Legendre collocation method for new generalized fractional advection-diffusion equation

Abstract

In this paper, the numerical method for solving a class of generalized fractional advection-diffusion equation (GFADE) is considered. The fractional derivative involving scale and weight factors is imposed for the temporal derivative and is analogous to the Caputo fractional derivative following an integration-after-differentiation composition. It covers many popular fractional derivatives by fixing different weights $w(t)$ and scale functions $z(t)$ inside. The numerical solution of such GFADE is derived via a collocation method, where conventional Legendre polynomials are implemented. Convergence and error analysis of polynomial expansions are studied theoretically. Numerical examples are considered with different boundary conditions to confirm the theoretical findings. By comparing the above examples with those from existing literature, we find that our proposed numerical method is simple, stable and easy to implement.

Keywords:

• Generalized fractional derivatives
• fractional advection-diffusion equation
• collocation method
• error analysis
• convergence analysis

Mathematics Subject Classifications:

• 35R11
• 26A33

Acknowledgments

All authors sincerely thank the reviewers for their constructive comments incorporated in the manuscript, which have significantly improved the original manuscript. In the meanwhile Dr. Yufeng Xu is grateful to Professors Qin Sheng (Baylor University, USA) and Hai-Wei Sun (University of Macau, Macao SAR) for their continuous encouragement and kind support on numerical combustion problems, numerical linear algebra, and numerical PDEs in recent years.

Disclosure statement

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

Additional information

Funding

The author Yufeng Xu is supported by the Natural Science Foundation of Hunan Province [grant number 2019JJ50755] and Natural Science Foundation of China [grant number 51974377].