Local discontinuous Galerkin method for distributed‐order time‐fractional diffusion‐wave equation: Application of Laplace transform

Abstract

In this paper, the Laplace transform combined with the local discontinuous Galerkin method is used for distributed‐order time‐fractional diffusion‐wave equation. In this method, at first, we convert the equation to some time‐independent problems by Laplace transform. Then, we solve these stationary equations by the local discontinuous Galerkin method to discretize diffusion operators at the same time. Next, by using a numerical inversion of the Laplace transform, we find the solution of the original equation. One of the advantages of this procedure is its capability to be implemented in a parallel environment. It's another advantage is that the number of stationary problems that should be solved is much less than that is needed in time‐marching methods. Finally, some numerical experiments have been provided to show the accuracy and efficiency of the method.

Stability and asymptotics for fractional delay diffusion‐wave equations

“…) , which contradicts (6). Thus, there exists no root s in a neighborhood of s = 0 for characteristic equation C 1 (s) = 0.…”

“…Fractional diffusion-wave equations, derived from the continuous time random walk [1,2], have been highly active in recent decades. The theoretical and numerical analysis of various types of fractional diffusion-wave equations were widely concerned by scholars, including time-fractional diffusion-wave equations [3][4][5][6][7], space-fractional tempered diffusion-wave equations [8], and time-space fractional damped diffusion-wave equations [9]. For time-dependent mod-els, delay happens everywhere because of the transportation of energies and materials [10][11][12].…”