Variational Homotopy Perturbation Method for Solving Fractional Initial Boundary Value Problems

Abstract

A variational homotopy perturbation method (VHPM) which is based on variational iteration method and homotopy perturbation method is applied to solve the approximate solution of the fractional initial boundary value problems. The nonlinear terms can be easily handled by the use of He's polynomials. It is observed that the variational iteration method is very efficient and easier to implements; illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm.

1. Introduction

Recently, it has turned out that many phenomena in engineering, physics, chemistry, and other sciences [1–3] can be described very successfully by models using mathematical tools for fractional calculus. 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, some semi-analytical techniques have also largely been used to solve these equations, such as, Adomian decomposition method [8, 9], variational iteration method [10–12], differential transform method [13, 14], Laplace decomposition method [15, 16], and homotopy perturbation method [17–23]. Most of these methods have their inbuilt deficiencies like the calculation of Adomian polynomials, the Lagrange multiplier, divergent results, and huge computational work.

Variational homotopy perturbation method [24–26] has a very simple solution procedure and absorbs all of the positive features of variational iteration and homotopy perturbation methods and is highly compatible with the diversity of the physical problems. In this work, we will use variational homotopy perturbation method to solve fractional partial differential equations with initial and boundary conditions. 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 variational homotopy perturbation method in solving fractional partial equations.

2. Description of the VHPM

To illustrate the basic idea of this method [24, 25], we consider a general fractional nonlinear nonhomogeneous partial differential equation with initial conditions of the following form:𝐷^𝛼_𝑡𝑢()()()(),𝑢()(𝑥)𝑥,𝑡+𝑅𝑢𝑥,𝑡+𝑁𝑢𝑥,𝑡=𝑔𝑥,𝑡𝑥,0=ℎ,𝑢_𝑡(𝑥,0)=𝑓(𝑥), (2.1) where 𝑔(𝑥,𝑡) is the inhomogeneous term, 𝑁 represents the general nonlinear differential operator, 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 as₀𝐷^𝛼_𝑡𝑢()=1𝑥,𝑡Γ(𝑛−𝛼)^𝑡₀𝑢^(𝑛)(𝑥,𝜏)𝑑𝜏(𝑡−𝜏)^{𝛼+1−𝑛},(𝑛−1<Re(𝛼)≤𝑛,𝑛∈𝑁), (2.2) where Γ(⋅) denotes the Gamma function. The properties of fractional derivative can be found in [1, 2]. According to variational iteration method [10, 11], we can construct a correction functional as follows:𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡₀𝜆(𝜉)𝜕^𝑚𝑢_𝑛𝜕𝜉^𝑚(𝑥,𝜉)+𝑅𝑢_𝑛(𝑥,𝜉)+𝑁𝑢_𝑛(𝑥,𝜉)−𝑔(𝑥,𝜉)𝑑𝜉, (2.3) where the values of the natural number 𝑚 can be 1 and 2 corresponding to 0<𝛼≤1 and 1<𝛼≤2, respectively, and 𝜆 is a Lagrange multiplier [11], which can be identified optimally via variational iteration method, for 𝑚=1 and𝜆=−1 and for 𝑚=2 and𝜆=𝜉−𝑡. The subscripts 𝑛 denote the 𝑛th approximation, and 𝑢_𝑛 is considered as a restricted variation. That is, 𝑢_𝑛𝛿=0, (2.3) is called a correction functional. In this method, it is required first to determine the Lagrange multiplier 𝜆 optimally. The successive approximation 𝑢_𝑛+1,𝑛≥0, of the solution 𝑢 will be readily obtained upon using the determined Lagrange multiplier and any selective function 𝑢₀; consequently, the solution is given by 𝑢=lim_𝑛→∞𝑢_𝑛. In the homotopy perturbation method, the basic assumption is that the solutions can be written as a power series in 𝑝:𝑢(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+𝑝³𝑢₃+⋯, (2.4) and the nonlinear term can be decomposed as𝑁𝑢(𝑥,𝑡)=^∞_𝑛=0𝑝^𝑛ℋ_𝑛(𝑢), (2.5) where 𝑝∈[0,1] is an embedding parameter. ℋ_𝑛(𝑢) is He’s polynomials [26] that can be generated byℋ_𝑛𝑢₀,…,𝑢_𝑛=1𝜕𝑛!𝜕𝑝^𝑛𝑛𝑁^∞_𝑖=0𝑝^𝑖𝑢_𝑖𝑝=0,𝑛=0,1,2…. (2.6) The variational homotopy perturbation method is obtained by the elegant coupling of correction functional (2.3) of variational iteration method with He’s polynomials and is given by^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑝^𝑡₀𝜆(𝜉)^∞_𝑛=0𝑝^𝑛𝜕^𝛼𝑢_𝑛𝜕𝜉^𝛼(𝑥,𝜉)+^∞_𝑛=0𝑝^𝑛𝑅𝑢(𝑥,𝜉)+^∞_𝑛=0𝑝^𝑛ℋ_𝑛(𝑢)−𝑔(𝑥,𝜉)𝑑𝜉. (2.7) Comparisons of like powers of 𝑝 give solutions of various orders.

This method does not resort to linearization or assumptions of weak nonlinearity, the solutions 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 variational homotopy perturbation method for fractional equations, we have applied it to the following several problems.

Case 1. Consider the one-dimensional fractional initial boundary value problem which describes the heat-like models [24]: 𝐷^𝛼_𝑡𝑢()=1𝑥,𝑡2𝑥²𝑢_𝑥𝑥,0<𝑥<1,𝑡>0, (3.1) with boundary conditions 𝑢(0,𝑡)=0,𝑢(1,𝑡)=𝑒^𝑡,𝑢(𝑥,0)=𝑥², (3.2) where 0<𝛼≤1, and the correct functional is given as 𝑢_𝑛+1(𝑥,𝑡)=𝑥²−^𝑡₀𝜕^𝛼𝑢_𝑛𝜕𝜉^𝛼()−1𝑥,𝜉2𝑥²𝑢_𝑛𝑥𝑥𝑑𝜉. (3.3)

Applying the modified variational iteration method, one has^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)=𝑥²−𝑝^𝑡₀𝜕^𝛼𝑢₀𝜕𝜉^𝛼𝜕+𝑝^𝛼𝑢₁𝜕𝜉^𝛼+𝑝²𝜕^𝛼𝑢₂𝜕𝜉^𝛼−1+⋯2𝜕²𝑢₀𝜕𝑥²𝜕+𝑝²𝑢₁𝜕𝑥²+𝑝²𝜕²𝑢₂𝜕𝑥²+⋯𝑑𝜉. (3.4) Comparing the coefficient of like powers of 𝑝,𝑝⁰∶𝑢₀(𝑥,𝑡)=𝑥²,𝑝¹∶𝑢₁(𝑥,𝑡)=𝑥²(),𝑝1+𝑡₂∶𝑢₂(𝑥,𝑡)=𝑥²(1+𝑡)+𝑡𝑥²+𝑡²𝑥²2−𝑡^2−𝛼𝑥²Γ(),𝑝3−𝛼₃∶𝑢₂(𝑥,𝑡)=𝑥²+3𝑡𝑥²+3𝑡²𝑥²2+𝑡³𝑥²6+𝑡^3−2𝛼𝑥²Γ()−4−2𝛼3𝑡^2−𝛼𝑥²Γ()−3−𝛼2𝑡^3−𝛽𝑥²Γ,⋮4−𝛽 (3.5) and so on, in this manner the rest of component of the solution can be obtained. If we take 𝛼=1, the first few components of the solution of (3.1) are as follows:𝑢₀(𝑥,𝑡)=𝑥²,𝑢₁(𝑥,𝑡)=𝑥²(),𝑝1+𝑡₂∶𝑢₂(𝑥,𝑡)=𝑥²𝑡1+𝑡+²,𝑝2!₃∶𝑢₂(𝑥,𝑡)=𝑥²𝑡1+𝑡+²+𝑡2!³,⋮3! (3.6) The solution of (3.1) in series form is given by𝑢(𝑥,𝑡)=𝑥²𝑡1+𝑡+²+𝑡2!3³!+⋯. (3.7) The series solution in a closed form is given by 𝑢(𝑥,𝑡)=𝑥²𝑒^𝑡, which was given in [24].

Case 2. Consider the following homogeneous fractional coupled Burger’s equation [26]: 𝐷^𝛼_𝑡𝑢−𝑢_𝑥𝑥−2𝑢𝑢_𝑥+(𝑢𝑣)_𝑥𝐷=0,_𝛽𝑡𝑣−𝑣_𝑥𝑥−2𝑣𝑣_𝑥+(𝑢𝑣)_𝑥=0, (3.8)(3.9) with initial conditions 𝑢()𝑣()𝑥,0=sin𝑥,𝑥,0=sin𝑥, (3.10)(3.11) where 0<𝛼,𝛽≤1. The correction functional for the previous fractional coupled system is given by 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−^𝑡₀𝜕^𝛼𝑢_𝑛𝜕𝜉^𝛼−𝜕²𝑢_𝑛𝜕𝑥²−2𝑢_𝑛𝑢_𝑛𝑥+𝑢_𝑛𝑣_𝑛𝑥𝑣𝑑𝜉,_𝑛+1(𝑥,𝑡)=𝑣_𝑛(𝑥,𝑡)−^𝑡₀𝜕^𝛽𝑣_𝑛𝜕𝜉^𝛽−𝜕²𝑣_𝑛𝜕𝑥²−2𝑣_𝑛𝑣_𝑛𝑥+𝑢_𝑛𝑣_𝑛𝑥𝑑𝜉. (3.12)(3.13)

Applying the variational homotopy perturbation method using He’s polynomials, we get𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯=𝑢₀(𝑥,𝑡)−𝑝^𝑡₀𝜕^𝛼𝑢₀𝜕𝜉^𝛼𝜕+𝑝^𝛼𝑢₁𝜕𝜉^𝛼−𝜕+⋯²𝑢₀𝜕𝑥²𝜕+𝑝²𝑢₁𝜕𝑥²+⋯𝑑𝜉,+𝑝^𝑡₀𝑢₀𝑢_0𝑥+𝑝𝑢₀𝑢_1𝑥+𝑢₁𝑢_0𝑥+𝑝²𝑢₀𝑢_2𝑥+𝑢₁𝑢_1𝑥+𝑢₂𝑢_0𝑥−𝑢+⋯₀𝑣_0𝑥𝑥+𝑝𝑢₁𝑣₀+𝑢₀𝑣_1𝑥𝑥+𝑝²𝑢₀𝑣₂+𝑢₁𝑣₁+𝑢₀𝑣_2𝑥𝑥𝑣𝑑𝜉,₀+𝑝𝑣₁+𝑝²𝑣₂+⋯=𝑣₀(𝑥,𝑡)−𝑝^𝑡₀𝜕^𝛽𝑣₀𝜕𝜉^𝛽𝜕+𝑝^𝛽𝑣₁𝜕𝜉^𝛽−𝜕+⋯²𝑣₀𝜕𝑥²𝜕+𝑝²𝑣₁𝜕𝑥²+⋯𝑑𝜉+𝑝^𝑡₀𝑣₀𝑣_0𝑥+𝑝𝑣₀𝑣_1𝑥+𝑣₁𝑣_0𝑥+𝑝²𝑣₀𝑣_2𝑥+𝑣₁𝑣_1𝑥+𝑣₂𝑣_0𝑥−𝑢+⋯₀𝑣_0𝑥𝑥+𝑝𝑢₁𝑣₀+𝑢₀𝑣_1𝑥𝑥+𝑝²𝑢₀𝑣₂+𝑢₁𝑣₁+𝑢₀𝑣_2𝑥𝑥𝑑𝜉. (3.14) Comparing the coefficient of like powders of 𝑝, one has𝑢₀()𝑣𝑥,0=sin𝑥,₀()𝑢𝑥,0=sin𝑥,₁()𝑣𝑥,0=sin𝑥−𝑡sin𝑥,₁()𝑢𝑥,0=sin𝑥−𝑡sin𝑥,₂()𝑡𝑥,0=sin𝑥−2𝑡sin𝑥+²sin𝑥2+𝑡^2−𝛼sin𝑥Γ(),𝑣3−𝛼₂()𝑡𝑥,0=sin𝑥−2𝑡sin𝑥+²sin𝑥2+𝑡^2−𝛽sin𝑥Γ,𝑢3−𝛽₃(𝑥,0)=sin𝑥−3𝑡sin𝑥+3𝑡²sin𝑥2−𝑡³sin𝑥6+𝑡^3−2𝛼sin𝑥Γ()+4−2𝛼3𝑡^2−𝛼sin𝑥Γ()−3−𝛼2𝑡^3−𝛼sin𝑥Γ(),𝑣4−𝛼₃(𝑥,0)=sin𝑥−3𝑡sin𝑥+3𝑡²sin𝑥2−𝑡³sin𝑥6+𝑡^3−2𝛽sin𝑥Γ+4−2𝛽3𝑡^2−𝛽sin𝑥Γ−3−𝛽2𝑡^3−𝛽sin𝑥Γ.4−𝛽 (3.15) If we take 𝛼=𝛽=1, the series solutions in closed-form are given by𝑢()𝑡𝑥,𝑡=sin𝑥1−𝑡+²−𝑡2!³+𝑡3!⁴()𝑣()𝑡4!+⋯=exp−𝑡sin𝑥,𝑥,𝑡=sin𝑥1−𝑡+²−𝑡2!³+𝑡3!⁴()4!+⋯=exp−𝑡sin𝑥, (3.16) which is in full agreement with the result given in [26].

Figure 1 depicts the third-order approximate solutions 𝑢(𝑥,𝑡) or 𝑣(𝑥,𝑡) for (3.8)–(3.11) by using the variational homotopy perturbation method when choosing 𝑥=0.8. From the figure, it is clear to see the time evolution of fractional Burger equation and we know that the approximate solution of the model is continuous with the fractional parameter 𝛼. Figure 2 shows the third-order approximate solutions 𝑢(𝑥,𝑡) or 𝑣(𝑥,𝑡) for (3.8)–(3.11) when 𝑡=0.8, and we also know that the solution of the fractional nonlinear equation changes with the parameters 𝛼.

Figure 1 

The surface of third-order approximate solution 𝑢(𝑥,𝑡)of (3.8) and (3.9) when 𝑥=0.8.

Figure 2 

The surface of third-order approximate solution 𝑢(𝑥,𝑡) of (3.8) and (3.9) when 𝑡=0.8.

Case 3. In this case, we consider the space fractional backward Kolmogorov equation as follows [27]: 𝜕𝑢(𝑥,𝑡)=()𝜕𝜕𝑡𝑥+1^𝛽𝑢𝜕𝑥^𝛽+𝑥²𝑒^𝑡𝜕^2𝛽𝑢𝜕𝑥^2𝛽, (3.17) subject to the initial condition: 𝑢(𝑥,0)=𝑥+1, (3.18) where 0<𝛽≤1. Space fractional derivative is also the Caputo definition 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. (3.19) The correction functional for the previous space fractional system is given by 𝑢_𝑛+1=𝑢_𝑛−^𝑡₀𝜕𝑢_𝑛(𝑥,𝜉)−()𝜕𝜕𝜉𝑥+1^𝛽𝑢_𝑛𝜕𝑥^𝛽−𝑥²𝑒^𝜉𝜕^2𝛽𝑢_𝑛𝜕𝑥^2𝛽𝑑𝜉. (3.20)

Applying the modified variational iteration method, one has^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)=𝑥+1−^𝑡₀𝜕∑^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)−()𝜕𝜕𝜉𝑥+1^𝛽∑^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)𝜕𝑥^𝛽−𝑥²𝑒^𝜉𝜕^2𝛽∑^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)𝜕𝑥^2𝛽𝑑𝜉. (3.21) Comparing the coefficient of like powders of 𝑃, we can obtain the following approximations:𝑢₀𝑢=𝑥+1,₁=1+𝑥−2𝑥^3−2𝛽Γ+2−2𝛽2𝑒^𝑡𝑥^3−2𝛽Γ+2−2𝛽𝑡𝑥^1−𝛽Γ+2−𝛽𝑡𝑥^2−𝛽Γ,𝑢2−𝛽₂=1+𝑥+6𝑥^5−4𝛽Γ−4−4𝛽12𝑒^𝑡𝑥^5−4𝛽Γ+4−4𝛽6𝑒^𝑡𝑥^5−4𝛽Γ−4−4𝛽10𝑥^5−4𝛽𝛽Γ+4−4𝛽20𝑒^𝑡𝑥^5−4𝛽𝛽Γ−4−4𝛽10𝑒^2𝑡𝑥^5−4𝛽𝛽Γ+4−4𝛽4𝑥^5−4𝛽𝛽²Γ−4−4𝛽8𝑒^𝑡𝑥^5−4𝛽𝛽²Γ+4−4𝛽4𝑒^2𝑡𝑥^5−4𝛽𝛽²Γ+𝑥4−4𝛽^3−3𝛽𝛽Γ−𝑒2−3𝛽^𝑡𝑥^3−3𝛽Γ+𝑒2−3𝛽_𝑡𝑡𝑥^3−3𝛽Γ+2−3𝛽2𝑥^4−3𝛽𝛽Γ−3−3𝛽2𝑒^𝑡𝑥^4−3𝛽Γ+3−3𝛽2𝑒^𝑡𝑡𝑥^4−3𝛽Γ−𝑥3−3𝛽^4−3𝛽𝛽Γ+𝑒3−3𝛽^𝑡𝑥^4−3𝛽𝛽Γ+3−3𝛽𝑡𝑥^2−𝛽Γ−𝑒2−𝛽^𝑡𝑡𝑥^4−3𝛽𝛽Γ+𝑡3−3𝛽²𝑥^1−2𝛽+𝑡2Γ2−2𝛽²𝑥^2−2𝛽−𝑥2Γ2−2𝛽Γ^3−2𝛽+𝑒2−2𝛽^𝑡𝑥^3−2𝛽Γ+2−2𝛽𝑡𝑥^1−𝛽Γ+2−𝛽2𝑒^𝑡𝑥^3−3𝛽Γ4−2𝛽ΓΓ−4−3𝛽2−2𝛽2𝑥^3−3𝛽Γ4−2𝛽ΓΓ−4−3𝛽2−2𝛽2𝑡𝑥^3−3𝛽Γ4−2𝛽ΓΓ−4−3𝛽2−2𝛽2𝑥^4−3𝛽Γ4−2𝛽ΓΓ+4−3𝛽2−2𝛽2𝑒^𝑡𝑥^4−3𝛽Γ4−2𝛽ΓΓ−4−3𝛽2−2𝛽2𝑡𝑥^4−3𝛽Γ4−2𝛽ΓΓ+𝑡4−3𝛽2−2𝛽₂𝑥^2−2𝛽Γ3−𝛽+𝑡2Γ3−2𝛽²𝑥^3−2𝛽Γ3−𝛽,2Γ3−2𝛽 (3.22) and so on; in the same manner the rest of components of the iteration formula (3.21) can be obtained using the Mathematica package. When fractional derivatives are 𝛽=1, the exact solution of (3.17) was given in [27] using homotopy perturbation method and the approximate solution of (3.17) is𝑢₀𝑢=𝑥+1,₁=(),𝑢𝑥+1)(1+𝑡₂=()𝑡𝑥+11+𝑡+²,⋮2! (3.23) and so on. Hence, we have the closed form𝑢()=()𝑡𝑥,𝑡𝑥+11+𝑡+²+𝑡2!³(𝑡)3!+⋯=exp)(𝑥+1, (3.24) which is the exact solution of the corresponding integer problem.

4. Conclusion

In this work, a variational homotopy perturbation method which is based on homotopy perturbation method and variational iteration method is used to solve fractional 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 VHPM can be applied for some other engineering system with less computational work.

Acknowledgment

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