A posteriori error estimation and adaptive strategy for a nonlinear fractional differential equation

Abstract

A nonlinear fractional differential equation with Caputo fractional derivative is considered. The problem is discretized by an upwind finite difference scheme for which a posteriori error analysis in the maximum norm is derived. A partly heuristic argument based on this a posteriori error estimation leads to several suitable monitor functions, and a new monitor function is constructed to design an adaptive grid algorithm. Numerical results are presented to illustrate the performance of the presented adaptive method. Compared with the other monitor functions, the presented adaptive grid method based on the new monitor function is more suitable to solve this type of nonlinearfractional differential equation.

Keywords:

• Caputo fractional derivative
• finite difference
• adaptive grid
• a posteriori error estimation
• monitor function

AMS Subject Classifications::

• 65L05
• 65L12
• 65L20

Acknowledgments

We would like to thank the anonymous referee for several suggestions for the improvement of this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the State Key Program of National Natural Science Foundation of China (11931003) and National Science Foundation of China (41974133, 11761015, 11671157, 11971410), the Natural Science Foundation of Guangxi (2020GXNSFAA159010, 2018GXNSFAA294079) and the key project of Natural Science Foundation of Guangxi Province (2017GXNSFDA198014, 2018GXNSFDA050014, AD20238065).