚𝛽ΓΓ()𝛽×𝑡1+𝑟𝛽1+𝑚−𝑟_𝑚𝛽Γ.1+𝑚𝛽 (41) We can point out that the approximate solutions of (11) and (29) by GDTM can be derived by the similar method compared with FCT. Without loss of generality, we only consider (29) with the initial conditions of (30). In fact, suppose that 𝑓(𝑥,𝑡)=^+∞_𝑚=0𝑓_𝑚(𝑥)𝑡^𝑚𝛽Γ,𝑔()=1+𝑚𝛽𝑥,𝑡^+∞_𝑚=0𝑔_𝑚(𝑥)𝑡^𝑚𝛽Γ,𝑝()=1+𝑚𝛽𝑥,𝑡^+∞_𝑚=0𝑝_𝑚(𝑥)𝑡^𝑚𝛽Γ,𝑞()=1+𝑚𝛽𝑥,𝑡^+∞_𝑚=0𝑞_𝑚(𝑥)𝑡^𝑚𝛽Γ.1+𝑚𝛽 (42) From (33), (40), and (41), we have the following iteration formulae: 𝑢₀(𝑥)=𝑢(𝑥,0),𝑣₀(𝑥)(),𝑓=𝑣𝑥,0₀(𝑥)=𝑣^′₀(𝑥),𝑔₀(𝑥)=𝑢^′₀(𝑥),𝑝₀(𝑥)=𝑔₀(𝑥)+3𝑣₀(𝑥),𝑞₀(𝑥)=𝑢²₀(𝑥)−𝑣₀(𝑥)𝑓+𝜆,_𝑚(𝑥)=𝑣^′_𝑚(𝑥),𝑔_𝑚(𝑥)=𝑢^′_𝑚(𝑥),𝑝_𝑚(𝑥)=𝑔_𝑚(𝑥)+3𝑣_𝑚(𝑥),𝑞_𝑚(𝑥)=^𝑚_𝑟=0𝑢_𝑟(𝑥)𝑢_𝑚−𝑟(𝑥)Γ1+𝑚𝛽Γ1+𝑟𝛽Γ1+(𝑚−𝑟)𝛽−𝑣_𝑚(𝑥),𝑢_𝑚+1(𝑥)=12𝑝_𝑚^′′(𝑥)−3^𝑚_𝑟=0𝑞_𝑟(𝑥)𝑔_𝑚−𝑟(𝑥)×Γ1+𝑚𝛽Γ1+𝑟𝛽Γ1+(𝑚−𝑟)𝛽+3^𝑚_𝑟=0𝑢_𝑟(𝑥)𝑓_𝑚−𝑟(𝑥)×Γ1+𝑚𝛽ΓΓ()𝛽,𝑣1+𝑟𝛽1+𝑚−𝑟_𝑚+1(𝑥)=−𝑓_𝑚^′′(𝑥)−^𝑚_𝑟=0𝑔_𝑟(𝑥)𝑝^′_𝑚−𝑟(𝑥)×Γ1+𝑚𝛽Γ1+𝑟𝛽Γ1+(𝑚−𝑟)𝛽+3^𝑚_𝑟=0𝑞_𝑟(𝑥)𝑓_𝑚−𝑟(𝑥)Γ1+𝑚𝛽ΓΓ()𝛽+1+𝑟𝛽1+𝑚−𝑟_𝑚𝑟=0𝑔_𝑟(𝑥)𝑔^′_𝑚−𝑟(𝑥)Γ1+𝑚𝛽ΓΓ()𝛽,1+𝑟𝛽1+𝑚−𝑟 (43) for 𝑚=0,1,2,3,….

Setting 𝑢_𝑚(𝑥,𝑡)=𝑢_𝑚(𝑥)(𝑡^𝑚𝛽/Γ(1+𝑚𝛽)) and 𝑣_𝑚(𝑥,𝑡)=𝑣_𝑚(𝑥)(𝑡^𝑚𝛽/Γ(1+𝑚𝛽)), and using (43), we can obtain the 3-order approximate solutions as (36) and (37) by using the symbol computational software Mathematica.

Acknowledgments

The project is supported by the NNSF of China (11061021), the NSF-IMU of China (2012MS0105, 2012MS0106), and the YSF-IMU of China (ND0811).