Solving Fractional Diffusion Equation via the Collocation Method Based on Fractional Legendre Functions

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.


Introduction
We consider the generalized time-fractional diffusion equation of the form     (, ) =  (, )  2   (, ) +  (, ) ,  ∈ (−1, 1) ,  ∈ (0, ) , with initial and boundary conditions  (−1, ) = ℎ 1 () , (1, ) = ℎ 2 () , (, 0) =  () , where ,  ∈  1 ([−1, 1] × [0, ]),  ∈ [−1, 1], ℎ 1 , ℎ 2 ∈ [0, ],  > 0, and 0 <  ≤ 1.For  = 1, the fractional diffusion equation is reduced to a conventional diffusion-reaction equation which is well studied, so we focus on 0 <  < 1.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][13][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 1 <  ≤ 2 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 ∑  =0     .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 2 Journal of Computational Methods in Physics 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.

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 (),  > 0, is said to be in the space   ,  ∈ R, if there exists a real number  > , such that () =    1 (), where  1 () ∈ [0, ∞), and it is said to be in the space Definition 2. The left Riemann-Liouville fractional integral of order  ≥ 0, of a function  ∈   ,  ≥ −1, is defined by Definition 3.For  > 0,  − 1 <  ≤ ,  ∈ N,  > 0, and  ∈   −1 , the left Caputo fractional derivative is defined by where Γ is the well-known Gamma function.
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  > 0 is using series expansion of the form ∑  =0     .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) In addition, {   () :  = 0, 1, 2, . ..} are orthogonal functions with respect to the weight function () =  −1 on (0, 1) 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.
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 L  +2 .Therefore, (15) leads to the elementwise equation: or for  = 0 : .

Numerical Results
In this section, we implement the proposed numerical technique for four examples.where   () is the Mittag-Leffler function.Since      (−  ) = −  (−  ) for 0 <  < 1, it is easy to see that the exact solution is  (, ) =   (−  ) sin . (48) The exact solution for  = 1 is The approximate solutions generated by the proposed method are presented in Figure 1, for different values of  and  =  = 10.
where  app (, ) is the approximate solution generated by the proposed method for  =  = 10.Table 1 presents the error for different values of .
Example 2. Consider the fractional diffusion equation: where (, ) =  3  2 is the exact solution.The approximate solutions generated by the proposed method are presented in Figure 3 for different values of  and  =  = 6. Figure 4 depicts the exact solution (red) and the approximate solution (green) for  = 1 and  =  = 6.
Table 2 presents the error for different values of .
Example 3. Consider the fractional diffusion equation presented in [17]: The exact solution is To apply the proposed method, we shall do the following change of variable  = 2 − 1.In this case, the -domain becomes [−1, 1].The approximate solutions generated by the proposed method and the exact solution are presented in Figure 5 for  = 0.92 and  = 0.98 at  = 0.01 and  =  = 10.Table 3 presents a comparison between the error in our results and the ones obtained by the finite difference method (FDM) [17] for  = 0.92, 0.98 and  = 0.01.To apply the proposed method, we will do the following change of variable  = 2 − 1.In this case, the -domain becomes [−1, 1].To make a comparison with the results of [18], assume that  I ,  II , and  III are the errors in [18] using uniform mesh, quasiuniform mesh, and nonuniform mesh for  = 1.Let  pro be the error in the proposed method for  = 1 and  =  = 10.Results are presented in Table 4.

Conclusion
In this paper, we use series expansion based on the shifted fractional Legendre functions to solve fractional diffusions equations of Caputo's type.We write the coefficients of the fractional derivative in terms of the shifted fractional Legendre functions as indicated in Theorem 5 and give explicit relationship between them.Then, we use the collocation method to compute these coefficients.To the best of our knowledge, the method has not been developed to integrate fractional diffusion equations of the form (1)- (2).We test the proposed technique for several examples and present four of them in this paper.These examples show the efficiency and the accuracy of the proposed method, where in few terms we achieved accuracy up to 10 −10 .In Examples 3 and 4, we compare our results with the ones obtained by FDM in [17,18].Both examples show that the proposed method works more efficiently and accurately than the methods in [17,18].

Figure 5 :
Figure 5: Exact and approximate solutions of Example 3.

Table 4 :
[18]lts of the method in[18]and the proposed method.