Local discontinuous Galerkin method for distributed-order time and space-fractional convection-diffusion and Schrödinger type equations

Abstract

Fractional partial differential equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we propose a local discontinuous Galerkin (LDG) method for the distributed-order time and Riesz space fractional convection-diffusion and Schrödinger type equations. We prove stability and optimal order of convergence $O(h^{N+1}+(Δt)^{1+\fracθ{2}}+θ^{2})$ for the distributed-order time and space-fractional diffusion and Schrödinger type equations, an order of convergence of $O(h^{N+\frac{1}{2}}+(Δt)^{1+\fracθ{2}}+θ^{2})$ is established for the distributed-order time and Riesz space fractional convection-diffusion equations where $Δt$, $h$ and $θ$ are the step sizes in time, space and distributed-order variables, respectively. Finally, the performed numerical experiments confirm the optimal order of convergence.