Fast compact finite difference schemes on graded meshes for fourth-order multi-term fractional sub-diffusion equations with the first Dirichlet boundary conditions

Abstract

In this paper, fast compact finite difference schemes are derived for fourth-order multi-term fractional sub-diffusion equations with initial singularity under the first Dirichlet boundary conditions. In contrast to the direct scheme, the fast algorithm adopted to approximate the Caputo derivative reduces the computation costs effectively. Sharp error estimate of the proposed scheme for linear models is rigorously presented by the energy method. Furthermore, to handle more intricate nonlinear models, a crucial Grönwall inequality is then deduced to analyse the stability and convergence of the linearized scheme. It is worth pointing out that the Grönwall inequality is also helpful in numerical analysis of multi-step schemes for other problems, such as, integro-differential equations with multiple fractional derivatives. Ultimately, numerical examples are provided to verify the efficiency of the established difference schemes.

Keywords:

• Fourth-order multi-term fractional sub-diffusion equation
• first Dirichlet boundary conditions
• non-smooth solution
• fast compact difference scheme
• stability and convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 35R11

Acknowledgments

The authors would like to thank the editor and reviewers for their constructive comments and suggestions, which helped the authors to improve the quality of the paper significantly.

Disclosure statement

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

Additional information

Funding

This research was partly supported by the National Natural Science Foundation of China [grant number 11701103], Young Top-notch Talent Program of Guangdong Province [grant number 2017GC010379], Natural Science Foundation of Guangdong Province [grant number 2022A1515012147], the Project of Science and Technology of Guangzhou [grant number 202102020704], and the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University [grant number 2021023].