Ultraspherical spectral collocation method for two-dimensional tempered space-fractional Zeldovich–Frank–Kamenetskii equations with exponential nonlinearities

M.A. Zaky; M.Z. Youssef; A. Al Kenany; S.S. Ezz-Eldien; M.A. Zaky; M.Z. Youssef; A. Al Kenany; S.S. Ezz-Eldien

doi:10.3934/math.2026196

AIMS Mathematics

2026, Volume 11, Issue 2: 4805-4817. doi: 10.3934/math.2026196

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Ultraspherical spectral collocation method for two-dimensional tempered space-fractional Zeldovich–Frank–Kamenetskii equations with exponential nonlinearities

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box-65892, Riyadh 11566, Saudi Arabia
2.
Mathematics Department, Al-Qunfudah University College, Umm Al-Qura University, Mecca, KSA
3.
Department of Mathematics, Faculty of Science, New Valley University, El-Kharga, 72511, Egypt
4.
Department of Information Technology, Faculty of Computer and Information, Socotra Archipelago University, Socotra, Yemen

Received: 07 October 2025 Revised: 30 December 2025 Accepted: 19 January 2026 Published: 27 February 2026
MSC : 34A08, 65M70, 33C45

This paper introduces a novel spectral collocation method for solving two-dimensional tempered space-fractional Zeldovich–Frank–Kamenetskii (ZFK) equations, which generalize the classical combustion model by incorporating tempered fractional diffusion operators. The tempered fractional ZFK equation is pivotal for modeling anomalous diffusion phenomena in thermal reaction and combustion systems, where nonlocal interactions and memory effects play a critical role. The proposed hybrid numerical scheme combines a spectral collocation method based on ultraspherical polynomials for spatial discretization with an implicit Runge–Kutta (IRK) technique for temporal integration. For the first time, we derive new tempered fractional differentiation matrices in physical space using ultraspherical polynomial bases, enabling efficient handling of the nonlocal tempered fractional operators. The results highlight the effectiveness of the new differentiation matrices in capturing anomalous diffusion phenomena while maintaining spectral accuracy, providing a robust framework for fractional combustion modeling.
- tempered fractional derivative,
- Zeldovich–Frank–Kamenetskii equations,
- spectral method,
- ultraspherical polynomials
Citation: M.A. Zaky, M.Z. Youssef, A. Al Kenany, S.S. Ezz-Eldien. Ultraspherical spectral collocation method for two-dimensional tempered space-fractional Zeldovich–Frank–Kamenetskii equations with exponential nonlinearities[J]. AIMS Mathematics, 2026, 11(2): 4805-4817. doi: 10.3934/math.2026196

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Abstract

This paper introduces a novel spectral collocation method for solving two-dimensional tempered space-fractional Zeldovich–Frank–Kamenetskii (ZFK) equations, which generalize the classical combustion model by incorporating tempered fractional diffusion operators. The tempered fractional ZFK equation is pivotal for modeling anomalous diffusion phenomena in thermal reaction and combustion systems, where nonlocal interactions and memory effects play a critical role. The proposed hybrid numerical scheme combines a spectral collocation method based on ultraspherical polynomials for spatial discretization with an implicit Runge–Kutta (IRK) technique for temporal integration. For the first time, we derive new tempered fractional differentiation matrices in physical space using ultraspherical polynomial bases, enabling efficient handling of the nonlocal tempered fractional operators. The results highlight the effectiveness of the new differentiation matrices in capturing anomalous diffusion phenomena while maintaining spectral accuracy, providing a robust framework for fractional combustion modeling.

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