Spectral collocation approach for solving the time-fractional Kuramoto-Sivashinsky equation using the Fibonacci coefficient polynomials

Waleed Mohamed Abd-Elhameed; Ahmed H. Al-Mehmadi; Naher Mohammed A. Alsafri; Omar Mazen Alqubori; Mohamed Adel; Ahmed Gamal Atta; Waleed Mohamed Abd-Elhameed; Ahmed H. Al-Mehmadi; Naher Mohammed A. Alsafri; Omar Mazen Alqubori; Mohamed Adel; Ahmed Gamal Atta

doi:10.3934/math.2025805

AIMS Mathematics

2025, Volume 10, Issue 8: 18070-18093. doi: 10.3934/math.2025805

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Spectral collocation approach for solving the time-fractional Kuramoto-Sivashinsky equation using the Fibonacci coefficient polynomials

1.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, Saudi Arabia
4.
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, Egypt

Received: 01 July 2025 Revised: 31 July 2025 Accepted: 04 August 2025 Published: 08 August 2025
MSC : 65M60, 11B39, 40A05, 34A08

This work presents an algorithm that applies the typical collocation method to solve the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Our suggested new basis functions are the Fibonacci coefficient polynomials. We develop new theoretical results of these polynomials, such as their integer and fractional derivatives. The proposed approach efficiently estimates spatial and temporal derivatives using the operational matrices (OMs) of derivatives of the introduced polynomials. The approach converts the TFKSE controlled by their conditions into a non-linear system of equations that may be handled numerically. A thorough error and convergence analysis study for the proposed Fibonacci coefficient expansion is presented. Several illustrative numerical examples, including those with known exact solutions, are presented to confirm the efficiency and applicability of the suggested approach, even with fewer terms of basis functions. In addition, comparisons with some numerical methods are presented to verify our numerical approach's high accuracy.
- time-fractional differential equations,
- Fibonacci numbers,
- collocation method,
- operational matrices,
- convergence analysis
Citation: Waleed Mohamed Abd-Elhameed, Ahmed H. Al-Mehmadi, Naher Mohammed A. Alsafri, Omar Mazen Alqubori, Mohamed Adel, Ahmed Gamal Atta. Spectral collocation approach for solving the time-fractional Kuramoto-Sivashinsky equation using the Fibonacci coefficient polynomials[J]. AIMS Mathematics, 2025, 10(8): 18070-18093. doi: 10.3934/math.2025805

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Abstract

This work presents an algorithm that applies the typical collocation method to solve the time-fractional Kuramoto–Sivashinsky equation (TFKSE). Our suggested new basis functions are the Fibonacci coefficient polynomials. We develop new theoretical results of these polynomials, such as their integer and fractional derivatives. The proposed approach efficiently estimates spatial and temporal derivatives using the operational matrices (OMs) of derivatives of the introduced polynomials. The approach converts the TFKSE controlled by their conditions into a non-linear system of equations that may be handled numerically. A thorough error and convergence analysis study for the proposed Fibonacci coefficient expansion is presented. Several illustrative numerical examples, including those with known exact solutions, are presented to confirm the efficiency and applicability of the suggested approach, even with fewer terms of basis functions. In addition, comparisons with some numerical methods are presented to verify our numerical approach's high accuracy.

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