Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method

Abstract

The homotopy analysis method (HAM) is applied to solve linear and nonlinear fractional partial differential equations (fPDEs). The fractional derivatives are described by Caputo's sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demonstrate the capability of HAM for nonlinear fPDEs.

1. Introduction

Fractional calculus has been given considerable popularity and importance during the past three decades, due mainly to its applications in numerous fields of science and engineering. For example, phenomena in the areas of fluid flow, rheology, electrical networks, probability and statistics, control theory of dynamical systems, electrochemistry of corrosion, chemical physics, optics and signal processing, and so on can be successfully modelled by linear or nonlinear fractional differential equations (fDEs) [1–4].

Finding accurate methods for solving nonlinear differential equations has become important. Some of the analytical methods for nonlinear differential equations are the Adomian decomposition method (ADM) [5–14], the homotopy-perturbation method (HPM) [15–19], variational iteration method (VIM) [12, 20–24], and the EXP-function method [25]. Another analytical approach that can be applied to solve nonlinear differential equations is to employ the homotopy analysis method (HAM) [26–29]. Some of the recent applications of HAM can be found in [30–41]. An account of the recent developments of HAM was given by Liao [42]. HAM has been successfully applied into engineering fields. The method has been applied to give an explicit solution for the Riemann problem of the nonlinear shallow-water equations [43]. The obtained Riemann solver has been implemented into a numerical model to simulate long waves, such as storm surge or tsunami, propagation and run-up.

Very recently, Song and Zhang [44] applied HAM to solve fractional KdV-Burgers-Kuramoto equation. Cang et al. [45] solved nonlinear Riccati differential equations of fractional order using HAM. Hashim et al. [46] employed HAM to solve fractional initial value problems (fIVPs) for ordinary differential equations. In [47], the applicability of the HAM was extended to construct numerical solution for the fractional BBM-Burgers equation. The HAM solutions for systems of nonlinear fractional differential equations were presented by Bataineh et al. [48].

A specific linear, nonhomogeneous time fractional partial differential equation (fPDE) with variable coefficients was first transformed to two fractional ordinary differential equations which were then solved by HAM in [49]. Recently, Xu et al. [50] applied the HAM to linear, homogeneous one- and two-dimensional fractional heat-like PDEs subject to the Neumann boundary conditions. Jafari and Seifi [51] applied HAM to linear and nonlinear homogeneous fractional diffusion-wave equations. Very recently, the HAM was shown to be capable of solving linear and nonlinear systems of fPDEs [52].

In this paper, we shall consider linear and nonlinear fPDEs of the form𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=𝑓𝑢,𝑢_𝑥,𝑢_𝑥𝑥,𝑛−1<𝛼≤𝑛,𝑡>0, (1.1)subject to the initial conditions𝑢^(𝑘)(𝑥,0)=𝑔_𝑘(𝑥),𝑘=0,1,2,…,𝑛−1, (1.2)where 𝑛 is an integer, 𝑓 is a linear/nonlinear function, and 𝐷^𝛼_𝑡(⋅)=𝜕^𝛼(⋅)/𝜕𝑡^𝛼 is a fractional differential operator. We shall demonstrate the applicability of HAM to fPDEs through several linear and nonlinear test examples.

2. Preliminaries

The fractional derivative is defined in the Caputo sense as in [53],𝐷^𝛼𝑤(𝑡)=𝐽^𝑛−𝛼𝐷^𝑛𝑤(𝑡). (2.1)Here 𝐷^𝑛 is the usual integer differential operator of order 𝑚 and 𝐽^𝛽 is the Riemann-Liouville fractional integral operator of order 𝛽>0, defined by𝐽^𝛽1𝑤(𝑡)=Γ(𝛽)^𝑡₀(𝑡−𝜏)^𝛽−1𝐽𝑤(𝜏)d𝜏,(𝑡>0),₀𝑤(𝑡)=𝑤(𝑡). (2.2) Some of the properties of the operator 𝐽^𝛽, which we will need in our work, are as follows [2, 3]:(1)𝐽^𝛽𝐽^𝜇𝑤(𝑡)=𝐽^𝛽+𝜇𝑤(𝑡),(2)𝐽^𝛽𝐽^𝜇𝑤(𝑡)=𝐽^𝜇𝐽^𝛽𝑤(𝑡),(3)𝐽^𝛽𝑡^𝛾=(Γ(𝛾+1)/Γ(𝛽+𝛾+1))𝑡^𝛽+𝛾. Caputo’s fractional derivative has a useful property [54]𝐽^𝜇𝐷^𝜇𝑤(𝑡)=𝑤(𝑡)−^𝑛−1_𝑘=0𝑤^(𝑘)0⁺𝑡^𝑘𝑘!,(𝑛−1<𝜇≤𝑛). (2.3)The operator form of the nonlinear fPDEs (1.1) can be written as follows:𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=𝐴𝑢,𝑢_𝑥,𝑢_𝑥𝑥+𝐵𝑢,𝑢_𝑥,𝑢_𝑥𝑥+𝐶(𝑥,𝑡),𝑛−1<𝛼≤𝑛,𝑡>0, (2.4)subject to the initial conditions𝑢^(𝑘)(𝑥,0)=𝑔_𝑘(𝑥),𝑘=0,1,2,…,𝑛−1, (2.5)where 𝐴 is a linear operator which might include other fractional derivatives of order less that 𝛼, 𝐵 is a nonlinear operator which also might include other fractional derivatives of order less that 𝛼 and 𝐶 is a known analytic function.

Applying the operator 𝐽^𝛼, the inverse operator of 𝐷^𝛼, to both sides of (2.4) with considering the initial conditions (2.5) according to (2.3), we obtain𝑢(𝑥,𝑡)=^𝑛−1_𝑘=0𝑔_𝑘𝑡(𝑥)^𝑘𝑘!+𝐽^𝛼𝐶(𝑥,𝑡)+𝐽^𝛼𝐴𝑢,𝑢_𝑥,𝑢_𝑥𝑥+𝐽^𝛼𝐵𝑢,𝑢_𝑥,𝑢_𝑥𝑥,𝑛−1<𝛼≤𝑛,𝑡>0. (2.6)

3. Homotopy Analysis Method (HAM)

3.1. The Zeroth-Order Deformation Equation

Let ℒ denote an auxiliary linear operator, 𝑢₀(𝑥,𝑡) is an initial approximation of 𝑢(𝑥,𝑡) which satisfies the initial conditions (2.5). Note that, in this paper, the auxiliary linear operator ℒ is not the same linear operator 𝐴 of (2.4).

Note that the original equation (1.1) contains the linear operator 𝐷^𝛼_𝑡. So, it is straightforward for us to choose the auxiliary linear operatorℒ(𝜙)=𝐷^𝛼_𝑡(𝜙). (3.1)According to (2.6), we can choose the initial approximation to be𝑢₀(𝑥,𝑡)=^𝑚−1_𝑘=0𝑔_𝑘𝑡(𝑥)^𝑘𝑘!+𝐽^𝛼𝐶(𝑥,𝑡). (3.2)

For simplicity, let us define, according to (2.4), the nonlinear operator𝒩(𝜙)=𝐷^𝛼_𝑡(𝜙)−𝐴𝜙,𝜙_𝑥,𝜙_𝑥𝑥−𝐵𝜙,𝜙_𝑥,𝜙_𝑥𝑥−𝐶(𝑥,𝑡). (3.3)Hence, in the frame of HAM [29], we can construct the so-called zeroth-order deformation(1−𝑞)ℒ𝑈(𝑥,𝑡;𝑞)−𝑢₀(𝑥,𝑡)=𝑞ℏ𝒩𝑈(𝑥,𝑡;𝑞), (3.4)subject to the following initial conditions:𝑈^(𝑘)(𝑥,0;𝑞)=𝑔_𝑘(𝑥),𝑘=0,1,2,…,𝑚−1, (3.5)where 𝑞∈[0,1] is the embedding parameter, ℏ≠0 is an auxiliary parameter, and 𝑈(𝑥,𝑡;𝑞) is an unknown function on the independent variables 𝑥,𝑡, and 𝑞.

When 𝑞=0, since 𝑢₀(𝑥,𝑡) satisfies all the initial conditions (2.5), and 𝜙=0 is a solution of ℒ𝜙=0, we have obviously𝑈(𝑥,𝑡;0)=𝑢₀(𝑥,𝑡), (3.6)and when 𝑞=1, the zeroth-order deformation equations (3.4) and (3.5) are equivalent to the original equations (2.4) and (2.5), provided𝑈(𝑥,𝑡;1)=𝑢(𝑥,𝑡). (3.7)Using the parameter 𝑞, we expand 𝑈(𝑥,𝑡;𝑞) in Taylor series as follows:𝑈(𝑥,𝑡;𝑞)=𝑢₀(𝑥,𝑡)+^∞_𝑚=1𝑢_𝑚(𝑥,𝑡)𝑞^𝑚, (3.8)where𝑢_𝑚1(𝑥,𝑡)=𝜕𝑚!^𝑚𝑈(𝑥,𝑡;𝑞)𝜕^𝑚𝑞_𝑞=0. (3.9)

Assume that the auxiliary linear operator ℒ, the initial guess 𝑢₀ and the auxiliary parameter ℏ are properly chosen such that the series (3.8) is convergent at 𝑞=1. Thus, due to (3.7) we have𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+^∞_𝑚=1𝑢_𝑚(𝑥,𝑡). (3.10)

3.2. The 𝑚th-Order Deformation Equation

Let us define the vector^→𝑢_𝑛=𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),…,𝑢_𝑛(𝑥,𝑡). (3.11)Following Liao [26–29], differentiating (3.4) 𝑚 times with respect to the embedding parameter 𝑞, then setting 𝑞=0, and finally dividing them by 𝑚!, we have the so-called 𝑚th-order deformation equationℒ𝑢_𝑚(𝑥,𝑡)−𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)=ℏℛ_𝑚^→𝑢_𝑚−1, (3.12)subject to the initial conditions𝑢_𝑚^(𝑘)(𝑥,0)=0,𝑘=0,1,2,…,𝑚−1, (3.13)whereℛ_𝑚^→𝑢_𝑚−1=1𝜕(𝑚−1)!^𝑚−1𝒩𝑈(𝑥,𝑡;𝑞)𝜕^𝑚−1𝑞_𝑞=0,𝜒_𝑚=0,𝑚≤1,1,𝑚>1. (3.14)(3.15)Substituting (3.3) into (3.14), and since 𝐴 is a linear operator, ℛ_𝑚(^→𝑢_𝑚−1) can be given byℛ_𝑚^→𝑢_𝑚−1=𝐷^𝛼_𝑡𝑢_𝑚−1−𝐴𝑢_(𝑚−1),𝑢_{(𝑚−1)𝑥},𝑢_{(𝑚−1)𝑥𝑥}−1𝜕(𝑚−1)!^𝑚−1𝐵𝑈,𝑈_𝑥,𝑈_𝑥𝑥𝜕^𝑚−1𝑞_𝑞=0−1−𝜒_𝑚𝐶(𝑥,𝑡). (3.16)

According to (3.1), we can apply the operator 𝐽^𝛼 to both sides of (3.12) to obtain𝐽^𝛼𝐷^𝛼𝑢_𝑚(𝑥,𝑡)−𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)=ℏ𝐽^𝛼ℛ_𝑚^→𝑢_𝑚−1. (3.17)Using the property (2.3) and the initial conditions (1.1), we have𝑢_𝑚(𝑥,𝑡)=𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)+ℏ𝐽^𝛼ℛ_𝑚^→𝑢_𝑚−1. (3.18)

Finally, for the purpose of computation, we will approximate the HAM solution (3.10) by the following truncated series:𝜙_𝑚(𝑥,𝑡)=^𝑚−1_𝑘=0𝑢_𝑘(𝑥,𝑡). (3.19)

3.3. Convergence Theorem

Theorem 3.1. As long as the series 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+∑^+∞_𝑚=1𝑢_𝑚(𝑥,𝑡) converges, where 𝑢_𝑚(𝑥,𝑡) is governed by (3.12) under the definitions (3.14) and (3.15), it must be a solution of (2.4).

Proof. If the series ∑^+∞_𝑚=0𝑢_𝑚(𝑥,𝑡) is convergent, we can write𝑆(𝑥,𝑡)=^+∞_𝑚=0𝑢_𝑚(𝑥,𝑡), (3.20)and it holdslim_𝑛→+∞𝑢_𝑛(𝑥,𝑡)=0. (3.21)From (3.12) and by using (3.15), it yieldsℏ^+∞_𝑚=1ℛ_𝑚^→𝑢_𝑚−1=^+∞_𝑚=1ℒ𝑢_𝑚(𝑥,𝑡)−𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)=lim^𝑛_{𝑛→+∞𝑚=1}ℒ𝑢_𝑚(𝑥,𝑡)−𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)=ℒlim^𝑛_{𝑛→+∞𝑚=1}𝑢_𝑚(𝑥,𝑡)−𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)=ℒlim_𝑛→+∞𝑢_𝑛(𝑥,𝑡)=0. (3.22)Since ℏ≠0, then^+∞_𝑚=1ℛ_𝑚^→𝑢_𝑚−1=0. (3.23)Substituting (3.16) into the above equation and simplifying it, due to the convergence of the series 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+∑^+∞_𝑚=1𝑢_𝑚(𝑥,𝑡) and since 𝐴 is a linear operator, yield^+∞_𝑚=1ℛ_𝑚^→𝑢_𝑚−1=^+∞_𝑚=1𝐷^𝛼_𝑡𝑢_𝑚−1−𝐴𝑢_(𝑚−1),𝑢_{(𝑚−1)𝑥},𝑢_{(𝑚−1)𝑥𝑥}−^+∞_𝑚=11𝜕(𝑚−1)!^𝑚−1𝐵𝑈,𝑈_𝑥,𝑈_𝑥𝑥𝜕^𝑚−1𝑞_𝑞=0−1−𝜒_𝑚𝐶(𝑥,𝑡)=𝐷^𝛼_𝑡^+∞_𝑚=0𝑢_𝑚−𝐴^+∞_𝑚=0𝑢_𝑚,^+∞_𝑚=0𝑢_𝑚𝑥,^+∞_𝑚=0𝑢_𝑚𝑥𝑥−^+∞_𝑚=01𝜕𝑚!^𝑚𝐵𝑈,𝑈_𝑥,𝑈_𝑥𝑥𝜕^𝑚𝑞_𝑞=0−𝐶(𝑥,𝑡). (3.24)Now, expanding the nonlinear term 𝐵(𝑈(𝑥,𝑡;𝑞),𝑈_𝑥(𝑥,𝑡;𝑞),𝑈_𝑥𝑥(𝑥,𝑡;𝑞)) by using the general Taylor theorem at 𝑞=0 yields𝐵𝑈(𝑥,𝑡;𝑞),𝑈_𝑥(𝑥,𝑡;𝑞),𝑈_𝑥𝑥(𝑥,𝑡;𝑞)=^+∞_𝑚=01𝜕𝑚!^𝑚𝐵𝑈,𝑈_𝑥,𝑈_𝑥𝑥𝜕^𝑚𝑞_𝑞=0𝑞^𝑚. (3.25)Setting 𝑞=1 in the above equation and using (3.8), we obtain𝐵^+∞_𝑚=0𝑢_𝑚,^+∞_𝑚=0𝑢_𝑚𝑥,^+∞_𝑚=0𝑢_𝑚𝑥𝑥=^+∞_𝑚=01𝜕𝑚!^𝑚𝐵𝑈,𝑈_𝑥,𝑈_𝑥𝑥𝜕^𝑚𝑞_𝑞=0. (3.26)Then^+∞_𝑚=1ℛ_𝑚^→𝑢_𝑚−1=𝐷^𝛼_𝑡^+∞_𝑚=0𝑢_𝑚−𝐴^+∞_𝑚=0𝑢_𝑚,^+∞_𝑚=0𝑢_𝑚𝑥,^+∞_𝑚=0𝑢_𝑚𝑥𝑥−𝐵^+∞_𝑚=0𝑢_𝑚,^+∞_𝑚=0𝑢_𝑚𝑥,^+∞_𝑚=0𝑢_𝑚𝑥𝑥−𝐶(𝑥,𝑡)=𝐷^𝛼_𝑡𝑆(𝑥,𝑡)−𝐴𝑆(𝑥,𝑡),𝑆_𝑥(𝑥,𝑡),𝑆_𝑥𝑥(𝑥,𝑡)−𝐵𝑆(𝑥,𝑡),𝑆_𝑥(𝑥,𝑡),𝑆_𝑥𝑥(𝑥,𝑡)−𝐶(𝑥,𝑡). (3.27)From the initial conditions (3.5) and (3.13), it holds that𝑆^(𝑘)(𝑥,0)=^+∞_𝑚=0𝑢_𝑚^(𝑘)(𝑥,0)=𝑢₀^(𝑘)(𝑥,0)+^+∞_𝑚=1𝑢_𝑚^(𝑘)(𝑥,0)=𝑔_𝑘(𝑥). (3.28)
Thus, 𝑆(𝑥,𝑡) is satisfied and also must be the exact solution for (2.4).

4. Test Examples

In this section, we shall illustrate the applicability of HAM to several linear and nonlinear fPDEs.

4.1. Problem 1

Let us consider the following linear time-fractional wave-like equations:𝐷^𝛼_𝑡1𝑢(𝑥,𝑡)=2𝑥²𝑢_𝑥𝑥(𝑥,𝑡),𝑡>0,𝑥∈𝑅,1<𝛼≤2,𝑢(𝑥,0)=𝑥,𝑢_𝑡(𝑥,0)=𝑥². (4.1)(4.2)We note that the heat-like counterpart of (4.1) was solved by HAM in [50] without direct comparison with the result by the ADM. According to (3.2), we can choose the initial guess to be𝑢₀(𝑥,𝑡)=𝑥+𝑥²𝑡. (4.3)From (3.18), we have𝑢_𝑚=𝜒_𝑚+ℏ𝑢_𝑚−1−ℏ2𝑥²𝐽^𝛼𝑢_{𝑚−1𝑥𝑥}. (4.4)Consequently, the first few terms of HAM series solutions are as follows:𝑢₁𝑡(𝑥,𝑡)=−ℏ^𝛼+1𝑥Γ(𝛼+2)²,𝑢₂𝑡(𝑥,𝑡)=−ℏ(ℏ+1)^𝛼+1𝑥Γ(𝛼+2)²+ℏ²𝑡^2𝛼+1𝑥Γ(2𝛼+2)²,𝑢₃(𝑥,𝑡)=−ℏ(ℏ+1)²𝑡^𝛼+1𝑥Γ(𝛼+2)²+2ℏ²𝑡(ℏ+1)^2𝛼+1𝑥Γ(2𝛼+2)²−ℏ³𝑡^3𝛼+1𝑥Γ(3𝛼+2)², (4.5)and so on. Hence, the HAM series solution is𝑢(𝑥,𝑡)=𝑢₀+𝑢₁+𝑢₂+𝑢₃+⋯=𝑥+𝑥²𝑡−ℏ1+(ℏ+1)+(ℏ+1)²𝑡+⋯^𝛼+1𝑥Γ(𝛼+2)²+ℏ²𝑡1+2(ℏ+1)+⋯^2𝛼+1𝑥Γ(2𝛼+2)²−ℏ³𝑡^3𝛼+1𝑥Γ(3𝛼+2)²+⋯. (4.6)Since we choose the initial guess 𝑢₀(𝑥,𝑡) to be the same initial guess used by ADM [12], we can notice that when ℏ=−1, the above expression gives the same solution given by ADM. Table 1 shows the HAM approximation solutions for (4.1)-(4.2) when 𝛼=1.5, 1.75, and 2 with ℏ=−1 and −1.0453. It is to be noted that the first four terms of the HAM series were used to evaluate the approximate solutions in Table 1.

Table 1 

Approximate solution of (4.1) for some values of 𝛼 using the 4-term HAM approximation, 𝜙₄, with ℏ=−1 and ℏ=−1.0453.

4.2. Problem 2

In this example, we consider the following one-dimensional linear inhomogeneous time-fractional equation:𝐷^𝛼𝑢(𝑥,𝑡)+𝑥𝑢_𝑥(𝑥,𝑡)+𝑢_𝑥𝑥(𝑥,𝑡)=2𝑡+2𝑥²+2,𝑡>0,𝑥∈𝑅,0<𝛼≤1, (4.7)subject to the initial condition𝑢(𝑥,0)=𝑥². (4.8)In Section 3, we chose the initial guess to contain the initial conditions and the source term 𝐶(𝑥,𝑡). In this example, due to the appearance of noise terms and also to get the exact solution, we will modify the way we choose the initial guess. The initial guess is set to contain only the initial condition (4.8), and the source term, 𝐶(𝑥,𝑡)=2𝑡+2𝑥²+2, will be added to 𝑢₁(𝑥,𝑡). The other terms are obtained the same as described in Section 3.

Hence, the initial guess is given by𝑢₀(𝑥,𝑡)=𝑥², (4.9)and according to (3.18), we have𝑢₁=𝐽^𝛼2𝑡+2𝑥²+2+ℏ𝐽^𝛼𝑥𝑢_0𝑥+𝑢_0𝑥𝑥,𝑢_𝑚=𝜒_𝑚+ℏ𝑢_𝑚−1+ℏ𝐽^𝛼𝑥𝑢_{𝑚−1𝑥𝑥}+𝑢_{𝑚−1𝑥𝑥},𝑚≥2. (4.10)The terms of the HAM solution series can be given by𝑢₁(𝑥,𝑡)=2𝑡^𝛼+1𝑥Γ(𝛼+2)+2(ℏ+1)²𝑡+1^𝛼,𝑢Γ(𝛼+1)₂𝑡(𝑥,𝑡)=2(ℏ+1)^𝛼+1𝑥Γ(𝛼+2)+2(ℏ+1)²+1(ℏ+1)𝑡^𝛼+Γ(𝛼+1)2ℏ𝑡^2𝛼,Γ(2𝛼+1) (4.11)and so on. Hence, the HAM series solution is𝑢(𝑥,𝑡)=𝑢₀+𝑢₁+𝑢₂+⋯=𝑥²𝑡+21+(ℏ+1)+⋯^𝛼+1𝑥Γ(𝛼+2)+2(ℏ+1)²𝑡+11+(ℏ+1)+⋯^𝛼𝑥Γ(𝛼+1)+4ℏ(ℏ+1)²𝑡+1^2𝛼+1𝑥Γ(2𝛼+1)²+⋯. (4.12)Taking ℏ=−1 in (4.12), we obtain the exact solution,𝑢(𝑥,𝑡)=𝑥²+2𝑡^𝛼.Γ(𝛼+2) (4.13)

4.3. Problem 3

Consider the following nonlinear time-fractional hyperbolic equation:𝐷^𝛼_𝑡𝜕𝑢(𝑥,𝑡)=𝜕𝑥𝑢(𝑥,𝑡)𝜕𝑢(𝑥,𝑡)𝜕𝑥,𝑡>0,𝑥∈𝑅,1<𝛼≤2, (4.14)subject to the initial conditions𝑢(𝑥,0)=𝑥²,𝑢_𝑡(𝑥,0)=−2𝑥². (4.15)Equation (4.14) can be rewritten as follows:𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=𝜕𝑢(𝑥,𝑡)𝜕𝑥²𝜕+𝑢(𝑥,𝑡)²𝑢(𝑥,𝑡)𝜕𝑥². (4.16)

From (3.4), construct the following zeroth-order deformation:(1−𝑞)ℒ𝑈(𝑥,𝑡;𝑞)−𝑢₀(𝑥,𝑡)=𝑞ℏ𝒩𝑈(𝑥,𝑡;𝑞), (4.17)subject to the following initial conditions:𝑈(𝑥,0)=𝑥²,𝑈_𝑡(𝑥,0)=−2𝑥², (4.18)where𝒩(𝜃)=𝐷^𝛼_𝑡𝜃−𝜃²_𝑥−𝜃𝜃_𝑥𝑥. (4.19)The auxiliary linear operator can be chosen as follows:ℒ(𝜃)=𝐷^𝛼_𝑡(𝜃), (4.20)with the propertyℒ(𝜃)=0when𝜃=0, (4.21)while, the initial guess is𝑢₀(𝑥,𝑡)=𝑥²−2𝑥²𝑡. (4.22)Again from (3.12), the high-order deformation equation can be given byℒ𝑢_𝑚(𝑥,𝑡)−𝜒_𝑚𝑢_𝑚−1(𝑥,𝑡)=ℏℛ_𝑚^→𝑢_𝑚−1, (4.23)subject to the initial conditions𝑢_𝑚^(𝑘)(𝑥,0)=0,𝑘=0,1, (4.24)whereℛ_𝑚^→𝑢_𝑚−1=1𝜕(𝑚−1)!^𝑚−1𝒩𝑈(𝑥,𝑡;𝑞)𝜕^𝑚−1𝑞_𝑞=0. (4.25)Then, ℛ_𝑚(^→𝑢_𝑚−1) can be given byℛ_𝑚^→𝑢_𝑚−1=𝐷^𝛼_𝑡𝑢_𝑚−1−^𝑚−1_𝑖=0𝑢_𝑖𝑥𝑢_{(𝑚−1−𝑖)𝑥}−^𝑚−1_𝑖=0𝑢_𝑖𝑢_{(𝑚−1−𝑖)𝑥𝑥}. (4.26)Accordingly, the governing equation is as follows:𝑢_𝑚=𝜒_𝑚𝑢_𝑚−1+ℏ𝐽^𝛼𝐷^𝛼_𝑡𝑢_𝑚−1−^𝑚−1_𝑖=0𝑢_𝑖𝑥𝑢_{(𝑚−1−𝑖)𝑥}−ℏ𝐽^𝛼𝑚−1_𝑖=0𝑢_𝑖𝑢_{(𝑚−1−𝑖)𝑥𝑥},𝑚≥1. (4.27)

Consequently, the first few terms of HAM series solutions are given by𝑢₁(𝑥,𝑡)=−6ℏ𝑥²𝑡^𝛼𝑡Γ(𝛼+1)−4^𝛼+1𝑡Γ(𝛼+2)+8^𝛼+2,𝑢Γ(𝛼+3)₂(𝑥,𝑡)=−6ℏ𝑥²𝑡(ℏ+1)^𝛼𝑡Γ(𝛼+1)−4(ℏ+1)^𝛼+1𝑡Γ(𝛼+2)+8(ℏ+1)^𝛼+2Γ(𝛼+3)+72ℏ²𝑥²𝑡^2𝛼Γ(2𝛼+1)−2Γ(𝛼+2)𝑡Γ(𝛼+1)+2^2𝛼+1Γ(2𝛼+2)+576ℏ²𝑥²Γ(𝛼+3)𝑡Γ(𝛼+2)+1^2𝛼+2𝑡Γ(2𝛼+3)+2Γ(𝛼+4)^2𝛼+3,Γ(2𝛼+4) (4.28)and so on. Hence, the HAM series solution is𝑢(𝑥,𝑡)=𝑢₀+𝑢₁+𝑢₂+𝑢₃+⋯=𝑥²−2𝑥²𝑡−6ℏ𝑥²𝑡1+(ℏ+1)+⋯^𝛼Γ(𝛼+1)+24ℏ𝑥²𝑡1+(ℏ+1)+⋯^𝛼+1Γ(𝛼+2)−48ℏ𝑥²𝑡1+(ℏ+1)+⋯^𝛼+2Γ(𝛼+3)+72ℎℏ²𝑥²𝑡^2𝛼Γ(2𝛼+1)+⋯. (4.29)The four-term HAM approximate solutions for (4.14)-(4.15), when 𝛼=1.5, 1.75, and 2 with ℏ=−1 and −1.0966, are shown in Table 2. Notice that the HAM approximate solution when 𝛼=2 with ℏ=−1.0966 is in good agreement with the exact solution, 𝑢(𝑥,𝑡)=(𝑥/(𝑡+1))².

Table 2 

Approximate solution of (4.14) for some values of 𝛼 using the 4-term HAM approximation, 𝜙₄, with ℏ=−1 and ℏ=−1.0966.

4.4. Problem 4

Consider the following nonlinear time-fractional Fisher’s equation:𝐷^𝛼𝑢(𝑥,𝑡)=𝑢_𝑥𝑥(𝑥,𝑡)+6𝑢(𝑥,𝑡)1−𝑢(𝑥,𝑡), (4.30)for 𝑡>0,𝑥∈𝑅,0<𝛼≤1, subject to the initial condition1𝑢(𝑥,0)=1+e^𝑥2. (4.31)According to (3.2), we can choose the initial guess to be𝑢₀1(𝑥,𝑡)=1+e^𝑥2, (4.32)and according to (3.18), we have𝑢_𝑚=𝜒_𝑚+ℏ𝑢_𝑚−1−ℏ𝐽^𝛼𝑢_{𝑚−1𝑥𝑥}+6𝑢_𝑚−1−6^𝑚−1_𝑖=0𝑢_𝑖𝑢_{𝑚−1−𝑖}. (4.33)

Consequently, the first few terms of HAM series solutions are as follows:𝑢₁(𝑥,𝑡)=−10ℏe^𝑥1+e^𝑥3𝑡^𝛼,𝑢Γ(𝛼+1)₂(𝑥,𝑡)=−10ℏe^𝑥1+e^𝑥3𝑡(ℏ+1)^𝛼Γ(𝛼+1)−5ℏ2e^𝑥−11+e^𝑥𝑡^2𝛼,Γ(2𝛼+1) (4.34)and so on. Hence, the HAM series solution is𝑢(𝑥,𝑡)=𝑢₀+𝑢₁+𝑢₂=1+⋯1+e^𝑥2−10ℏe^𝑥1+e^𝑥3𝑡1+(ℏ+1)+⋯^𝛼Γ(𝛼+1)+50ℏ²e^𝑥2e^𝑥−11+e^𝑥4𝑡^2𝛼Γ(2𝛼+1)+⋯. (4.35)Table 3 shows the 3-term HAM approximate solutions for (4.30)-(4.31), 𝜙₃, when 𝛼=0.5, 0.75, and 1 with ℏ=−1 and −1.05133. We notice that the HAM approximate solution when 𝛼=2 with ℏ=−1.05133 is in good agreement with the exact solution, 𝑢(𝑥,𝑡)=1/(1+exp(𝑥−5𝑡))².

Table 3 

Approximate solution of (4.30) for some values of 𝛼 using the 3-term HAM approximation, 𝜙₃, with ℏ=−1 and −1.05133.

5. Conclusions

In this work, the homotopy analysis method (HAM) was implemented to derive exact and approximate analytical solutions for both linear and nonlinear partial differential equations of fractional order. The convergence region of the series solution obtained by HAM can be controlled and adjusted by the auxiliary parameter ℏ. We give some examples to show the efficiency and accuracy of the suggested method. It was also demonstrated that the Adomian decomposition method (ADM) is a special case of HAM for the first and second test examples.

Acknowledgment

The financial support received from the Academy of Sciences Malaysia under tSAGA Grant no. P24c (STGL-011-2006) is gratefully acknowledged.