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Open AccessArticle
Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation
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Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia
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Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt
3
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
*
Author to whom correspondence should be addressed.
Fractal Fract. 2024, 8(7), 427; https://doi.org/10.3390/fractalfract8070427 (registering DOI)
Submission received: 18 June 2024
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Revised: 18 July 2024
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Accepted: 18 July 2024
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Published: 20 July 2024
Abstract
This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs.
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MDPI and ACS Style
Alharbi, M.H.; Sunayh, A.F.A.; Atta, A.G.; Abd-Elhameed, W.M.
Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation. Fractal Fract. 2024, 8, 427.
https://doi.org/10.3390/fractalfract8070427
AMA Style
Alharbi MH, Sunayh AFA, Atta AG, Abd-Elhameed WM.
Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation. Fractal and Fractional. 2024; 8(7):427.
https://doi.org/10.3390/fractalfract8070427
Chicago/Turabian Style
Alharbi, Mohammed H., Abdullah F. Abu Sunayh, Ahmed Gamal Atta, and Waleed Mohamed Abd-Elhameed.
2024. "Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation" Fractal and Fractional 8, no. 7: 427.
https://doi.org/10.3390/fractalfract8070427
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