Efficient methods for nonlinear time fractional diffusion-wave equations and their fast implementations

Recently, numerous numerical schemes for solving linear time fractional diffusion-wave equations have been developed. However, most of these methods require relatively high smoothness in time and need extensive computational work and large storage due to the nonlocal property of fractional derivatives. In this paper, an efficient scheme and an alternating direction implicit (ADI) scheme are constructed for one-dimensional and two-dimensional nonlinear time fractional diffusion-wave equations based on their equivalent partial integro-differential equations. The proposed methods require weaker smoothness in time compared to the methods based on discretizing fractional derivative directly. They are proved to be unconditionally stable and convergent with first-order of accuracy in time and second order of accuracy in space. Fast implementations of the proposed methods are presented by the sum-of-exponentials (SOE) approximation for the kernel t− 2+α on the interval [τ,T], where 1 < α < 2. Finally, numerical experiments are carried out to illustrate the theoretical results of our direct schemes and demonstrate their powerful computational performances.