A Finite Difference/Finite Volume Method For Solving The Fractional Diffusion Wave Equation

In this paper, we present and analyze a fully discrete numerical method for solving the time-fractional diffusion wave equation: partial derivative(beta)(t) u - div(a del u) = f, 1 < beta< 2. We first construct a difference formula to approximate partial derivative(beta)(t) u by using an interpolation of derivative type. The truncation error of this formula is of O(Delta t(2+delta-beta))-order if function u(t) is an element of C-2;delta [0; T] where 0 <= delta <= 1 is the Holder continuity index. This error order can come up to O(Delta t(3-beta)) if u(t) is an element of C-3[0; T]. Then, in combinination with the linear finite volume discretization on spatial domain, we give a fully discrete scheme for the fractional wave equation. We prove that the fully discrete scheme is unconditionally stable and the discrete solution admits the optimal error estimates in the H-1 -norm and L-2-norm, respectively. Numerical examples are provided to verify the effectiveness of the proposed numerical method.