A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients

We consider the numerical solutions of the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients in a bounded domain. The time fractional derivatives are described in the Caputo sense. A unified numerical scheme based on finite difference method in time and Legendre spectral method in space is proposed. Detailed error analysis is given for the fully discrete scheme. The convergence rate of the proposed scheme in L2 norm is O(τ2+N1−m), where τ, N, and m are the time-step size, polynomial degree, and regularity in the space variable of the exact solution, respectively. Numerical examples are presented to illustrate the theoretical results.