Fourth-order effective approximation of the normalized Riemann-Liouville tempered fractional derivatives and its applications

Jianxin Li; Zeshan Qiu; Jianxin Li; Zeshan Qiu

doi:10.3934/math.2025794

AIMS Mathematics

2025, Volume 10, Issue 8: 17801-17831. doi: 10.3934/math.2025794

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Research article

Fourth-order effective approximation of the normalized Riemann-Liouville tempered fractional derivatives and its applications

Jianxin Li ,
Zeshan Qiu ^,

Department of Basic Education, Xinjiang University of Political Science and Law, Tumushuke 843900, China

Received: 07 June 2025 Revised: 29 July 2025 Accepted: 31 July 2025 Published: 06 August 2025
MSC : 65M06, 65M12

In this paper, a fourth-order quasi-compact approximation for the normalized Riemann-Liouville tempered fractional derivatives was proposed. Its effectiveness was proved by using the generating function method, and it was applied to the numerical solution of the two-sided space tempered fractional diffusion equation with the time Caputo tempered fractional derivative. For the time Caputo tempered fractional derivative, we transformed the Caputo tempered fractional derivative into the Riemann-Liouville tempered fractional derivative through the relationship between them, and then employed the tempered weighted and shifted Grünwald difference operator to approximate the Riemann-Liouville tempered fractional derivative in the time direction. Thus, an efficient numerical scheme with second-order accuracy in time and fourth-order accuracy in space was derived. The stability and convergence of the numerical scheme were rigorously and elaborately proved, and the effectiveness of the numerical scheme was verified by a series of simulations conducted on numerical examples.
Citation: Jianxin Li, Zeshan Qiu. Fourth-order effective approximation of the normalized Riemann-Liouville tempered fractional derivatives and its applications[J]. AIMS Mathematics, 2025, 10(8): 17801-17831. doi: 10.3934/math.2025794

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Abstract

In this paper, a fourth-order quasi-compact approximation for the normalized Riemann-Liouville tempered fractional derivatives was proposed. Its effectiveness was proved by using the generating function method, and it was applied to the numerical solution of the two-sided space tempered fractional diffusion equation with the time Caputo tempered fractional derivative. For the time Caputo tempered fractional derivative, we transformed the Caputo tempered fractional derivative into the Riemann-Liouville tempered fractional derivative through the relationship between them, and then employed the tempered weighted and shifted Grünwald difference operator to approximate the Riemann-Liouville tempered fractional derivative in the time direction. Thus, an efficient numerical scheme with second-order accuracy in time and fourth-order accuracy in space was derived. The stability and convergence of the numerical scheme were rigorously and elaborately proved, and the effectiveness of the numerical scheme was verified by a series of simulations conducted on numerical examples.

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