A second-order finite difference scheme for quasilinear time fractional parabolic equation based on new fractional derivative

ABSTRACT

Recently, Caputo and Fabrizio introduce a new derivative with fractional order which has the ability to describe the material heterogeneities and the fluctuations of different scales. In this article, a finite difference scheme to solve a quasilinear fractal mobile/immobile transport model based on the new fractional derivative is introduced and analysed. This equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times. Some a priori estimates of discrete $L^{\mathrm{\infty }}\left(L^2\right)$ errors with optimal order of convergence $O({\tau }^2+h^2)$ are established on uniform partition. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Keywords:

• Second order
• new derivative
• quasilinear
• fractional mobile/immobile transport model
• estimates

2010 Mathematics Subject Classifications:

• 65M06
• 65M12
• 65M15
• 26A33

Acknowledgments

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped them to improve the results 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 [grant number 91630207], [grant number 11471194], [grant number 11571115], by the National Science Foundation [grant number DMS-1620194], by the OSD/ARO MURI [grant number W911NF-15-1-0562], by the National Science and Technology Major Project of China [grant number 2011ZX05052], [grant number 2011ZX05011-004], and by Shandong Provincial Natural Science Foundation, China [grant number ZR2011AM015].