Numerical study of the variable-order time-fractional mobile/immobile advection-diffusion equation using direct meshless local Petrov-Galerkin methods

Abstract

In this paper, we use direct meshless local Petrov-Galerkin (DMLPG) methods for solving the variable-order time-fractional mobile/immobile advection-diffusion equation in two dimensions. The basis of the DMLPG methods is on the generalized moving least-square (GMLS) approximation and local weak form of the equation. Since the GMLS approximation uses basic polynomials as shape functions, there is no need to construct complex shape functions in the moving least-square (MLS) approximation. In addition, the calculation of numerical integrals in the local weak form of the equation is simple. We also give proof of the convergence and stability of the time-discrete scheme. The analytical results show that the semi-discrete time scheme is unconditionally stable and the convergence order is O(Δt2). Moreover, to show the accuracy and efficiency of the methods, we use the regular and irregular domains with distributions of uniform and scattered nodes in numerical examples.