Local discontinuous Galerkin method for distributed-order time and space-fractional convection-diffusion and Schrödinger type equations
This work provides a rigorous numerical analysis for a class of fractional PDEs with distributed-order derivatives, which is important for modeling complex physical phenomena, but the contribution is incremental as it extends existing LDG methods to a new type of fractional operator.
The paper proposes a local discontinuous Galerkin method for distributed-order time and space-fractional convection-diffusion and Schrödinger type equations, proving stability and optimal convergence rates (e.g., O(h^{N+1}+(Δt)^{1+θ/2}+θ^2) for diffusion and Schrödinger equations). Numerical experiments confirm the theoretical results.
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.