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Numerical Treatment of the Time-Fractional Kuramoto–Sivashinsky Equation Using a Combined Chebyshev-Collocation Approach

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Yayımlandı:Fractal and Fractional vol. 9, no. 11 (2025), p. 727-750
Yazar: Abd-Elhameed, Waleed Mohamed
Diğer Yazarlar: Abdelkawy, Mohamed A, Alsafri Naher Mohammed A., Atta Ahmed Gamal
Baskı/Yayın Bilgisi:
MDPI AG
Konular:
Applied mathematics
Physics
Partial differential equations
Mathematical analysis
Numerical analysis
Polynomials
Collocation methods
Chebyshev approximation
Approximation
Algorithms
Error analysis
Methods
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Özet:In this paper, we present a collocation algorithm for numerically treating the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Certain orthogonal polynomials, which are expressed as combinations of Chebyshev polynomials, and their shifted polynomials are introduced. Some new theoretical formulas regarding these polynomials have been developed, including their operational matrices of both integer and fractional derivatives. The derived formulas will be the foundation for designing the proposed numerical algorithm, which relies on converting the governing problem with its underlying conditions into a nonlinear algebraic system, which can be solved using Newton’s iteration technique. A rigorous error analysis for the proposed combined Chebyshev expansion is presented. Some numerical examples are given to ensure the applicability and efficiency of the presented algorithm. These results demonstrate that the proposed algorithm attains superior accuracy with fewer expansion terms.
ISSN:2504-3110
DOI:10.3390/fractalfract9110727
Kaynak:Engineering Database