High-Order Numerical Method for Solving a Space Distributed-Order Time-Fractional Diffusion Equation

This article proposes a high-order numerical method for a space distributed-order time-fractional diffusion equation. First, we use the mid-point quadrature rule to transform the space distributed-order term into multi-term fractional derivatives. Second, based on the piecewise-quadratic polynomials, we construct the nodal basis functions, and then discretize the multi-term fractional equation by the finite volume method. For the time-fractional derivative, the finite difference method is used. Finally, the iterative scheme is proved to be unconditionally stable and convergent with the accuracy O(σ2 + τ2−β + h3), where τ and h are the time step size and the space step size, respectively. A numerical example is presented to verify the effectiveness of the proposed method.