A fast compact finite difference method for quasilinear time fractional parabolic equation without singular kernel

Abstract

ABSTRACTIn this paper, a fast compact finite difference method for quasilinear time fractional parabolic equation without singular kernel is developed and analysed. Compact difference scheme is used as a high order approximation for spatial derivative in the fractional parabolic equation, and the Caputo–Fabrizio (C–F) fractional derivative is discretized by a second-order approximation. We have proved that the proposed scheme has fourth-order spatial accuracy and second-order temporal accuracy.However, due to the nonlocal nature of fractional operator, numerically solving the time fractional parabolic equation with traditional direct solvers generally require memories and computational complexity, where N and M represent the number of time steps and grid points in space, respectively. We developed a fast evaluation scheme for the new C–F fractional derivative, which significantly reduced the computational complexity to , and the memory requirement to . Numerical experiments are given to verify the effectiveness and high order convergence of the proposed scheme.