Tarek Aboelenen

Tarek Aboelenen

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.

Tarek Aboelenen

We propose a nodal discontinuous Galerkin method for solving the nonlinear Riesz space fractional Schrödinger equation and the strongly coupled nonlinear Riesz space fractional Schrödinger equations. These problems have been expressed as a system of low order differential/integral equations. Moreover, we prove, for both problems, $L^{2}$ stability and optimal order of convergence $O(h^{N+1})$, where $h$ is space step size and $N$ is polynomial degree. Finally, the performed numerical experiments confirm the optimal order of convergence.

Tarek Aboelenen

We provide a mathematical framework for studying different versions of discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville fractional elliptic problems on a finite domain. The boundedness and stability analysis of the primal bilinear form are provided. A priori error estimate under energy norm and optimal error estimate under $L^{2}$ norm are obtained for DG methods of the different formulations. Finally, the performed numerical examples confirm the optimal convergence order of the different formulations.

Tarek Aboelenen

A direct discontinuous Galerkin (DDG) finite element method is developed for solving fractional convection-diffusion and Schrödinger type equations with a fractional Laplacian operator of order $α$ $(1<α<2)$. The fractional operator of order $α$ is expressed as a composite of second order derivative and a fractional integral of order $2-α$. These problems have been expressed as a system of parabolic equation and low order integral equation. This allows us to apply the DDG method which is based on the direct weak formulation for solutions of fractional convection-diffusion and Schrödinger type equations in each computational cell, letting cells communicate via the numerical flux $(\partial_{x}u)^{*}$ only. Moreover, we prove stability and optimal order of convergence $O(h^{N+1})$ for the general fractional convection-diffusion and Schrödinger problems where $h$, $N$ are the space step size and polynomial degree. The DDG method has the advantage of easier formulation and implementation as well as the high order accuracy. Finally, numerical experiments confirm the theoretical results of the method.