@article{Li2025, 
author = {Jianxin Li and Zeshan Qiu},
title = {Fourth-order effective approximation of the normalized Riemann-Liouville tempered fractional derivatives and its applications},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {8},
pages = {17801-17831},
keywords = {normalized Riemann-Liouville tempered fractional derivatives, effective fourth-order quasi-compact approximation, space-time tempered fractional diffusion equation, tempered weighted and shifted Grünwald difference operator, stability and convergence},
url = {https://www.sciopen.com/article/10.3934/math.2025794},
doi = {10.3934/math.2025794},
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.}
}