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

We develop a local discontinuous Galerkin finite element method for the distributed-order time and Riesz space-fractional convection-diffusion and Schrodinger-type equations. The stability of the presented schemes is proved and optimal order of convergence for the Riesz space-fractional diffusion and Schrodinger-type equations with distributed order in time, an order of convergence of is provided for the Riesz space-fractional convection-diffusion equations with distributed order in time where h, and are space step size, the distributed-order variables and the step sizes in time, respectively. Finally, the performed numerical examples confirm the optimal convergence order and illustrate the effectiveness of the method.