The MLPG with improved weight function for two-dimensional heat equation with non-local boundary condition

In this paper, a meshless local Petrov–Galerkin (MLPG) method is presented to treat the heat equation with the Dirichlet, Neumann, and non-local boundary conditions on a square domain. The Moving Least Square (MLS) approximation is a classical MLS method, in which the Gaussian weight function is the most common shape function. However, shape functions for the classical MLS approximation lack the Kronecker delta function property. Thus in this method, the boundary conditions cannot have a penalty parameter imposed easily and directly. In the method we choose a weight function that leads to the MLS approximation shape functions approximating the Kronecker delta function property, and nodes on the Dirichlet boundary conditions, which enables a direct application of essential boundary conditions without the additional numerical method. The improved weight function in MLS approximation has been successfully implemented in solving the diffusion equation problem. Two test problems are presented to verify the efficiency, easiness and accuracy of the method. Also Ne and root mean square errors are obtained to show the convergence of the method.