Variational Homotopy Perturbation Method for Solving Fractional Initial Boundary Value Problems

and Applied Analysis 3 and the nonlinear term can be decomposed as


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 semianalytical 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.

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: where g x, t is the inhomogeneous term, N represents the general nonlinear differential operator, R is the linear differential operator, and D α t u x, t is the Caputo fractional derivative of function u x, t which is defined as 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: 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 and λ −1 and for m 2 and λ ξ − t. The subscripts n denote the nth approximation, and u n is considered as a restricted variation. That is, u n δ 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 u 0 ; 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: Abstract and Applied Analysis 3 and the nonlinear term can be decomposed as where p ∈ 0, 1 is an embedding parameter. H n u is He's polynomials 26 that can be generated by 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

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.

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 : where 0 < α ≤ 1, and the correct functional is given as Applying the modified variational iteration method, one has ∞ n 0 3.4 Comparing the coefficient of like powers of p, . . .

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:

3.6
The solution of 3.1 in series form is given by The series solution in a closed form is given by u x, t x 2 e t , which was given in 24 .
Abstract and Applied Analysis 5 Case 2. Consider the following homogeneous fractional coupled Burger's equation 26 : where 0 < α, β ≤ 1. The correction functional for the previous fractional coupled system is given by Applying the variational homotopy perturbation method using He's polynomials, we get

3.14
Comparing the coefficient of like powders of p, one has

3.15
If we take α β 1, the series solutions in closed-form are given by 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.
Case 3. In this case, we consider the space fractional backward Kolmogorov equation as follows 27 : subject to the initial condition: where 0 < β ≤ 1. Space fractional derivative is also the Caputo definition with respect to x and is defined as

3.19
Abstract and Applied Analysis The correction functional for the previous space fractional system is given by 3.20 Applying the modified variational iteration method, one has ∞ n 0 x 2 e ξ ∂ 2β ∞ n 0 p n u n x, t ∂x 2β dξ.

8 Abstract and Applied Analysis
Comparing the coefficient of like powders of P , we can obtain the following approximations: u 0 x 1,

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 is u 0 x 1,

3.23
and so on. Hence, we have the closed form u x, t x 1 1 t t 2 2! t 3 3! · · · exp t x 1 , 3.24 which is the exact solution of the corresponding integer problem.

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.