A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation

Abstract

This paper is concerned with the moving least squares (MLS) meshless approach for the numerical solution of two-dimensional (2D) variable-order time fractional nonlinear diffusion-wave equation (V-OTFND-WE) on arbitrary domains. The variable-order fractional derivative of this equation is expressed in the Caputo type. The proposed method is based on the MLS approximation in conjunction with the finite difference technique. For spatial derivatives, the MLS approximation and for temporal derivative, the finite difference technique are utilized to discretize the equation. The aim of the present paper is to show that the meshfree method based on the MLS approximation and collocation scheme is suitable for solving the variable-order fractional partial differential equations (PDEs). The proposed method is validated, using some different nonlinear numerical examples with complex geometries. The results obtained demonstrate the accuracy and easy implementation of the method for nonlinear variable-order time fractional PDEs.