A coupled numerical method for time-fractional convection-diffusion equations based on compressed Legendre polynomials

Hui Zhu; Lei Yang; Ruimin Zhang; Yuntao Jia; Hui Zhu; Lei Yang; Ruimin Zhang; Yuntao Jia

doi:10.3934/math.2026792

AIMS Mathematics

2026, Volume 11, Issue 7: 19488-19515. doi: 10.3934/math.2026792

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Research article

A coupled numerical method for time-fractional convection-diffusion equations based on compressed Legendre polynomials

1.
School of Computer Science and Engineering, Faculty of Innovation Engineering, Macau University of Science and Technology, Macau 999078, China
2.
Beijing Institute of Technology, Zhuhai 519088, China
3.
Innovation Technology Research Institute, Macau University of Science and Technology, Zhuhai, 519000, Guangdong, China
4.
Guangzhou City University of Technology, Guangzhou, 510800

Received: 12 April 2026 Revised: 06 June 2026 Accepted: 17 June 2026 Published: 02 July 2026
MSC : 35R11, 65M12

In this paper, we develop a novel coupled numerical method for time-fractional convection-diffusion equations. First, in the temporal dimension, a derivative-based linear interpolation is utilized to approximate the time partial derivative. Then, we construct a high-order approximate scheme for the temporal fractional derivative, which transforms the original partial differential equation into a semi-discrete ordinary differential system involving only spatial variables. In the spatial dimension, a standard orthogonal basis is constructed by using compressed Legendre polynomials. Combining the concept of $ \varepsilon $-approximate solutions, we derive the numerical solution of the semi-discrete scheme. Convergence analysis and numerical experiments demonstrate that the proposed algorithm achieves second-order convergence in time, and the spatial convergence accuracy improves as the Legendre polynomial degree and compression level increase.
- coupled numerical method,
- compressed Legendre polynomials,
- semi-discrete scheme,
- time fractional convection-diffusion equations
Citation: Hui Zhu, Lei Yang, Ruimin Zhang, Yuntao Jia. A coupled numerical method for time-fractional convection-diffusion equations based on compressed Legendre polynomials[J]. AIMS Mathematics, 2026, 11(7): 19488-19515. doi: 10.3934/math.2026792

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Abstract

In this paper, we develop a novel coupled numerical method for time-fractional convection-diffusion equations. First, in the temporal dimension, a derivative-based linear interpolation is utilized to approximate the time partial derivative. Then, we construct a high-order approximate scheme for the temporal fractional derivative, which transforms the original partial differential equation into a semi-discrete ordinary differential system involving only spatial variables. In the spatial dimension, a standard orthogonal basis is constructed by using compressed Legendre polynomials. Combining the concept of $ \varepsilon $-approximate solutions, we derive the numerical solution of the semi-discrete scheme. Convergence analysis and numerical experiments demonstrate that the proposed algorithm achieves second-order convergence in time, and the spatial convergence accuracy improves as the Legendre polynomial degree and compression level increase.

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