A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of twodimensional fractionalorder Legendre functions (2DFLFs). The operational matrices of integration and derivative for 2DFLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2DFLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.
Fractional partial differential equations play a significant role in modeling physical and engineering processes. Therefore, there is an urgent need to develop efficient and fast convergent methods for FPDEs. Recently, several different techniques, including Adomian’s decomposition method (ADM) [
The method based on the orthogonal functions is a wonderful and powerful tool for solving the FDEs and has enjoyed many successes in this realm. The operational matrix of fractional integration has been determined for some types of orthogonal polynomials, such as Chebyshev polynomials [
Benefiting from their “exponentialconvergence” property when smooth solutions are involved, spectral methods have been widely and effectively used for the numerical solution of partial differential equations. The basic idea of spectral methods is to expand a function into sets of smooth global functions, called the trial functions. Because of their special properties, the orthogonal polynomials are usually chosen to be trial functions. Spectral methods can obtain very accurate approximations for a smooth solution while only need a few degrees of freedom. Recently, Chebyshev spectral method [
Motivated and inspired by the ongoing research in orthogonal polynomials methods and spectral methods, we construct twodimensional fractionalorder Legendre functions and derive the operational matrices of integration and derivative for the solution of FPDEs. To the best of the authors’ knowledge, such approach has not been employed for solving FPDEs.
The rest of the paper is organized as follows. In Section
Some necessary definitions and Lemma of the fractional calculus theory [
A real function
RiemannLiouville fractional integral operator (
The fractional derivative of
Let
In this section, we introduce the fractionalorder Legendre functions which were first proposed by Kazem et al. [
In this section, the definitions and theorems of 2DFLFs are given by Liu’s method described in [
Let
The basis
Let
Consider
A function of two independent variables
If the series
By multiplying
If the infinite series in (
According to the definition of FLFs, one can find that fractional Legendre polynomials are identical to Legendre polynomials shifted to
If the function
Let
If two continuous functions defined on
Suppose that
If the 2DFLFs expansion of a continuous function
Theorem
If the sum of the absolute values of the 2DFLFs coefficients of a continuous function
Consider
If a continuous function
Let
The RiemannLiouville fractional integration of order
Consider
Let
Using previous Lemma
Let
Using (
Now by substituting above equations in (
In a similar way as previous, one can obtain the operational matrix of RiemannLiouville fractional integration with respect to variable
Let
The FLFs Caputo fractional derivative of
Consider
Let
Using previous Lemma
Let
Using (
Now by substituting above equations in (
In a similar way as above, one can get Caputo fractional derivative of order
Let
Consider the following FPDEs:
By employing operator
We first express unknown function
Now for the nonlinear part, by employing the nonlinear term approximation method described in [
For the linear part, we have
After substituting (
Now, the present method is applied to solve the linear and nonlinear FPDEs, and their results are compared with the solution of other methods. The accuracy of our approach is estimated by the following error functions:
Consider the onedimensional linear inhomogeneous fractional Burger’s equation [
By employing 2DFLFs method, one can get
Figures
Numerical results for Example
Consider nonlinear fractional KleinGordon equation [
By employing 2DFLFs method with
The numerical results of Example
Errors of Example
Error 















Numerical results of Example
Consider the nonlinear timefractional advection partial differential equation [
Figure
Numerical values when







FVIM  2DFLFs  FVIM  2DFLFs  FVIM  2DFLFs  Exact  
0.25  0.25  0.12422501  0.12225461  0.09230374  0.09224583  0.06250058  0.062500  0.062500 
0.50  0.24845002  0.24450922  0.18460748  0.18449165  0.12500117  0.125000  0.125000  
0.75  0.37267504  0.36676383  0.27691122  0.27673748  0.18750175  0.187500  0.187500  
1.00  0.49690005  0.48901844  0.36921496  0.36898331  0.25000234  0.250000  0.250000  
 
0.50  0.25  0.18377520  0.16584130  0.15148283  0.14985508  0.12507592  0.125000  0.125000 
0.50  0.36755040  0.33168259  0.30296566  0.29971016  0.25015184  0.250000  0.250000  
0.75  0.55132559  0.49752389  0.45444848  0.44956524  0.37522776  0.375000  0.375000  
1.00  0.73510079  0.66336518  0.60593131  0.59942032  0.50030368  0.500000  0.500000  
 
0.75  0.25  0.27227270  0.20678964  0.21407798  0.20119503  0.18881843  0.187500  0.187500 
0.50  0.54454540  0.41357929  0.42815596  0.40239005  0.37763687  0.375000  0.375000  
0.75  0.81681810  0.62036893  0.64223394  0.60358508  0.56645530  0.562500  0.562500  
1.00  1.08909080  0.82715857  0.85631192  0.80478011  0.75527373  0.750000  0.750000 
Numerical results of Example
We finally consider the linear timefractional wave equation:
Table
Numerical values when







FVIM  2DFLFs  FVIM  2DFLFs  FVIM  Exact  
0.25  0.25  0.26622298  0.26622021  0.26593959  0.26594005  0.26578827  0.26578827 
0.50  0.56489190  0.56488083  0.56375836  0.56376020  0.56315308  0.56315308  
0.75  0.89600678  0.89598187  0.89345630  0.89346046  0.89209443  0.89209443  
1.00  1.25956762  1.25952332  1.25503343  1.25504082  1.25261232  1.25261232  
 
0.50  0.25  0.28474208  0.28474415  0.28340402  0.28340659  0.28256846  0.28256846 
0.50  0.63896831  0.63897662  0.63361610  0.63362636  0.63027383  0.63027383  
0.75  1.06267869  1.06269739  1.05063622  1.05065931  1.04311611  1.04311611  
1.00  1.55587323  1.55590647  1.53446439  1.53450544  1.52109530  1.52109531  
 
0.75  0.25  0.30690489  0.30690747  0.30361709  0.30361656  0.30139478  0.30139480 
0.50  0.72761955  0.72762986  0.71446834  0.71446625  0.70557913  0.70557918  
0.75  1.26214400  1.26216719  1.23255378  1.23254905  1.21255304  1.21255316  
1.00  1.91047821  1.91051944  1.85787338  1.85786498  1.82231652  1.82231673 
Numerical results of Example
We define a basis of 2DFLFs and derived its operational matrices of fractional derivative and integration, which are used to approximate the numerical solution of FPDEs. Compared with other numerical methods, 2DFLFs method can accurately represent properties of fractional calculus. Moreover, only a small number of 2DFLFs are needed to obtain a satisfactory result. The obtained results demonstrate the validity and applicability of proposed method for solving the FPFEs.
This work is supported by the National Natural Science Foundation of China (Grant no. 11272352). The authors are grateful to the anonymous referees for their comments which substantially improved the quality of this paper.