Efficient Jacobi spectral-IRK method for one- and two-dimensional tempered fractional Allen-Cahn equations

Maged Z. Youssef; Mahmoud A. Zaky; Faizah A.H. Alomari; Amra Al Kenany; Samer Ezz-Eldien; Maged Z. Youssef; Mahmoud A. Zaky; Faizah A.H. Alomari; Amra Al Kenany; Samer Ezz-Eldien

doi:10.3934/math.2025883

AIMS Mathematics

2025, Volume 10, Issue 8: 19795-19815. doi: 10.3934/math.2025883

Previous Article Next Article

Research article

Efficient Jacobi spectral-IRK method for one- and two-dimensional tempered fractional Allen-Cahn equations

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Al-Baha University, Al-Baha 65779, Saudi Arabia
3.
Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Mecca, KSA
4.
Department of Mathematics, Faculty of Science, New Valley University, El-Kharga 72511, Egypt

Received: 20 June 2025 Revised: 07 August 2025 Accepted: 19 August 2025 Published: 28 August 2025
MSC : 65L05, 65R20, 65N35, 65L03

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.
- tempered fractional Allen-Cahn equation,
- Jacobi polynomials,
- spectral collocation,
- implicit Runge-Kutta,
- exponential tempering
Citation: Maged Z. Youssef, Mahmoud A. Zaky, Faizah A.H. Alomari, Amra Al Kenany, Samer Ezz-Eldien. Efficient Jacobi spectral-IRK method for one- and two-dimensional tempered fractional Allen-Cahn equations[J]. AIMS Mathematics, 2025, 10(8): 19795-19815. doi: 10.3934/math.2025883

Related Papers:

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.

References

[1]	S. M. Allen, J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall., 27 (1979), 1085–1095. https://doi.org/10.1016/0001-6160(79)90196-2 doi: 10.1016/0001-6160(79)90196-2
[2]	A. A. Wheeler, M. J. Boettinger, G. B. Mcfadden, Phase-field model for isothermal phase transitions in binary alloys, Phys. Rev. A, 45 (1992), 7424–7439. https://doi.org/10.1103/PhysRevA.45.7424 doi: 10.1103/PhysRevA.45.7424
[3]	L. Golubovic, A. Levandovsky, D. Moldovan, Interface dynamics and far-from-equilibrium phase transitions in multilayer epitaxial growth and erosion on crystal surfaces: continuum theory insights, East Asian J. Appl. Math., 1 (2011), 297–371. https://doi.org/10.4208/eajam.040411.030611a doi: 10.4208/eajam.040411.030611a
[4]	J. Kim, Phase field models for multi-component fuid flows, Commun. Comput. Phys., 12 (2012), 613–661. https://doi.org/10.4208/cicp.301110.040811a doi: 10.4208/cicp.301110.040811a
[5]	Y. Yu, W Zhou, Z. Zhang, Q. Bi, Analysis on the motion of nonlinear vibration with fractional order and time variable mass, Appl. Math. Lett., 124 (2022), 107621. https://doi.org/10.1016/j.aml.2021.107621 doi: 10.1016/j.aml.2021.107621
[6]	B. Yin, Y. Liu, H. Li, S. He, Fast algorithm based on TT-M FE system for space fractional Allen-Cahn equations with smooth and non-smooth solutions, J. Comput. Phys., 379 (2019), 351–372. https://doi.org/10.1016/j.jcp.2018.12.004 doi: 10.1016/j.jcp.2018.12.004
[7]	H. Li, Y. Li, Y. Zeng, Z. Luo, A reduced-dimension extrapolation two-grid Crank-Nicolson finite element method of unknown solution coefficient vectors for spatial fractional nonlinear Allen-Cahn equations, Comput. Math. Appl., 167 (2024), 110–122. https://doi.org/10.1016/j.camwa.2024.05.007 doi: 10.1016/j.camwa.2024.05.007
[8]	H. Yuan, Z. Hui, A class of higher-order time-splitting Monte Carlo method for fractional Allen-Cahn equation, Appl. Math. Lett., 163 (2025), 109467. https://doi.org/10.1016/j.aml.2025.109467 doi: 10.1016/j.aml.2025.109467
[9]	L. Bu, L. Mei, Y. Hou, Stable second-order schemes for the space-fractional Cahn-Hilliard and Allen-Cahn equations, Comput. Math. Appl., 78 (2019), 3485–3500. https://doi.org/10.1016/j.camwa.2019.08.015 doi: 10.1016/j.camwa.2019.08.015
[10]	B. Zhang, Y. Yang, An adaptive unconditional maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation, Comput. Math. Appl., 139 (2023), 28–37. https://doi.org/10.1016/j.camwa.2023.02.022 doi: 10.1016/j.camwa.2023.02.022
[11]	S. Zhai, C. Ye, Z. Weng, A fast and efficient numerical algorithm for fractional Allen-Cahn with precise nonlocal mass conservation, Appl. Math. Lett., 103 (2020), 106190. https://doi.org/10.1016/j.aml.2019.106190 doi: 10.1016/j.aml.2019.106190
[12]	W. Chen, D. Lu, H. Chen, H. Liu, Crank-Nicolson Legendre spectral approximation for space-fractional Allen-Cahn equation, Elect. J. Differ. Equ. Texas State Uni., 2019 (2019), 76. https://doi.org/10.37218/ejde.2019.76.1 doi: 10.37218/ejde.2019.76.1
[13]	T. Hou, T. Tang, J. Yang, Numerical analysis of fully discretized Crank-Nicolson scheme for fractional-in-space Allen-Cahn equations, J. Sci. Comput., 72 (2017), 1214–1231. https://doi.org/10.1007/s10915-017-0396-9 doi: 10.1007/s10915-017-0396-9
[14]	B. Zhang, Y. Yang, A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen‑Cahn equation, Numer. Algor., 93 (2023), 179–202. https://doi.org/10.1007/s11075-022-01411-x doi: 10.1007/s11075-022-01411-x
[15]	M. M. Meerschaert, F. Sabzikar, M. S. Phanikumar, A. Zeleke, Tempered fractional timeseries model for turbulence in geophysical flows, J. Stat. Mech.: Theory Exper., 14 (2014), 1742–5468. https://doi.org/10.1088/1742-5468/2014/09/P09023 doi: 10.1088/1742-5468/2014/09/P09023
[16]	M. M. Meerschaert, Y. Zhang, B. Baeumer, Tempered anomalous diffusion in heteroge-neous systems, Geophys. Res. Lett., 35 (2008), L17403. https://doi.org/10.1029/2008GL034899 doi: 10.1029/2008GL034899
[17]	A. Cartea, D. Del-Castillo-Negrete, Fractional diffusion models of option prices in marketswith jumps, Phys. A: Stat. Mech. Appl., 374 (2007), 749–763. https://doi.org/10.1016/j.physa.2006.08.071 doi: 10.1016/j.physa.2006.08.071
[18]	A. Hanyga, Wave propagation in media with singular memory, Math. Comput. Model., 34 (2001), 1399–1421. https://doi.org/https://doi.org/10.1016/S0895-7177(01)00137-6 doi: 10.1016/S0895-7177(01)00137-6
[19]	M. A. Zaky, Existence, uniqueness and numerical analysis of solutions of tempered fractional boundary value problems, Appl. Numer. Math., 145 (2019), 429–457. https://doi.org/10.1016/j.apnum.2019.05.008 doi: 10.1016/j.apnum.2019.05.008
[20]	R. Almeida, N. Martins, J. V. C. Sousa, Fractional tempered differential equations depending on arbitrary kernels, AIMS Math., 9 (2024), 9107–9127. https://doi.org/10.3934/math.2024443 doi: 10.3934/math.2024443
[21]	I. Lamrani, H. Zitane, D. F. M. Torres, Controllability and observability of tempered fractional differential systems, Commun. Nonlinear Sci. Numer. Simul., 142 (2025), 108501. https://doi.org/0.1016/j.cnsns.2024.108501
[22]	M. S. Heris, M. Javidi, A predictor-corrector scheme for the tempered fractional diferential equations with uniform and non-uniform meshes, J. Supercomput., 75 (2019), 8168–8206. https://doi.org/10.1007/s11227-019-02979-3 doi: 10.1007/s11227-019-02979-3
[23]	T. Zhao, Efficient spectral collocation method for tempered fractional differential equations, Fractal Fract., 7 (2023), 277. https://doi.org/10.3390/fractalfract7030277 doi: 10.3390/fractalfract7030277
[24]	H. K. Dwivedi, Rajeev, A novel fast tempered algorithm with high-accuracy scheme for 2D tempered fractional reaction-advection-subdiffusion equation, Comput. Math. Appl., 176 (2024), 371–397. https://doi.org/10.1016/j.camwa.2024.11.004 doi: 10.1016/j.camwa.2024.11.004
[25]	A. Bibi, M. Rehman, A numerical method for solutions of tempered fractional differential equations, J. Comput. Appl. Math., 443 (2024), 115772. https://doi.org/10.1016/j.cam.2024.115772 doi: 10.1016/j.cam.2024.115772
[26]	M. Partohaghighi, E. Asante-Asamani, O. S. Iyiola, A robust numerical scheme for solving Riesz-tempered fractional reaction–diffusion equations, J. Comput. Appl. Math., 450 (2024), 115992. https://doi.org/10.1016/j.cam.2024.115992 doi: 10.1016/j.cam.2024.115992
[27]	S. Duo, Y. Zhang, Numerical approximations for the tempered fractional Laplacian: error analysis and applications, J. Sci. Comput., 81 (2019), 569–593. https://doi.org/10.1007/s10915-019-01029-7 doi: 10.1007/s10915-019-01029-7
[28]	H. Zhang, J. Yan, X. Qian, X. Gu, S. Song, On the preserving of the maximum principle and energy stability of high-order implicit-explicit Runge-Kutta schemes for the space-fractional Allen-Cahn equation, Numer. Algor., 88 (2021), 1309–1336. https://doi.org/10.1007/s11075-021-01077-x doi: 10.1007/s11075-021-01077-x

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)