A self-singularity-capturing scheme for fractional differential equations

Abstract

We develop a two-stage computational framework for robust and accurate time-integration of multi-term linear/nonlinear fractional differential equations. In the first stage, we formulate a self-singularity-capturing scheme, given available/observable data for diminutive time, experimentally obtained or sampled from an approximate numerical solution utilizing a fine grid nearby the initial time. The fractional differential equation provides the necessary knowledge/insight on how the hidden singularity can bridge between the initial and the subsequent short-time solution data. In the second stage, we utilize the multi-singular behaviour of solution in a variety of numerical methods, without resorting to making any ad-hoc/uneducated guesses for the solution singularities. Particularly, we employed an implicit finite-difference method, where the captured singularities, in the first stage, are taken into account through some Lubich-like correction terms, leading to an accuracy of order $\mathcal{O}\left(\mathrm{\Delta }{t}^{3-\alpha }\right)$. We show that this novel framework can control the error even in the presence of strong multi-singularities.

Keywords:

• Multi-fractal singularities
• random singularities
• correction terms
• short-time data
• long-time integration
• automated algorithms

2010 Mathematics Subject Classification:

• 34A08

Acknowledgments

The authors would like to thank Dr Ehsan Kharazmi and Dr Yongtao Zhou for productive discussions along the development of this work.

Disclosure statement

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

Additional information

Funding

This work was supported by the Army Research Office (ARO) Young Investigator Program Award [grant number W911NF-19-1-0444], the MURI/ARO [grant number W911NF- 15-1-0562], and partially by the National Science Foundation Award [grant number DMS-1923201] and Air Force Office of Scientific Research (AFOSR) Young Investigator Program Award [grant number FA9550-17-1-0150]