A meshless method for two-dimensional diffusion equation with an integral condition

This paper presents a new approach based on the meshless local Petrov–Galerkin (MLPG) and collocation methods to treat the parabolic partial differential equations with non-classical boundary conditions. In the presented method, the MLPG method is applied to the interior nodes while the meshless collocation method is applied to the nodes on the boundaries, and so the Dirichlet boundary condition is imposed directly. To treat the complicated integral boundary condition appearing in the problem, Simpsonʹs composite numerical integration rule is applied. A time stepping scheme is employed to approximate the time derivative. Finally, two numerical examples are presented showing the behavior of the solution and the efficiency of the proposed method.