Solving Time-fractional Diffusion Equations By The Matrix Mittag-Leffler Function

Over the past few decades,time-fractional diffusion equations have been widely used in many fields such as engineering,physics,biology and so on.In the paper,we first briefly review the most relevant existing recent numerical methods.Then a numerical scheme is constructed for solving the time-fractional diffusion equation.The main idea consists of two steps.In the first step,a spectral method is used to discretize the equation in space to obtain a fractional ordinary differential equation system.Then in the second step,the analytical solution is deduced for this ordinary equation system.Finally different methods are proposed to approximate the integral operators in the time direction for the different types of source terms.The key point in making the algorithm efficent is to use the properties of the matrix Mittag-Leffler function.For the evaluation of matrix MittagLeffler functions,two fast algorithms are discussed:Caratheodory-Fejer approximation and Schur-Parlett algorithm,both of which are highly efficient.Several numerical examples are provided to verify the efficiency of the proposed method.