@article{Youssef2025, 
author = {Maged Z. Youssef and Mahmoud A. Zaky and Faizah A.H. Alomari and Amra Al Kenany and Samer Ezz-Eldien},
title = {Efficient Jacobi spectral-IRK method for one- and two-dimensional tempered fractional Allen-Cahn equations},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {8},
pages = {19795-19815},
keywords = {tempered fractional Allen-Cahn equation, Jacobi polynomials, spectral collocation, implicit Runge-Kutta, exponential tempering},
url = {https://www.sciopen.com/article/10.3934/math.2025883},
doi = {10.3934/math.2025883},
abstract = {Tempered fractional Allen-Cahn equations have many vital applications in science and engineering. The presence of a tempered fractional derivative enables these equations to efficiently explain more complex systems where long-range effects exist with an exponential decay. In this paper, we provide a numerical approach to tempered space-fractional Allen-Cahn equations. The spectral collocation method is implemented, based on Jacobi polynomials, to reduce one- and two-dimensional tempered space-fractional Allen-Cahn equations to a system of ordinary differential equations in the time direction. Then, the implicit Runge-Kutta method is applied to approximate the resulting system. This is the first work that uses the implicit Runge-Kutta method to solve one- and two-dimensional tempered space-fractional Allen-Cahn equations. High accuracy of the spectral collocation method, together with the simplicity and low computational cost of the implicit Runge-Kutta method, represent key advantages of the proposed scheme when applied to such a problem. Numerical results for two test problems are performed to test the validity and superiority of the suggested numerical scheme over other numerical schemes.}
}