Bell polynomial-based semi-discretization approach for the extended Fisher-Kolmogorov equations

M. J. Huntul; M. J. Huntul

doi:10.3934/math.20251286

AIMS Mathematics

2025, Volume 10, Issue 12: 29263-29284. doi: 10.3934/math.20251286

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Bell polynomial-based semi-discretization approach for the extended Fisher-Kolmogorov equations

M. J. Huntul ^,

Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Saudi Arabia

Received: 07 October 2025 Revised: 21 November 2025 Accepted: 04 December 2025 Published: 12 December 2025
MSC : 41A10, 41A58, 65L60, 65M12

This work introduces an integrated numerical framework for obtaining numerical approximations to the extended Fisher-Kolmogorov (EFK) equation model. The methodology entails a two-stage process: initial linearization of the governing equation through a Taylor series approach, followed by the application of a spectral collocation method utilizing Bell polynomials to resolve the resultant linear system. Comprehensive theoretical considerations, including a detailed error estimate in the weighted $ L^2 $-norm, are established. The numerical investigation, comprising three illustrative test cases, confirms the scheme's computational efficiency and demonstrates a marked improvement in accuracy when compared to current state-of-the-art results and exact solutions.
- Bell polynomials,
- Taylor approach,
- extended Fisher-Kolmogorov equation,
- collocation nodes,
- error analysis
Citation: M. J. Huntul. Bell polynomial-based semi-discretization approach for the extended Fisher-Kolmogorov equations[J]. AIMS Mathematics, 2025, 10(12): 29263-29284. doi: 10.3934/math.20251286

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Abstract

This work introduces an integrated numerical framework for obtaining numerical approximations to the extended Fisher-Kolmogorov (EFK) equation model. The methodology entails a two-stage process: initial linearization of the governing equation through a Taylor series approach, followed by the application of a spectral collocation method utilizing Bell polynomials to resolve the resultant linear system. Comprehensive theoretical considerations, including a detailed error estimate in the weighted $ L^2 $-norm, are established. The numerical investigation, comprising three illustrative test cases, confirms the scheme's computational efficiency and demonstrates a marked improvement in accuracy when compared to current state-of-the-art results and exact solutions.

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