Abstract
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 two-dimensional fractional-order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.
Highlights
Fractional partial differential equations play a significant role in modeling physical and engineering processes
The Riemann-Liouville fractional integration of order γ > 0 of the 2D-FLFs ψij can be obtained in the form of
In a similar way as previous, one can obtain the operational matrix of Riemann-Liouville fractional integration with respect to variable y
Summary
Fractional partial differential equations play a significant role in modeling physical and engineering processes. Kazem et al [28] presented the orthogonal fractional order Legendre functions based on shifted Legendre polynomials to find the numerical solution of FDEs and drew a conclusion that their method is accurate, effective, and easy to implement Benefiting from their “exponential-convergence” property when smooth solutions are involved, spectral methods have been widely and effectively used for the numerical solution of partial differential equations. Motivated and inspired by the ongoing research in orthogonal polynomials methods and spectral methods, we construct two-dimensional fractional-order 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.
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