MLPG method for two-dimensional diffusion equation with Neumannʹs and non-classical boundary conditions

In this paper, a meshless local Petrov–Galerkin (MLPG) method is presented to treat parabolic partial differential equations with Neumannʹs and non-classical boundary conditions. A difficulty in implementing the MLPG method is imposing boundary conditions. To overcome this difficulty, two new techniques are presented to use on square domains. These techniques are based on the finite differences and the Moving Least Squares (MLS) approximations. Non-classical integral boundary condition is approximated using Simpsonʹs composite numerical integration rule and the MLS approximation. Two test problems are presented to verify the efficiency and accuracy of the method.