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.
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 distributedorder time and Riesz space fractional convection-diffusion and Schrödinger type equations. We prove stability and optimal order of convergence O(h N +1 + (∆t) 1+ θ 2 + θ 2 ) for the distributed-order time and space-fractional diffusion and Schrödinger type equations, an order of convergence of O(h N + 1 2 +(∆t) 1+ θ 2 +θ 2 ) is established for the distributed-order time and Riesz space fractional convection-diffusion equations where ∆t, 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.Keywords: time distributed order and space-fractional convection-diffusion equations, time distributed order and space-fractional Schrödinger type equations, local discontinuous Galerkin method, stability, error estimates.
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