Spectral solutions for nonlinear static beam and fractional Riccati problems using new Lucas coefficient polynomials

Waleed Mohamed Abd-Elhameed; Shuja'a Ali Alsulami; Omar Mazen Alqubori; Naher Mohammed A. Alsafri; Mohamed Adel; Ahmed Gamal Atta; Waleed Mohamed Abd-Elhameed; Shuja'a Ali Alsulami; Omar Mazen Alqubori; Naher Mohammed A. Alsafri; Mohamed Adel; Ahmed Gamal Atta

doi:10.3934/math.2025932

AIMS Mathematics

2025, Volume 10, Issue 9: 20862-20890. doi: 10.3934/math.2025932

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Spectral solutions for nonlinear static beam and fractional Riccati problems using new Lucas 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: 07 July 2025 Revised: 28 August 2025 Accepted: 02 September 2025 Published: 10 September 2025
MSC : 11B83, 65M70, 35R11

This study proposes novel spectral algorithms employing the Lucas coefficient polynomials to solve two significant nonlinear models: The static beam problem and the fractional Riccati equation. Our suggested approaches are based on developing novel theoretical findings that we derive for the introduced polynomials. These results include formulae for inversion, moment, derivatives, and linearization. Two methodologies are followed to treat the two nonlinear problems. The nonlinear fourth-order integro-differential static beam problem is treated using the collocation method, while the nonlinear fractional Riccati equation is treated using the tau method. Rigorous convergence and error analysis for the Lucas coefficient expansions are given. Compared to previous methods, the suggested algorithms exhibit exponential convergence and high accuracy, as verified by numerical testing. The findings demonstrate that spectral algorithms may effectively handle nonlinear differential equations using Lucas coefficient polynomials.
- Lucas numbers,
- spectral methods,
- nonlinear fractional Riccati equation,
- nonlinear fourth-order integro-differential equation,
- convergence analysis
Citation: Waleed Mohamed Abd-Elhameed, Shuja'a Ali Alsulami, Omar Mazen Alqubori, Naher Mohammed A. Alsafri, Mohamed Adel, Ahmed Gamal Atta. Spectral solutions for nonlinear static beam and fractional Riccati problems using new Lucas coefficient polynomials[J]. AIMS Mathematics, 2025, 10(9): 20862-20890. doi: 10.3934/math.2025932

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Abstract

This study proposes novel spectral algorithms employing the Lucas coefficient polynomials to solve two significant nonlinear models: The static beam problem and the fractional Riccati equation. Our suggested approaches are based on developing novel theoretical findings that we derive for the introduced polynomials. These results include formulae for inversion, moment, derivatives, and linearization. Two methodologies are followed to treat the two nonlinear problems. The nonlinear fourth-order integro-differential static beam problem is treated using the collocation method, while the nonlinear fractional Riccati equation is treated using the tau method. Rigorous convergence and error analysis for the Lucas coefficient expansions are given. Compared to previous methods, the suggested algorithms exhibit exponential convergence and high accuracy, as verified by numerical testing. The findings demonstrate that spectral algorithms may effectively handle nonlinear differential equations using Lucas coefficient polynomials.

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