DMLPG solution of the fractional advection–diffusion problem

The aim of this work is application of the direct meshless local Petrov–Galerkin (DMLPG) method for solving a two-dimensional time fractional advection–diffusion equation. This method is based on the generalized moving least squares (GMLS) approximation, and makes a considerable reduction in the cost of numerical integrations in weak forms. In fact, DMLPG shifts the integrals over the close form polynomials rather than the complicated MLS shape functions. Moreover, the values of integrals on subdomains with the same shapes are equal. Thus DMLPG is a weak-based meshless technique in the cost-level of collocation or integration-free methods. In time domain, a simple and suitable finite difference approximation is employed. Some examples show the advantages of the new method in comparison with the traditional MLPG method.