Local discontinuous Galerkin method for distributed-order time and space-fractional convection-diffusion and Schr\"odinger type equations

TL;DR

This paper introduces a local discontinuous Galerkin method for solving distributed-order fractional PDEs, proving stability and optimal convergence rates, with numerical experiments confirming theoretical results.

Contribution

The paper develops a novel LDG method specifically for distributed-order fractional PDEs, providing stability analysis and convergence proofs.

Findings

Proves stability and optimal convergence of the LDG method.

Establishes specific convergence orders for different fractional equations.

Numerical experiments confirm theoretical convergence rates.

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 Schr\"odinger type equations. We prove stability and optimal order of convergence $O(h^{N+1}+(\Delta t)^{1+\frac{\theta}{2}}+\theta^{2})$ for the distributed-order time and space-fractional diffusion and Schr\"odinger type equations, an order of convergence of $O(h^{N+\frac{1}{2}}+(\Delta t)^{1+\frac{\theta}{2}}+\theta^{2})$ is established for the distributed-order time and Riesz space fractional convection-diffusion equations where $\Delta t$, $h$ and $\theta$ are the step sizes in time, space and distributed-order variables, respectively. Finally, the performed numerical experiments confirm the optimal order of convergence.