Abstract

A formulation of the fractional Legendre functions is constructed to solve the generalized time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre and path following methods. Analysis for the presented method is given and numerical results are presented.

1. Introduction

We consider the generalized time-fractional diffusion equation of the form with initial and boundary conditions where , , , , , , and . For , the fractional diffusion equation is reduced to a conventional diffusion-reaction equation which is well studied, so we focus on . Some existence and uniqueness results of Problem (1)-(2) were established in [1]. In recent years, great interests were devoted to the analytical and numerical treatments of fractional differential equations (FDEs). Usually, FDEs appear as generalizations to existing models with integer derivative and they also present new models for some physical problems [2, 3]. In general, FDEs do not possess exact solutions in closed forms, and, therefore, numerical methods such as the variational iteration (VIM) [4, 5], the homotopy analysis method (HAM) [6, 7], and the Adomian decomposition method (ADM) [8, 9] have been implemented for several types of FDEs. Also, the maximum principle and the method of lower and upper solutions have been extended to deal with FDEs and obtain analytical and numerical results [10, 11]. The Tau method, the pseudospectral method, and the wavelet method based on the Legendre polynomials have been implemented for several types of FDEs [12–14]. Kazem et al. [12] have constructed the Legendre functions of fractional order and discussed some of their properties. The resulting Legendre function operational and product matrices, together with the Tau method, have been implemented to solve linear and nonlinear fractional differential equations. The effectiveness of the approach has been examined through several examples. In [13], a fractional diffusion equation is considered, where the fractional derivative of order refers to the spatial variable . The Legendre pseudospectral method is implemented to solve the problem, where the solution is expanded with regular Legendre polynomials. As a result, a system of linear equation has been obtained and integrated using the finite difference method. However, in solving fractional differential equations of order using series expansions, it is common and more efficient to expand the solution with fractional functions of the form . Rawashdeh [14] has implemented the Legendre wavelets method for integrodifferential equations with fractional order.

The Legendre collocation method has been implemented for wide classes of differential equations and the effectiveness of the method is illustrated [15]. In the recent work, we intend to apply the collocation method based on the shifted fractional Legendre functions to integrate the Problem (1)-(2). To the best of our knowledge, the method has not been developed to integrate fractional diffusion equations of the form (1)-(2). We organize this paper as follows. In Section 2, we present basic definitions and results of fractional derivative. In Section 3, we present the numerical technique for solving Problem (1)-(2). In Section 4, we present some numerical results to illustrate the efficiency of the presented method. Finally we conclude with some comments in Section 5.

2. Preliminaries

In this section, we present the definition and some preliminary results of the Caputo fractional derivative, as well as the definition of the fractional-order Legendre functions and their properties.

Definition 1. A real function , , is said to be in the space , , if there exists a real number , such that , where , and it is said to be in the space if , .

Definition 2. The left Riemann-Liouville fractional integral of order , of a function , , is defined by

Definition 3. For , , , , and , the left Caputo fractional derivative is defined by where is the well-known Gamma function.

The Caputo derivative defined in (4) is related to the Riemann-Liouville fractional integral, , of order , by . The Caputo fractional derivative satisfies the following properties for , , and (see [16]):(1), (2), (3), where is constant,(4), (5), where are constants.

The basic concept of this paper is the Legendre polynomials. For this reason, we study some of their properties.

Definition 4. The Legendre polynomials are the eigenfunctions of the Sturm-Liouville problem:

Among the properties of the Legendre polynomials we list the following [19]:(1), where ,(2), for ,(3).

In order to use these polynomials on the interval , we define the shifted Legendre polynomials by . Using the change of variable , has the following properties:(1), (2), for ,(3) and .

The analytic closed form of the shifted Legendre polynomials of degree is given by

One of the common and efficient methods for solving fractional differential equations of order is using series expansion of the form . For this reason, we define the fractional-order Legendre function by . Using the properties of the shifted Legendre polynomials, it is easy to verify that [20](1), ,(2), for ,(3) and ,(4) and .

In addition, are orthogonal functions with respect to the weight function on with

The closed form of is given by

Using properties (4) and (5) of the Caputo fractional derivative, we have

The following result is important, since it facilitates applying the collection method.

Theorem 5. Let and let be a piecewise continuous function on . Then,(i) can be represented by infinite series expansion as , where (ii) converges uniformly on to , where and , for and  .

Proof. (i) Since and is a piecewise continuous function on ,   converges uniformly to on , where can be computed by the orthogonality relation of the Legendre polynomials; see [11]. Since , defined by , is a bijection continuous function, the infinite series converges uniformly to on , where the value of follows from the orthogonality relation of with respect to the weight function on , which completes the proof.
(ii) Let for . From Part (i), converges uniformly to on . Since and is a piecewise continuous function on , and converges uniformly to on . Thus, converges uniformly to on which gives the result of the second part. The value of follows from the orthogonality relation of with respect to the weight function on .

3. Collocation Method

In this section, we use the fractional-order Legendre collocation method to discretize Problem (1)-(2). For simplicity, we assume that and . If , we use the change of variable to make the -domain (0,1). Expand the solution in terms of the fractional-order Legendre function as follows: Thus, where for . Therefore, for , the residual is given by

Orthogonalize the residual with respect to the Dirac delta function as follows: where are the collocation points. We choose the collocation points to be the roots of . Therefore, (15) leads to the elementwise equation: or Let

Thus, we can rewrite (17) in the matrix form as where Since for , it is easy to see that , where is matrix. Therefore, System (19) becomes

Now, we study the boundary conditions on the variable . From (12), one can see that which implies that where . From Systems (21) and (23), we obtain the following fractional system: where

From (2) and (12), we see that

Using the orthogonality property of the Legendre polynomials, we get

Therefore, our initial fractional value problem becomes where

To solve the System (28)-(29), we use the fractional-order Legendre collocation method. Approximate the solution in terms of the fractional-order Legendre function as follows: Thus, where , and , for and . Therefore, for , the residual is given by

Orthogonalize the residual with respect to the Dirac delta function as follows: where are the collocation points. We choose the collocation points to be where are the roots of . Therefore, (34) leads to the elementwise equation: or Let

Thus, we can rewrite (37) in the matrix form as where