An operational treatment for two-dimensional time-fractional Gray-Scott models

Samer S. Ezz-Eldien; Ali H. Tedjani; Amra Al Kenany; Marwa Alzubaidi; Samer S. Ezz-Eldien; Ali H. Tedjani; Amra Al Kenany; Marwa Alzubaidi

doi:10.3934/math.2026215

AIMS Mathematics

2026, Volume 11, Issue 2: 5246-5269. doi: 10.3934/math.2026215

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An operational treatment for two-dimensional time-fractional Gray-Scott models

1.
Department of Mathematics, Faculty of Science, New Valley University, El-Kharga, 72511, Egypt
2.
Department of İnformation Technology, Faculty of Computer and İnformation, Archipelago University, Socotra, Yemen
3.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
4.
Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Mecca, KSA
5.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia

Received: 08 October 2025 Revised: 16 January 2026 Accepted: 20 January 2026 Published: 28 February 2026
MSC : 65L05, 65R20, 65N35, 65L03

Due to the poor regularity of solutions to time-fractional partial differential equations, traditional spectral approaches typically suffer from significant accuracy reductions when applied in the time domain. To overcome this problem, in this paper, we applied a spectral approach based on a new set of basis functions, which are smooth in the spatial direction and non-smooth in the time direction. The spectral collocation approach was used combined with the operational matrix approach based on the new set of basis functions to solve two-dimensional time-fractional Gray-Scott models. Applying the operational technique reduces the computations of the full scheme and achieves significant accuracy by using a small number of these functions. Numerical results confirmed the high accuracy of the proposed approach when applied for smooth and non-smooth solutions.
- Legendre polynomial,
- spectral collocation method,
- operational matrix,
- fractional differential equations,
- Gray-Scott model
Citation: Samer S. Ezz-Eldien, Ali H. Tedjani, Amra Al Kenany, Marwa Alzubaidi. An operational treatment for two-dimensional time-fractional Gray-Scott models[J]. AIMS Mathematics, 2026, 11(2): 5246-5269. doi: 10.3934/math.2026215

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Abstract

Due to the poor regularity of solutions to time-fractional partial differential equations, traditional spectral approaches typically suffer from significant accuracy reductions when applied in the time domain. To overcome this problem, in this paper, we applied a spectral approach based on a new set of basis functions, which are smooth in the spatial direction and non-smooth in the time direction. The spectral collocation approach was used combined with the operational matrix approach based on the new set of basis functions to solve two-dimensional time-fractional Gray-Scott models. Applying the operational technique reduces the computations of the full scheme and achieves significant accuracy by using a small number of these functions. Numerical results confirmed the high accuracy of the proposed approach when applied for smooth and non-smooth solutions.

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