Numerical simulation of a generalized anomalous electro‐diffusion process in nerve cells by a localized meshless approach in Pseudospectral mode

Abstract

Pseudospectral techniques are known as powerful tools and highly accurate solvers to deal with partial differential equations. In the current work, a local meshless technique in Pseudospectral mode is employed to deal with an interesting and general mathematical model which describes anomalous electrodiffusion of ions in spiny dendrites, the two-dimensional variable-order time fractional nonlinear cable equation. For this purpose, at first step a second-order implicit difference method and a modified second-order weighted and shifted Grünwald difference scheme are used to discretize the appearing integer and variable order fractional time derivatives, respectively. Then a local Pseudospectral meshless method based on the sufficiently smooth compactly supported radial point interpolation basis functions is formulated for solving the semidiscretized problem. The main advantage of the proposed computational technique is that it leads to a sparse and better-conditioned system of algebraic equations. Finally, some numerical experiments are presented to demonstrate and verify the performance and accuracy of the method.