A fast numerical scheme for a variably distributed-order time-fractional diffusion equation and its analysis. (15th February 2022) - Jia, Jinhong; Wang, Hong; Zheng, Xiangcheng
Abstract: We develop a fast numerical method for a variably distributed-order time-fractional diffusion equation modeling the anomalous diffusion with uncertainties in inhomogeneous medium. Different from the analysis techniques of the commonly-used schemes for time-fractional problems like L1 methods, error estimates are proved based on a novel discretization coefficient splitting. The proposed method has the same accuracy as traditional schemes, while only O ( N 3 / 2 log ⁡ N ) computations and O ( N log ⁡ N ) storage are required for generating and storing temporal discretization coefficients, where N refers to the number of time steps. We further design a corresponding fast divide and conquer algorithm to reduce the computational complexity of solving the linear system from O ( M N 2 ) of the time-stepping method to O ( M N log 3 ⁡ N ) with M being the number of spatial nodes. Numerical experiments are presented to substantiate the theoretical results.