Nonlinear Dynamics

, Volume 92, Issue 2, pp 395–413

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

Original Paper

## Abstract

We develop a local discontinuous Galerkin finite element method for the distributed-order time and Riesz space-fractional convection–diffusion and Schrödinger-type equations. The stability of the presented schemes is proved and optimal order of convergence $$\mathcal {O}(h^{N+1}+(\Delta t)^{1+\frac{\theta }{2}}+\theta ^{2})$$ for the Riesz space-fractional diffusion and Schrödinger-type equations with distributed order in time, an order of convergence of $$\mathcal {O}(h^{N+\frac{1}{2}}+(\Delta t)^{1+\frac{\theta }{2}}$$ $$+\theta ^{2})$$ is provided for the Riesz space-fractional convection–diffusion equations with distributed order in time where h, $$\theta$$ and $$\Delta t$$ 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.

## Keywords

Fractional convection–diffusion equations with distributed order in time Fractional Schrödinger-type equations with distributed order in time Local discontinuous Galerkin method Stability Optimal convergence

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