A local weak form meshless method to simulate a variable order time-fractional mobile-immobile transport model

In this study, a simulation of an anomalous mobile-immobile transport process in complex systems is produced numerically. The process is mathematically described as a two-dimensional time-fractional mobile-immobile advection-diffusion equation in Coimbra variable order derivative sense. A local weak form meshless method combined with a time stepping approach is discussed and implemented to simulate the model. For this purpose, firstly, an implicit difference stepping method is used to semi-discretize the model in time direction. For full discretization, a set of regularly distributed nodes is created in the primary spatial domain and the local radial point interpolation method is used to construct the spatial shape functions on the distributed data-sites. Then an efficient meshless procedure based on combination of local Petrov-Galerkin method and collocation technique is formulated to treat in the interior and on the boundary of the primal spatial domain, respectively. Finally, some benchmark problems are presented on regular and irregular domains to verify the validity, efficiency and accuracy of the method.