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Peer-reviewedCC BY 4.0Mathematics

Two-Parameter General-Shift Lucas Polynomial Spectral Collocation for Time-Fractional Nonlinear Partial Integro-Differential Equations

  • Waleed Mohamed Abd-Elhameed1
  • Naher Mohammed A. Alsafri2
  • Omar Mazen Alqubori2
  • Amr Kamel Amin3
  • Mohammed Atef Abdallah4
  • Ahmed Gamal Atta5
5 affiliations
  1. 1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
  2. 2Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
  3. 3Department of Mathematics, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi Arabia
  4. 4Department of Mathematics, Al-Lith University College, Umm Al-Qura University, Al-Lith 24382, Saudi Arabia
  5. 5Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt
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Abstract

The present study introduces a two-parameter general-shift Lucas basis from the classical Lucas family via an affine argument transformation, which then forms the basis of a spectral collocation approach for the numerical treatment of nonlinear time-fractional partial integro-differential equations. This parametrized development of the ordinary Lucas basis provides a unified framework for both the classical Lucas polynomials and the standard shifted forms of the Lucas basis. We obtain closed-form power and inverse-power representations in terminating hypergeometric form, which are used to derive the operational matrices of derivatives for ordinary and Caputo fractional differentiation. A tensor-product expansion discretizes the space–time variables. Quadrature approximation is applied to the Volterra integral using Gauss–Legendre quadrature, while collocation of the governing equation, along with its underlying conditions, leads to a system of nonlinear algebraic equations that can be handled by Newton’s method. To evaluate the performance of our proposed basis in terms of accuracy, sensitivity to parameters, and computational efficiency, we provide error bounds and conduct numerical experiments. Our results demonstrate that our proposed basis achieves high levels of accuracy while offering the flexibility provided by its two affine parameters.

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Cite

BibTeXRIS
BibTeX
@article{abdelhameed2026twoparameter,
  title = {{Two-Parameter General-Shift Lucas Polynomial Spectral Collocation for Time-Fractional Nonlinear Partial Integro-Differential Equations}},
  author = {Waleed Mohamed Abd-Elhameed and Naher Mohammed A. Alsafri and Omar Mazen Alqubori and Amr Kamel Amin and Mohammed Atef Abdallah and Ahmed Gamal Atta},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14213912},
  url = {https://www.mdpi.com/2227-7390/14/21/3912},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Two-Parameter General-Shift Lucas Polynomial Spectral Collocation for Time-Fractional Nonlinear Partial Integro-Differential Equations
AU  - Waleed Mohamed Abd-Elhameed
AU  - Naher Mohammed A. Alsafri
AU  - Omar Mazen Alqubori
AU  - Amr Kamel Amin
AU  - Mohammed Atef Abdallah
AU  - Ahmed Gamal Atta
PY  - 2026
DA  - 2026/10/26
JO  - Mathematics
DO  - 10.3390/math14213912
UR  - https://www.mdpi.com/2227-7390/14/21/3912
AB  - The present study introduces a two-parameter general-shift Lucas basis from the classical Lucas family via an affine argument transformation, which then forms the basis of a spectral collocation approach for the numerical treatment of nonlinear time-fractional partial integro-differential equations. This parametrized development of the ordinary Lucas basis provides a unified framework for both the classical Lucas polynomials and the standard shifted forms of the Lucas basis. We obtain closed-form power and inverse-power representations in terminating hypergeometric form, which are used to derive the operational matrices of derivatives for ordinary and Caputo fractional differentiation. A tensor-product expansion discretizes the space–time variables. Quadrature approximation is applied to the Volterra integral using Gauss–Legendre quadrature, while collocation of the governing equation, along with its underlying conditions, leads to a system of nonlinear algebraic equations that can be handled by Newton’s method. To evaluate the performance of our proposed basis in terms of accuracy, sensitivity to parameters, and computational efficiency, we provide error bounds and conduct numerical experiments. Our results demonstrate that our proposed basis achieves high levels of accuracy while offering the flexibility provided by its two affine parameters.
SN  - 2227-7390
ER  -