Numerical investigation of reproducing kernel particle Galerkin method for solving fractional modified distributed-order anomalous sub-diffusion equation with error estimation

Abstract

In the Galerkin weak form technique based on various kernels that they do not have δ-Kronecker property, in order to apply the essential boundary condition, there are two straight strategies that one of them is the Lagrange multiplier method and another one is the penalty method. In the penalty method the main boundary value problem (BVP) is converted to a new BVP with Robin boundary condition. So, we obtain a new BVP that it must be solved. The main purpose of this paper is to propose an error analysis to verify that the solutions of penalty method obtained by applying the essential boundary condition are convergent to the solution of main BVP with essential boundary condition. For this aim, we select fractional modified distributed-order anomalous sub-diffusion equation. At the first stage, we propose a second-order difference scheme for the temporal variable. The convergence and stability analysis for the time-discrete scheme are proposed. At the second stage, we derive the full-discrete scheme based on the Galerkin weak form and shape functions of reproducing kernel particle method (RKPM) as the mentioned shape functions do not have the δ-Kronecker property. Furthermore, it is shown that when the penalty parameter goes to infinity then the solutions of BVP with Robin boundary condition are convergent to the solutions of BVP based on the essential boundary condition. The proposed examples verify that the present error estimate is true.