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

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 + (∆t) θ 2 + θ) for the distributed-order time and space-fractional diffusion and Schrödinger type equations, an order of convergence of O(h 1 2 +(∆t) θ 2 +θ) 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.