Numerical simulation of a modified anomalous diffusion process with nonlinear source term by a local weak form meshless method

Abstract

In the current work, a modified anomalous diffusion process in two dimensional space with nonlinear source term is investigated numerically. The process is modeled as a two dimensional nonlinear time-fractional sub-diffusion equation in sense of Riemann–Liouville fractional derivatives. An efficient and accurate computational technique is performed for solving the governing problem. In the proposed method, firstly, a second-order accurate formulation is proposed to discretize the problem in the temporal direction. It has been proved that the proposed time discretization scheme is unconditionally stable. Then a meshfree method based on a combination of the local Petrov–Galerkin weak form and a collocation approach is implemented for spatial discretization. In our implementation both regularly and irregularly distributed field nodes are used to represent the primary spatial domain and its boundary. The radial point interpolation method is used for constructing meshfree shape functions on the distributed field nodes. Some numerical experiments have been carried out to verify the accuracy and performance of the proposed method. The numerical findings demonstrate efficiency and high accuracy of the technique and confirm the theoretical prediction.