Numerical approach for solving two dimensional fractal-fractional PDEs using peridynamic method

Abstract

This paper develops a numerical approach for solving two-dimensional fractal-fractional parabolic partial differential equations. The fractal-fractional derivative is defined in the Atangana-Riemann-Liouville sense with Mittage-Leffler kernel. To solve this equation, we first eliminate the spatial derivatives using peridynamic differential operators. Then, we derive an operational matrix (OM) of fractal-fractional derivative in terms of the Legendre polynomials to simplify the time derivative. The aim of the formulated method is to transform the original problem into an uncomplicated system of linear algebraic equations which can be solved by mathematical software. The applicability of the approach is examined for several examples and numerical results show the computational efficiency of the method.