Spectral and pseudospectral schemes for the distributed order time fractional reaction-diffusion equation with Neumann boundary conditions

In this paper, two efficient numerical algorithms for the distributed order time fractional reaction-diffusion equation with Neumann boundary conditions are proposed, combining the finite difference method in time with Legendre spectral and Gauss-Lobatto-Legendre-Birkhoff (GLLB) pseudospectral method in space, respectively. It is proved that both of the schemes are unconditionally stable and have the same convergent order O(τ2+ Δα2+ N1-m), where τ, Δα, N and m are the temporal step, step size in distributed-order variable, polynomial degree and spatial regularity of the exact solution. Numerical results are presented to support the theoretical analysis.