Fully discrete spectral method for solving a novel multi-term time-fractional mixed diffusion and diffusion-wave equation

A novel multi-term time-fractional mixed diffusion and diffusion-wave equation will be considered in this work. Different from the general multi-term time-fractional mixed diffusion and diffusion-wave equations, this new multi-term equation possesses a special time-fractional operator on the spatial derivative. We use a new discrete scheme to approximate the time-fractional derivative, which can improve the temporal accuracy. Then, a fully discrete spectral scheme is developed based on finite difference discretization in time and Legendre spectral approximation in space. Meanwhile, a very important lemma is proposed and proved, to obtain the unconditional stability and convergence of the fully discrete spectral scheme. Finally, four numerical experiments are presented to confirm our theoretical analysis. Both of our analysis and numerical test indicate that the fully discrete scheme is accurate and efficient in solving the generalized multi-term time-fractional mixed diffusion and diffusion-wave equation.