A numerical treatment of the two-dimensional multi-term time-fractional mixed sub-diffusion and diffusion-wave equation

Abstract

In this paper, we consider an important kind of fractional partial differential equations, namely multi-term time-fractional mixed sub-diffusion and diffusion-wave equation. The crucial importance of the considered equation is due to the fact that it generalizes some substantial types of fractional differential equations that can be widely used in describing many real-life phenomena, some of these equations are the time-fractional sub-diffusion, time-fractional diffusion-wave and time-fractional diffusion equations. In this study, the 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation is transformed to its integrated form with respect to time. An extension of the operational matrix of second-order derivative to the 2D case is used in combination with the operational matrix of fractional-order integrals and the time-space spectral collocation method to reduce such equations to systems of algebraic equations, which are solved using any suitable solver. As far as the authors know, this is the first attempt to deal with 2D multi-term time-fractional mixed sub-diffusion and diffusion-wave equation via a spectral approach. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. Our results confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.