Legendre Polynomials Operational Matrix Method for Solving Fractional Partial Differential Equations with Variable Coefficients

A numerical method for solving a class of fractional partial differential equations with variable coefficients based on Legendre polynomials is proposed. A fractional order operational matrix of Legendre polynomials is also derived. The initial equations are transformed into the products of several matrixes by using the operational matrix. A system of linear equations is obtained by dispersing the coefficients and the products of matrixes. Only a small number of Legendre polynomials are needed to acquire a satisfactory result. Results obtained using the scheme presented here show that the numerical method is very effective and convenient for solving fractional partial differential equations with variable coefficients.


Introduction
The subject of factional calculus was found over 300 years ago.The theory of integrals and derivatives of noninteger order goes back to Leibnitz, Liouville, and Letnikov.In recent years, fractional derivative and fractional differential equations have played a very significant role in many areas in fluid flow, physics, mechanics, and other applications.A lot of practical problems can be elegantly modeled with the help of the fractional derivative [1][2][3][4][5].Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes.Due to the increasing applications, a lot of attention has been paid to numerical and exact solution of fractional differential equations and fractional partial equations.The analytical solutions of fractional differential equations are still in a preliminary stage.Except in a limited number of these equations, we have difficulty in seeking their analytical as well as numerical solutions.Thus there have been attempts to develop the methods for getting analytical and numerical solutions of fractional differential equations.Recently, some methods have drawn attention, such as Adomian decomposition method (ADM) [6,7], variational iteration method (VIM) [8], generalized differential transform method (GDTM) [9][10][11], finite difference method (FDM) [12], and wavelet method [13,14].
In this paper, our study focuses on a class of fractional partial differential equations as follows:
There have been several methods for solving the fractional partial differential equation.Doha et al. used Jacobi tau approximation to solve the numerical solution of the space fractional diffusion equation [15].Yi et al. [16] applied block pulse functions method to obtain the fractional partial equations.Podlubny [17] obtained the numerical solution of the fractional partial differential equations with constant coefficients by using Laplace transform method.

Definitions of Fractional Derivatives and Integrals
Definition 1. Riemann-Liouville fractional integral of order , ( ≥ 0) is defined as follows [17]: where is the gamma function.The Riemann-Liouville fractional integral satisfies the following properties: Definition 2. Caputo's fractional derivative of order , ( ≥ 0) is defined as follows [17]: Particularly, the operator   * satisfies the following properties ( is a constant): (6)
The numerical solutions for  = 4,  = 5 are displayed in Figures 4 and 5 and the exact solution is shown in Figure 6.
Example 3. Consider this equation: where 40Γ (2/3) .From Examples 1-3, we can see that the method in this paper can be effectively used to solve the numerical solution of fractional partial differential equation with variable coefficients.From the above results, the absolute errors between   the numerical solutions and the exact solution are rather small.What is more, due to the absolute error in this paper is about 10 −15 , the Legendre polynomials method can reach higher degree of accuracy by comparing the approximations obtained by block pulse method [16].

Conclusion
In this paper, we use the Legendre polynomials method to solve a class of fractional partial differential equations with variable coefficients.The Legendre polynomials operational matrix of fractional differentiation is derived from the property of Legendre polynomials.The initial equation is translated into the product of some relevant matrixes, which can also be regarded as the system of linear equations.The error analysis of Legendre polynomials is also given.The numerical results show that numerical solutions obtained by our method are in very good agreement with the exact solution.

Table 1 :
Absolute error for t = 1/4 s and different values of .

Table 2 :
Absolute error for t = 1/2 s and different values of .

Table 3 :
Absolute error for t = 3/4 s and different values of .