Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method

Abstract

A homotopy perturbation transformation method (HPTM) which is based on homotopy perturbation method and Laplace transform is first applied to solve the approximate solution of the fractional nonlinear equations. The nonlinear terms can be easily handled by the use of He's polynomials. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm.

1. Introduction

In recent years, system of fractional nonlinear partial differential equations [1–3] have attracted much attention in a variety of applied sciences. The importance of obtaining the exact and approximate solutions of fractional nonlinear equations in physics and mathematics is still a significant problem that needs new methods to discover exact and approximate solutions. But these nonlinear fractional differential equations are difficult to get their exact solutions [4–7]. So, numerical methods have been used to handle these equations [8–11], and some semianalytical techniques have also largely been used to solve these equations. Such as, Adomian decomposition method [12, 13], variational iteration method [14, 15], differential transform method [16, 17], Laplace decomposition method [18, 19], and homotopy perturbation method [20–25]. Most of these methods have their inbuilt deficiencies like the calculation of Adomian's polynomials, the Lagrange multiplier, divergent results, and huge computational work.

In this work, we will use homotopy perturbation transformation method introduced by Khan [26, 27] to solve fractional nonlinear partial differential equations. This new method basically illustrates how two powerful algorithms, homotopy perturbation method and Laplace transform method, can be combined and used to approximate the solutions of nonlinear equation. The proposed algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form. This paper considers the effectiveness of the homotopy perturbation transformation method in solving fractional nonlinear equations.

2. Description of the HPTM

To illustrate the basic idea of this method [26, 27], we consider a general fractional nonlinear nonhomogeneous partial differential equation with initial conditions of the form 𝐷^𝛼_𝑡𝑢()()()(),𝑢()(𝑥)𝑥,𝑡+𝑅𝑢𝑥,𝑡+𝑁𝑢𝑥,𝑡=𝑔𝑥,𝑡𝑥,0=ℎ,𝑢_𝑡(𝑥,0)=𝑓(𝑥), (2.1)(2.2) where 𝑔(𝑥,𝑡) is the source term, 𝑁 represents the general nonlinear differential operator and 𝑅 is the linear differential operator, and 𝐷^𝛼_𝑡𝑢(𝑥,𝑡) is the Caputo fractional derivative of function 𝑢(𝑥,𝑡) which is defined as ₀𝐷^𝛼_𝑡𝑢()=1𝑥,𝑡Γ(𝑛−𝛼)^𝑡₀𝑢^(𝑛)(𝑥,𝜏)𝑑𝜏(𝑡−𝜏)^{𝛼+1−𝑛},(𝑛−1<𝑅𝑒(𝛼)𝑛,𝑛∈𝑁), (2.3) where Γ(⋅) denotes the Gamma function. The properties of fractional derivative can be found in [1, 2]. Laplace transform (denoted throughout this paper by 𝐿) of the Caputo operator is an important property will be used in this paper 𝐿₀𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=𝑠^𝛼𝑢(𝑥,𝑠)−^𝑛−1_𝑘=0𝑢^𝑘𝑥,0⁺𝑠^{𝛼−1−𝑘},(𝑛−1<𝛼⩽𝑛). (2.4)

Taking the Laplace transform on both sides of (2.1), 𝐿₀𝐷^𝛼_𝑡𝑢(𝑥,𝑡)+𝐿[𝑅𝑢(𝑥,𝑡)]+𝐿[𝑁𝑢(𝑥,𝑡)]=𝐿𝑔(𝑥,𝑡), (2.5) Using the property of the laplace transform, we have 𝐿[𝑢(=ℎ(𝑥)𝑥,𝑡)]𝑠+𝑓(𝑥)𝑠²−1𝑠^𝛼𝐿[(−1𝑅𝑢𝑥,𝑡)]𝑠^𝛼𝐿[(+1𝑁𝑢𝑥,𝑡)]𝑠^𝛼𝐿𝑔(𝑥,𝑡). (2.6) Operating with the Laplace inverse on both sides of (2.6) gives 𝑢(𝑥,𝑡)=𝐺(𝑥,𝑡)−𝐿⁻¹1𝑠^𝛼𝐿[(+1𝑅𝑢𝑥,𝑡)]𝑠^𝛼𝐿[𝑁𝑢(𝑥,𝑡)], (2.7) where 𝐺(𝑥,𝑡) represent the term arising from the source term and the prescribed initial conditions. Then, we apply the homotopy perturbation method; the basic assumption is that the solutions can be written as a power series in 𝑝𝑢(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+𝑝³𝑢₃+⋯, (2.8) and the nonlinear term can be decomposed as 𝑁𝑢(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛ℋ_𝑛(𝑢), (2.9) where 𝑝∈[0,1] is an embedding parameter. ℋ_𝑛(𝑢) is He's polynomials [28, 29] and can be generated by ℋ_𝑛𝑢₀,…,𝑢_𝑛=1𝜕𝑛!𝜕𝑝^𝑛𝑛𝑁^∞_𝑖=0𝑝^𝑖𝑢_𝑖𝑝=0,𝑛=0,1,2…. (2.10) Substituting (2.8) and (2.9) in (2.7) we get ^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)=𝐺(𝑥,𝑡)−𝑝𝐿⁻¹1𝑠^𝛼𝐿𝑅^∞_𝑛=0𝑝^𝑛𝑢_𝑛(+1𝑥,𝑡)𝑠^𝛼𝐿^∞_𝑛=0𝑝^𝑛ℋ_𝑛(𝑢). (2.11) Equating the terms with identical powers in 𝑝, we obtain the following approximations: 𝑝⁰∶𝑢₀()(),𝑝𝑥,𝑡=𝐺𝑥,𝑡₁∶𝑢₁(𝑥,𝑡)=−𝐿⁻¹1𝑠^𝛼𝐿𝑅𝑢₀(+1𝑥,𝑡)𝑠^𝛼𝐿ℋ₀(𝑢,𝑝₂)∶𝑢₂(𝑥,𝑡)=−𝐿⁻¹1𝑠^𝛼𝐿𝑅𝑢₁(+1𝑥,𝑡)𝑠^𝛼𝐿ℋ₁(𝑢,⋮) (2.12) The best approximations for the solution are 𝑢(𝑥,𝑡)=𝑢₀+𝑢₁+𝑢₂+𝑢₃+⋯. (2.13)

This method does not resort to linearization or assumptions of weak nonlinearity, the solution generated in the form of general solution, and it is more realistic compared to the method of simplifying the physical problems.

3. Approximate Solutions of Fractional Equations

In order to assess the advantages and the accuracy of the homotopy perturbation transform method for fractional nonlinear equations, we have applied it to the following several problems.

Case 1. Consider the following time fractional advection nonhomogeneous equation [27]: 𝐷^𝛼_𝑡𝑢(𝑥,𝑡)+𝑢𝑢_𝑥=2𝑡+𝑥+𝑡³+𝑥𝑡²,𝑢(𝑥,0)=0, (3.1)(3.2) where 0<𝛼⩽1, taking the Laplace transform on both sides of (3.1)-(3.2) 𝐿𝐷^𝛼_𝑡𝑢(𝑥,𝑡)+𝐿𝑢𝑢_𝑥=𝐿2𝑡+𝑥+𝑡³+𝑥𝑡². (3.3) Using the property of the Laplace transform, we have 𝐿[𝑢(=1𝑥,𝑡)]𝑠^𝛼2𝑠²+𝑥𝑠+6𝑠⁴+2𝑥𝑠³−1𝑠^𝛼𝐿𝑢𝑢_𝑥. (3.4) Operating with the Laplace inverse on both sides of (3.4) gives 𝑢(𝑥,𝑡)=2𝑡^𝛼+1Γ()+2+𝛼𝑥𝑡^𝛼Γ()+1+𝛼6𝑡^𝛼+3Γ()+4+𝛼2𝑥𝑡^𝛼+2Γ(3+𝛼)−𝐿⁻¹1𝑠^𝛼𝐿𝑢𝑢_𝑥. (3.5) Then, we apply the homotopy perturbation method, and substituting (2.8) and (2.9) in (3.5) we get ^∞_𝑛=0𝑝^𝑛𝑢_𝑛=2𝑡^𝛼+1Γ()+2+𝛼𝑥𝑡^𝛼Γ()+1+𝛼6𝑡^𝛼+3Γ()+4+𝛼2𝑥𝑡^𝛼+2Γ()𝐿3+𝛼−𝑝⁻¹1𝑠^𝛼𝐿^∞_𝑛=0𝑝^𝑛ℋ_𝑛(𝑢), (3.6) where ℋ_𝑛(𝑢) is He's polynomials that represents nonlinear term 𝑢𝑢_𝑥; we have a few terms of the He's polynomials for 𝑢𝑢_𝑥 which are given by ℋ₀(𝑢)=𝑢₀𝑢_0𝑥,ℋ₁(𝑢)=𝑢₀𝑢_1𝑥+𝑢₁𝑢_0𝑥,ℋ₂(𝑢)=𝑢₀𝑢_2𝑥+𝑢₁𝑢_1𝑥+𝑢₂𝑢_0𝑥,⋮ (3.7) Comparing the coefficient of like powers of 𝑝, we have 𝑢₀(𝑥,𝑡)=2𝑡^𝛼+1Γ()+2+𝛼𝑥𝑡^𝛼Γ()+1+𝛼6𝑡^𝛼+3Γ()+4+𝛼2𝑥𝑡^𝛼+2Γ(),𝑢3+𝛼₁(𝑥,𝑡)=−𝐿⁻¹1𝑠^𝛼𝐿𝑢₀𝑢_0𝑥()𝑡=−𝑥Γ1+2𝛼^3𝛼Γ²()Γ()−()𝑡1+𝛼1+3𝛼2Γ2+2𝛼^1+3𝛼Γ()Γ()Γ()−()𝑡1+𝛼2+𝛼2+3𝛼4𝑥Γ3+2𝛼^2+3𝛼Γ()Γ()Γ()−()𝑡1+𝛼3+𝛼3+3𝛼4Γ3+3𝛼^4+2𝛼Γ()Γ()Γ()−()𝑡2+𝛼3+𝛼4+3𝛼6Γ4+2𝛼^3+3𝛼Γ()Γ()Γ()−()𝑡1+𝛼4+𝛼4+3𝛼4𝑥Γ5+2𝛼^4+3𝛼Γ²()Γ()−()𝑡3+𝛼5+3𝛼12Γ6+2𝛼^5+3𝛼Γ()Γ()Γ(),𝑢3+𝛼4+𝛼6+3𝛼₂(𝑥,𝑡)=−𝐿⁻¹1𝑠^𝛼𝐿𝑢₀𝑢_1𝑥+𝑢₁𝑢_0𝑥=2𝑥Γ(1+2𝛼)Γ(1+4𝛼)𝑡^5𝛼Γ²()Γ()Γ()+()Γ()𝑡1+𝛼1+3𝛼1+5𝛼2Γ1+2𝛼2+4𝛼^1+5𝛼Γ²()Γ()Γ()Γ()+()Γ()𝑡1+𝛼2+𝛼1+3𝛼2+5𝛼2Γ2+2𝛼2+4𝛼^1+5𝛼Γ²()Γ()Γ()Γ()+()Γ()𝑡1+𝛼2+𝛼2+3𝛼2+5𝛼4𝑥Γ1+2𝛼3+4𝛼^2+5𝛼Γ²()Γ()Γ()Γ()⋮1+𝛼3+𝛼1+3𝛼3+5𝛼+… (3.8) and so on; in this manner the rest of component of the solution can be obtained. The solution of (3.1) in series form is given by 𝑢(𝑥,𝑡)=2𝑡^𝛼+1Γ()+2+𝛼𝑥𝑡^𝛼Γ()+1+𝛼6𝑡^𝛼+3Γ()+4+𝛼2𝑥𝑡^𝛼+2Γ()−()𝑡3+𝛼𝑥Γ1+2𝛼^3𝛼Γ²(1+𝛼)Γ(1+3𝛼)+⋯. (3.9) If we take 𝛼=1, the first few components the solution of (3.1) are as follows: 𝑢₀(𝑥,𝑡)=𝑡²𝑡+𝑥𝑡+⁴4+𝑥𝑡³3,𝑢₁()𝑡𝑥,𝑡=−⁴4−𝑥𝑡³3−2𝑥𝑡⁵−157𝑡⁶−72𝑥𝑡⁷−𝑡63⁸,𝑢96₂(𝑥,𝑡)=5𝑡¹2+80642𝑥𝑡¹1+20792783𝑡¹0+30240038𝑥𝑡⁹+2835143𝑡⁸+288022𝑥𝑡⁷+3157𝑡⁶+122𝑥𝑡⁵,⋮15 (3.10) The noise terms −𝑡⁴/4−𝑥𝑡³/3 between the components 𝑢₀ and 𝑢₁ can be canceled and the remaining term of 𝑢₀ still satisfies the equation. For this special case, the exact solution is therefore 𝑢(𝑥,𝑡)=𝑡²+𝑥𝑡. (3.11) which was given in [27].

Case 2. Consider the following time-space fractional nonlinear Fokker-Planck equation [30]: 𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=−𝐷^𝛽_𝑥4𝑢²𝑥−𝑥𝑢3+𝐷_𝑥^2𝛽𝑢²,𝑢(𝑥,0)=𝑥², (3.12)(3.13) where 0<𝛼,𝛽⩽1, and 𝛼 and 𝛽 are parameters describing the order of the time- and space-fractional derivatives, respectively. 𝐷^𝛽_𝑥 is also the Caputo fractional derivative with respect to 𝑥 and is defined as 𝐷^𝛽_𝑥𝑢()=1𝑥,𝑡Γ𝑚−𝛽^𝑥₀𝑢^(𝑚)(𝜉,𝑡)𝑑𝜉(𝑥−𝜉)^{𝛽+1−𝑚},𝑚−1<𝑅𝑒𝛽⩽𝑚,𝑚∈𝑁 (3.14) taking the Laplace transform on both sides of (3.12)-(3.13) 𝑠^𝛼𝐿[𝑢(𝑥,𝑡)]−𝑥²𝑠^𝛼−1=𝐿−𝐷^𝛽_𝑥4𝑢²𝑥−𝑥𝑢3+𝐷_𝑥^2𝛽𝑢². (3.15) We have 𝐿[𝑢(=𝑥𝑥,𝑡)]²𝑠+1𝑠^𝛼𝐿−𝐷^𝛽_𝑥4𝑢²𝑥−𝑥𝑢3+𝐷_𝑥^2𝛽𝑢². (3.16) Operating with the Laplace inverse on both sides of (3.16) gives 𝑢(𝑥,𝑡)=𝑥²+𝐿⁻¹1𝑠^𝛼𝐿−𝐷^𝛽_𝑥4𝑢²𝑥−𝑥𝑢3+𝐷_𝑥^2𝛽𝑢². (3.17) Then, we apply the homotopy perturbation method, and substituting (2.8) and (2.9) in (3.17) we get ^∞_𝑛=0𝑝^𝑛𝑢_𝑛=𝑥²+𝑝𝐿⁻¹1𝑠^𝛼𝐿𝐷_𝑥^2𝛽∞_𝑛=0𝑝^𝑛ℋ_𝑛(𝑢)−𝐷^𝛽_𝑥4∑^∞_𝑛=0𝑝^𝑛ℋ_𝑛(𝑢)𝑥−𝑥∑^∞_𝑛=0𝑝^𝑛𝑢_𝑛3, (3.18) where ℋ_𝑛(𝑢) is He's polynomials that represents nonlinear term 𝑢²; we have a few terms of the He's polynomials for 𝑢² which are given by ℋ₀(𝑢)=𝑢²₀,ℋ₁(𝑢)=2𝑢₀𝑢₁,ℋ₂(𝑢)=𝑢²₁+2𝑢₀𝑢₂,⋮ (3.19)(3.20)(3.21) Comparing the coefficient of like powers of 𝑝, we have (3.22) 𝑢₀(𝑥,𝑡)=𝑥²,𝑢₁(𝑥,𝑡)=𝐿⁻¹1𝑠^𝛼𝐿𝐷_𝑥^2𝛽𝑢²₀−𝐷^𝛽_𝑥4𝑢²₀𝑥−𝑥𝑢₀3,=24𝑡^𝛼𝑥^4−2𝛽Γ()Γ−1+𝛼5−2𝛽22𝑡^𝛼𝑥^3−𝛽Γ()Γ,𝑢1+𝛼4−𝛽₂(𝑥,𝑡)=𝐿⁻¹1𝑠^𝛼𝐿𝐷_𝑥^2𝛽2𝑢₀𝑢₁−𝐷^𝛽_𝑥8𝑢₀𝑢₁𝑥−𝑥𝑢₁3=−184𝑡^2𝛼𝑥^5−3𝛽Γ6−2𝛽Γ()ΓΓ+1+2𝛼6−3𝛽5−2𝛽48𝑡^2𝛼𝑥^6−4𝛽Γ7−2𝛽Γ()ΓΓ+1+2𝛼7−4𝛽5−2𝛽506𝑡^2𝛼𝑥^4−2𝛽Γ5−𝛽()ΓΓ−3Γ1+2𝛼5−2𝛽4−𝛽44𝑡^2𝛼𝑥^5−3𝛽Γ6−𝛽Γ()ΓΓ,⋮1+2𝛼6−3𝛽4−𝛽 (3.22) and so on; in this manner the rest of component of the solution can be obtained. The solution of (3.12) in series form is given by 𝑢(𝑥,𝑡)=𝑥²+24𝑡^𝛼𝑥^4−2𝛽Γ()Γ−1+𝛼5−2𝛽22𝑡^𝛼𝑥^3−𝛽Γ()Γ−1+𝛼4−𝛽184𝑡^2𝛼𝑥^5−3𝛽Γ6−2𝛽Γ(1+2𝛼)Γ6−3𝛽Γ5−2𝛽+⋯. (3.23) If we take 𝛼=𝛽=1, the first few components the solution of (3.12) are as follows: 𝑢₀(𝑥,𝑡)=𝑥²,𝑢₁(𝑥,𝑡)=𝑥²𝑢𝑡,₂(𝑥,𝑡)=𝑥²𝑡²,⋮2! (3.24) and so on. Hence, for this special case, we have 𝑢(𝑥,𝑡)=𝑥²𝑡1+𝑡+²2!+⋯=𝑥²𝑒^𝑡 (3.25) which was given in [30].
Figure 1 shows the approximate solution for (3.12)-(3.13) by using the homotopy perturbation transformation method when choosing 𝑥=0.8, 𝛼=1. From the figure, it is clear to see the time evolution of fractional Fokker-Planck equation and we also know the approximate solution of the model is continuous with the fractional parameter 𝛽. Figure 2 shows the approximate solution for (3.12)-(3.13) when 𝑡=0.8, 𝛼=1, and the approximate solution of the model is continuous with the fractional parameter 𝛽. Figures 3 and 4 show the approximate solution for (3.12)-(3.13) when the parameter 𝛽=1, and from the figures, we also know that the solution of the fractional nonlinear equation changes with the parameters 𝛼 and 𝑥,𝑡.
Table 1 shows the approximate solutions for (3.12) by using the homotopy perturbation transformation method, Adomian decomposition method, variational iteration method and the exact solution 𝑢(𝑥,𝑡)=𝑥²𝑒^𝑡 when𝛼=𝛽=1. It is noted that only the third-order of the homotopy perturbation transformation solution is used in evaluating the approximate solutions in Table 1, and it is evident that the method used in this paper and the Adomian decomposition method have high accuracy compare with the variational iteration method, and we take 15 terms of the VIM solution. And for nonlinear equations, Adomian's polynomials are very difficult to calculate. In brief, the homotopy perturbation transformation method is an effectiveness tool to solve fractional nonlinear equation only using Mathematica symbol computing software.

Table 1 

Approximate values and exact solutions when 𝛼=1,𝛽=1 for (3.12).

Figure 1 

The surface of second-order approximate solution of (3.12) when 𝑥=0.8, 𝛼=1.

Figure 2 

The surface of second-order approximate solution of (3.12) when 𝑡=0.8, 𝛼=1.

Figure 3 

The surface of second-order approximate solution of (3.12) when 𝑥=0.8, 𝛽=1.

Figure 4 

The surface of second-order approximate solution of (3.12) when 𝑡=0.8, 𝛽=1.

Case 3. Consider the following time fractional nonhomogeneous nonlinear system [31]: 𝐷^𝛼_𝑡𝑢−𝑢_𝑥𝐷𝑣−𝑢=1,_𝛽𝑡𝑣+𝑢𝑣_𝑥+𝑣=1, (3.26)(3.27) with the initial conditions 𝑢(𝑥,0)=𝑒^−𝑥,𝑣(𝑥,0)=𝑒^𝑥, (3.28) where 0<𝛼, 𝛽⩽1; in a similar way as above we have ^∞_𝑛=0𝑝^𝑛𝑢_𝑛=𝑒^−𝑥+𝑡^𝛼Γ()𝐿𝛼+1+𝑝⁻¹1𝑠^𝛼𝐿^∞_𝑛=0𝑝^𝑛ℋ_1𝑛(𝑢,𝑣)+^∞_𝑛=0𝑝^𝑛𝑢_𝑛,_∞𝑛∑₌₀𝑝^𝑛𝑣_𝑛=𝑒^𝑥+𝑡^𝛽Γ𝐿𝛽+1−𝑝⁻¹1𝑠^𝛼𝐿∑^∞_𝑛=0𝑝^𝑛ℋ_2𝑛(𝑢,𝑣)+∑^∞_𝑛=0𝑝^𝑛𝑣_𝑛, (3.29) where ℋ_1𝑛(𝑢,𝑣) and ℋ_2𝑛(𝑢,𝑣) are He's polynomials that represent nonlinear term 𝑣𝑢_𝑥 and 𝑢𝑣_𝑥 respectively, and we have a few terms of the He's polynomials for 𝑣𝑢_𝑥 and 𝑢𝑣_𝑥 which are given by ℋ₁₀(𝑢,𝑣)=𝑣₀𝑢_0𝑥,ℋ₁₁(𝑢,𝑣)=𝑣₁𝑢_0𝑥+𝑣₀𝑢_1𝑥,ℋ₁₂(𝑢,𝑣)=𝑣₂𝑢_0𝑥+𝑣₁𝑢_1𝑥+𝑣₀𝑢_2𝑥,⋮ℋ₂₀(𝑢,𝑣)=𝑢₀𝑣_0𝑥,ℋ₂₁(𝑢,𝑣)=𝑢₁𝑢_0𝑥+𝑢₀𝑣_1𝑥,ℋ₂₂(𝑢,𝑣)=𝑢₂𝑣_0𝑥+𝑢₁𝑣_1𝑥+𝑢₀𝑣_2𝑥,⋮ (3.30) Comparing the coefficient of like powers of 𝑝, we have 𝑢₀(𝑥,𝑡)=𝑒^−𝑥+𝑡^𝛼Γ(),𝑣1+𝛼₀(𝑥,𝑡)=𝑒^𝑥+𝑡^𝛽Γ,𝑢1+𝛽₁(𝑥,𝑡)=𝐿⁻¹1𝑠^𝛼𝐿𝑣₀𝑢_0𝑥+𝑢₀,𝑡=−^𝛼Γ()+𝑒1+𝛼^−𝑥𝑡^𝛼Γ()+𝑡1+𝛼Γ(^2𝛼)−𝑒1+2𝛼^−𝑥𝑡^𝛼+𝛽Γ,𝑣1+𝛼+𝛽₁(𝑥,𝑡)=−𝐿⁻¹1𝑠^𝛼𝐿𝑢₀𝑣_0𝑥+𝑣₀,𝑡=−^𝛽Γ−𝑒1+𝛽^𝑥𝑡^𝛽Γ−𝑡1+𝛽Γ^2𝛽−𝑒1+2𝛽^𝑥𝑡^𝛼+𝛽Γ,𝑢1+𝛼+𝛽₂(𝑥,𝑡)=𝐿⁻¹1𝑠^𝛼𝐿𝑣₁𝑢_0𝑥+𝑣₀𝑢_1𝑥+𝑢₁=−2𝑡^2𝛼Γ()+𝑒1+2𝛼^−𝑥𝑡^2𝛼Γ()+𝑡1+2𝛼Γ(^3𝛼)+𝑡1+3𝛼Γ^𝛼+𝛽+𝑒1+𝛼+𝛽_−𝑥𝑡^𝛼+𝛽Γ+1+𝛼+𝛽2𝑡^2𝛼+𝛽Γ−𝑒1+2𝛼+𝛽^−𝑥𝑡^2𝛼+𝛽Γ+𝑒1+2𝛼+𝛽^−𝑥𝑡^𝛼+2𝛽Γ−𝑒1+𝛼+2𝛽_−𝑥𝑡^2𝛼+𝛽Γ1+𝛼+𝛽Γ()ΓΓ+𝑒1+𝛼1+𝛽1+2𝛼+𝛽^−𝑥𝑡^2𝛼+2𝛽Γ1+𝛼+2𝛽ΓΓΓ,𝑣1+𝛽1+𝛼+𝛽1+2𝛼+2𝛽₂(𝑥,𝑡)=−𝐿⁻¹1𝑠^𝛼𝐿𝑢₁𝑣_0𝑥+𝑢₀𝑣_1𝑥+𝑣₁𝑡=−^𝛼+𝛽Γ+𝑒1+𝛼+𝛽^𝑥𝑡^𝛼+𝛽Γ−𝑒1+𝛼+𝛽^𝑥𝑡^2𝛼+𝛽Γ+1+2𝛼+𝛽2𝑡^2𝛽Γ+𝑒1+2𝛽_𝑥𝑡^2𝛽Γ+1+2𝛽2𝑡^𝛼+2𝛽Γ+𝑒1+𝛼+2𝛽^𝑥𝑡^𝛼+2𝛽Γ+𝑡1+𝛼+2𝛽Γ^3𝛽+𝑒3𝛽_𝑥𝑡^𝛼+2𝛽Γ1+𝛼+𝛽Γ()ΓΓ+𝑒1+𝛼1+𝛽1+𝛼+2𝛽^𝑥𝑡^2𝛼+2𝛽Γ1+2𝛼+𝛽Γ()ΓΓ,⋮1+𝛼1+𝛼+𝛽1+2𝛼+2𝛽 (3.31) and so on; in this manner the rest of component of the solution can be obtained. The solution of (3.26) and (3.27) in series form is given by 𝑢(𝑥,𝑡)=𝑒^−𝑥+𝑒^−𝑥𝑡^𝛼Γ()−𝑡1+𝛼Γ(^2𝛼)−𝑒1+2𝛼^−𝑥𝑡^𝛼+𝛽Γ+𝑒1+𝛼+𝛽^−𝑥𝑡^2𝛼Γ()𝑣()1+2𝛼+⋯,𝑥,𝑡=𝑒^𝑥−𝑒^𝑥𝑡^𝛽Γ+𝑡1+𝛽Γ^2𝛽−𝑒1+2𝛽^𝑥𝑡^𝛼+𝛽Γ+𝑒1+𝛼+𝛽^𝑥𝑡^2𝛽Γ1+2𝛽+⋯. (3.32) If we take 𝛼=𝛽=1, the first few components the solution of (3.26) and (3.27) are as follows: 𝑢₀(𝑥,𝑡)=𝑒^−𝑥𝑣+𝑡,₀(𝑥,𝑡)=𝑒^𝑥𝑢+𝑡,₁(𝑥,𝑡)=−𝑡+𝑒^−𝑥𝑡+𝑡²2−𝑒^−𝑥𝑡²2,𝑣₁(𝑥,𝑡)=−𝑡−𝑒^𝑥𝑡−𝑡²2−𝑒^𝑥𝑡²2,𝑢₂(𝑥,𝑡)=−𝑡²2+𝑒^−𝑥𝑡²+𝑡³2−𝑒^−𝑥𝑡³3+𝑒^−𝑥𝑡⁴8,𝑣₂()=𝑡𝑥,𝑡²2+𝑒^𝑥𝑡²+𝑡³2+𝑒^𝑥𝑡³3+𝑒^𝑥𝑡⁴8,⋮ (3.33) and so on. Hence, for this special case, we have 𝑢(𝑥,𝑡)=𝑒^−𝑥𝑡1+𝑡+²+𝑡2!3³!+⋯=𝑒^−𝑥+𝑡,𝑣(𝑥,𝑡)=𝑒^𝑥𝑡1−𝑡+²−𝑡2!3³!+⋯=𝑒^𝑥−𝑡, (3.34) which was given in [31].

4. Conclusion

In this work, a homotopy perturbation transformation method which is based on homotopy perturbation method and Laplace transform is used to solve fractional nonlinear partial equations. The nonlinear terms can be easily handled by the use of He's polynomials. It is worth mentioning that the method is capable of reducing the volume of the computational work as compared to the classical methods while still maintaining the high accuracy of the result, and the size reduction amounts to an improvement of the performance of the approach. The HPTM can be applied for some other engineering system with less computational work.

Acknowledgments

The authors express our thanks to the referees for their fruitful advices and comments. This paper is supported by the National Science Foundation of Shandong Province (Grant no. Y2007A06 and ZR2010Al019) and the China Postdoctoral Science Foundation (Grant no. 20100470783.)