Numerical Approach Based on Two-Dimensional Fractional-Order Legendre Functions for Solving Fractional Differential Equations

In this paper, a robust, effective, and accurate numerical approach is proposed to obtain the numerical solution of fractional differential equations. The principal characteristic of the approach is the new orthogonal functions based on shifted Legendre polynomials to the fractional calculus. Also the fractional differential operational matrix is driven. Then the matrix with the Tau method is utilized to transform this problem into a system of linear algebraic equations. By solving the linear algebraic equations, the numerical solution is obtained. The approach is tested via some examples. It is shown that the FLF yields better results. Finally, error analysis shows that the algorithm is convergent.


Introduction
Fractional-order calculus has a long history.Compared with integer order, fractional differential equations show many advantages over the simulation of physical phenomena [1].In recent decades, fractional partial differential equations (FPDEs) [2] are widely used to describe engineering processes and dynamical systems; more and more researchers have devoted to study a variety of methods for solving fractional differential equations.The most commonly used methods are generalized differential transform method [3], Adomian decomposition method [4], finite difference method [5], and so on.Recently, the authors have proposed wavelet methods for solving a class of fractional convectiondiffusion equations.In [6], different families of wavelets [7][8][9][10][11] have been widely applied for solving problems of partial differential equations.In [12], the authors have obtained the approximate solutions of fractional differential equations using the generalized block pulse operational matrix.
The methods based on the orthogonal functions [13][14][15] are powerful and wonderful for solving FPDEs and have achieved great success in this field.Now the operational matrices of fractional derivative and integral [14][15][16] for Bernstein polynomials and Jacobi polynomials [15] have been derived.However, these polynomials use integer power series to approximate fractional ones; it cannot accurately represent properties of fractional integral and differential.Recently, in [17], Rida and Yousef have proposed a fractional extension of classical Legendre polynomials by replacing the integer order derivative in Rodrigues formula [18] with fractional-order derivatives.These functions are complex, so they are not unsuitable for solving FPDEs.Subsequently, Kazem et al. put forward the orthogonal fractional-order Legendre functions based on shifted Legendre polynomials.In [19], they used these functions to obtain the numerical solutions of fractional partial differential equations.Numerical results show that their methods are effective, accurate, and easy to implement.In this paper, FLF is suitable to characterize the properties of fractional partial differential equations; thus we attempt to use fractional-order Legendre functions to acquire numerical solution of the FPDEs.
The article is organized as follows.In Section 2, we introduce some basic definitions of fractional derivatives and integrals.In Section 3, we construct the FLFs and introduce some properties.In Section 4, the fractional operational matrices of FLFs are obtained.In Section 5, we use the proposed method to solve FPDEs.In Section 6, the proposed method is tested through several numerical examples.In Section 7, the error analysis is given.Finally Section 8 concludes the paper.

Preliminaries and Notations
In this section, we recall the essentials of the fractional calculus theory that will be used in this article.
In case  = 1, the generalized Taylor's formula (6) is the classical Taylors formula.

3.2.
Fractional-Order Legendre Definition.We define the fractional-order Legendre functions (FLFs) by introducing the change of variable  =   and  > 0 on shifted Legendre polynomials.The fractional-order Legendre function   (  ) is denoted by    ().The fractional-order Legendre functions are a particular solution of the normalized eigenfunctions of the singular Sturm-Liouville problem.
For arbitrary function (, ) ∈  2 ([0, 1]×[0, 1]), they can be expanded as the following formula: where Proof.Using contradiction, let Then there is at least one coefficient such that   ̸ =   ; however Proof.Suppose that (, ) and (, ) can be expended by FLFs as follows: By subtracting the above two equations with each other, we have If we consider truncated series in (25), we obtain where In this paper, we use the Tau method [20] to compute the coefficients   .

Fractional Differential Operational Matrix of FLFs
The derivative of the Φ() can be approximated where D  is called the FLFs operational matrix of derivative. where Proof.With the properties of the derivative (ii) and the orthogonally of FLFs, we have Let Multiplying    ()   () on both sides of (40), we have Substituting ( 40) and ( 41) into (39), we have Hence, ,  = 0, 1, . . .,  − 1. (43)

Description of the Proposed Method
We consider a class of fractional partial differential equations as follows: subject to the initial conditions where 0 ≤ ,  ≤ 1,  − 1 < V,  ≤ ,  V (, )/ V and   (, )/  are fractional derivative in Caputo sense, (, ) is the known continuous function, and (, ) is the unknown function.
Therefore,   (, ) satisfies the following problem: where   (, ) is the residual function, We find an approximation    (, ) to the error function   (, ) in the same way as we did before for the solution of the problem, the error function satisfies the problem We should note that, in order to construct the approximate    (, ) to   (, ), only (55) needs to be recalculated in the same way as we did before for the solution of (44).where (, ) = 8 2  3/2 /3√ + 32 2  7/4 /21Γ(3/4); using the method presented in Section 5, the equation is transformed as follows:

Numerical Examples
By solving (57), we can obtain the numerical solution of the problem.
Therefore, in order to show the efficiency of the present method, we define the error function as follows:

Conclusions
In this paper, we define a basis of FLFs and have derived its fractional differential operational matrix.The fractional derivatives are described in the Caputo sense.The matrix together with Tau method is used to effectively approximate the exact solution of fractional partial differential equations.Compared with other numerical approach, the FLFs-Tau method can accurately represent properties of fractional calculus.Moreover, only a small number of FLFs are needed to obtain a satisfactory result.Numerical results which are given in the previous section demonstrate good accuracy of the algorithm.Finally, error analysis shows that the algorithm is convergent.

Theorem 11 .
Suppose D  is the  ×  operational matrix of Caputo fractional derivatives of order  > 0,  > /2, when  ∉ N; then the elements of D  are obtained as{  }