A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials
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 Horada...
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2025-01-01
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| author | Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Ahmed Gamal Atta |
| author_facet | Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Ahmed Gamal Atta |
| author_sort | Waleed Mohamed Abd-Elhameed |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-e5fc58fcb91e446f8a8b5ef4c1c99fbf |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-e5fc58fcb91e446f8a8b5ef4c1c99fbf2025-01-24T13:40:05ZengMDPI AGMathematics2227-73902025-01-0113230010.3390/math13020300A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam PolynomialsWaleed Mohamed Abd-Elhameed0Omar Mazen Alqubori1Ahmed Gamal Atta2Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, EgyptDepartment of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23218, Saudi ArabiaDepartment of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, EgyptThis 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.https://www.mdpi.com/2227-7390/13/2/300generalized Fibonacci polynomialsKorteweg–de Vries equationsoperational matricesconvergence analysiscollocation method |
| spellingShingle | Waleed Mohamed Abd-Elhameed Omar Mazen Alqubori Ahmed Gamal Atta A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials Mathematics generalized Fibonacci polynomials Korteweg–de Vries equations operational matrices convergence analysis collocation method |
| title | A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials |
| title_full | A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials |
| title_fullStr | A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials |
| title_full_unstemmed | A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials |
| title_short | A Collocation Approach for the Nonlinear Fifth-Order KdV Equations Using Certain Shifted Horadam Polynomials |
| title_sort | collocation approach for the nonlinear fifth order kdv equations using certain shifted horadam polynomials |
| topic | generalized Fibonacci polynomials Korteweg–de Vries equations operational matrices convergence analysis collocation method |
| url | https://www.mdpi.com/2227-7390/13/2/300 |
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