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

Highlights：

• A new meshless method is employed to solve fractional modified distributed-order anomalous sub-diffusion equation.

• The mentioned method is based on the shape functions of reproducing kernel particle method and Galekrin idea.

• The stability and convergence of the developed method have been studied.

• 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.

• Some examples are studied that they show the efficiency of the proposed method.

• The numerical results confirm the theoretical concepts.

摘要

•A new meshless method is employed to solve fractional modified distributed-order anomalous sub-diffusion equation.•The mentioned method is based on the shape functions of reproducing kernel particle method and Galekrin idea.•The stability and convergence of the developed method have been studied.•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.•Some examples are studied that they show the efficiency of the proposed method.•The numerical results confirm the theoretical concepts.

论文关键词：Fractional PDEs,Modified distributed-order anomalous sub-diffusion equation fractional derivative,Convergence analysis and error estimate,Reproducing kernel particle method (RKPM),Riemann–Liouville derivative,Meshless Galerkin method,Kernel based method

论文评审过程：Received 5 December 2019, Revised 14 May 2020, Accepted 24 September 2020, Available online 8 November 2020, Version of Record 8 November 2020.