A fast compact finite difference method for quasilinear time fractional parabolic equation without singular kernel. Issue 7 (3rd July 2019)

• Record Type: Journal Article
• Authors: Liu, Huan; Cheng, Aijie; Yan, Hongjie; Liu, Zhengguang; Wang, Hong
• Is Part Of: International journal of computer mathematics. Volume 96:Issue 7(2019)
• Page Start: 1444
• Page End: 1460
• Publication Date: 2019-07-03
• Subjects: Anomalous diffusion -- fast solution method -- Caputo–Fabrizio fractional derivative -- error estimates -- nonsingular kernel; 26A33 -- 35K20 -- 65N06 -- 65M12; Computers -- Periodicals; Numerical analysis -- Periodicals; Automation -- Periodicals; 004.0151
• DOI: 10.1080/00207160.2018.1501479
• Languages: English
• ISSNs: 0020-7160

Abstract

ABSTRACT: In 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 requireO ( M N ) memories andO ( M N 2 ) 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 toO ( M N ), and the memory requirement toO ( M ) . Numerical experiments are given to verify the effectiveness and high order convergence of the proposed scheme.