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

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 require $O\left(MN\right)$ memories and $O\left(M{N}^2\right)$ 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 $O\left(MN\right)$, and the memory requirement to $O\left(M\right)$. Numerical experiments are given to verify the effectiveness and high order convergence of the proposed scheme.

KEYWORDS:

• Anomalous diffusion
• fast solution method
• Caputo–Fabrizio fractional derivative
• error estimates
• nonsingular kernel

2010 AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 35K20
• 65N06
• 65M12

Acknowledgements

The authors would like to express their sincere thanks to the editor and referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China under Grants 91630207, 11471194 and 11571115, by the Division of Mathematical Sciences under Grant DMS-1620194, by the OSD/ARO MURI Grant W911NF-15-1-0562, by the National Science and Technology Major Project of China under Grants 2011ZX05052 and 2011ZX05011-004, and by Natural Science Foundation of Shandong Province under Grant ZR2011AM015, and by the Taishan Scholars Program of Shandong Province of China.