A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials

Waleed Mohamed Abd-Elhameed (), Omar Mazen Alqubori and Ahmed Gamal Atta
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Waleed Mohamed Abd-Elhameed: Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
Omar Mazen Alqubori: Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi Arabia
Ahmed Gamal Atta: Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt

Mathematics, 2025, vol. 13, issue 2, 1-26

Abstract: This paper proposes a numerical algorithm for the nonlinear fifth-order Korteweg–de Vries equations. This class of equations is known for its significance in modeling various complex wave phenomena in physics and engineering. The approximate solutions are expressed in terms of certain shifted Horadam polynomials. A theoretical background for these polynomials is first introduced. The derivatives of these polynomials and their operational metrics of derivatives are established to tackle the problem using the typical collocation method to transform the nonlinear fifth-order Korteweg–de Vries equation governed by its underlying conditions into a system of nonlinear algebraic equations, thereby obtaining the approximate solutions. This paper also includes a rigorous convergence analysis of the proposed shifted Horadam expansion. To validate the proposed method, we present several numerical examples illustrating its accuracy and effectiveness.

Keywords: generalized Fibonacci polynomials; Korteweg–de Vries equations; operational matrices; convergence analysis; collocation method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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