An efficient numerical method for the distributed-order time-fractional diffusion equation with the error analysis and stability properties

Abstract

In this manuscript, we study and examine the time-fractional modified anomalous sub-diffusion model of distributed-order. Two numerical approaches are used to study the approximate solutions of the presented model. For the first approach, we use a second-order difference method based on the L1 formula for the temporal variable. In this case, stability and convergence analysis for time discretization using the L1 formula is displayed. For the second approach, we display and demonstrate the numerical method for the full-discrete based on the Galerkin weak form based on various kernels and shape functions of reproducing kernel particle method which do not have the $\delta$-Kronecker property. Moreover, in this manuscript, in order to be able to use the necessary boundary conditions, the two straight strategies are used: one is the Lagrange multiplier method, and the other one is the penalty method. By using penalty method, we can the main boundary value model is changed to a new BVP with Robin boundary condition. At the end of the article, some numerical examples are presented and analyzed for the efficiency of the proposed numerical method. Also, the presented numerical method is compared with other numerical approaches, and the results of this comparison are reported in the form of a table.