Afast numerical scheme for avariably distributed-order time-fractional diffusion equation and its analysis

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)logN) computations and O(N logN) storage are required for generating and storing temporal discretization coefficients, where.. 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(MN2) of the time-stepping method to O(MN log(3)N) with.. being the number of spatial nodes. Numerical experiments are presented to substantiate the theoretical results.

Automated summary

A fast numerical method is developed for a variably distributed-order time-fractional diffusion equation modeling anomalous diffusion with uncertainties in inhomogeneous medium. The method has the same accuracy as traditional schemes but requires less computations and storage. Additionally, a fast divide and conquer algorithm is designed to reduce computational complexity when solving the linear system.