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Peer-reviewedCC BY 4.0Fractal and Fractional

A Shifted One-Parameter Horadam Polynomial Framework for the Time-Fractional Supergeneralized Viscous Burgers Equation

  • Waleed Mohamed Abd-Elhameed1
  • Hussam Salah Alahmadi2
  • Omar Mazen Alqubori2
  • Naher Mohammed A. Alsafri2
  • Amr Kamel Amin3
  • Ahmed Gamal Atta4
4 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, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt
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Abstract

A compact operational-matrix collocation algorithm is introduced and analyzed for the numerical treatment of the time-fractional supergeneralized viscous Burgers equation. The approximation space is generated by the shifted one-parameter Horadam polynomials Hjc, for a positive real number c, and the algebraic properties required by the algorithm are derived explicitly. In particular, we obtain a power form representation, a monomial inversion relation, and closed-form operational matrices for integer and Caputo fractional derivatives. These relations allow the nonlinear fractional equation to be discretized directly in a tensor-product polynomial space and converted into a finite nonlinear system for the expansion coefficients. The resulting system is treated by the standard Newton iteration method. The convergence discussion and the error estimates support the proposed approximation, while the numerical experiments show that accurate solutions are obtained with a relatively small number of collocation points. Comparisons with existing methods indicate the efficiency of the present approach for Burgers-type fractional models.

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Cite

BibTeXRIS
BibTeX
@article{abdelhameed2026shifted,
  title = {{A Shifted One-Parameter Horadam Polynomial Framework for the Time-Fractional Supergeneralized Viscous Burgers Equation}},
  author = {Waleed Mohamed Abd-Elhameed and Hussam Salah Alahmadi and Omar Mazen Alqubori and Naher Mohammed A. Alsafri and Amr Kamel Amin and Ahmed Gamal Atta},
  journal = {Fractal and Fractional},
  year = {2026},
  month = oct,
  doi = {10.3390/fractalfract10100687},
  url = {https://www.mdpi.com/2504-3110/10/10/687},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - A Shifted One-Parameter Horadam Polynomial Framework for the Time-Fractional Supergeneralized Viscous Burgers Equation
AU  - Waleed Mohamed Abd-Elhameed
AU  - Hussam Salah Alahmadi
AU  - Omar Mazen Alqubori
AU  - Naher Mohammed A. Alsafri
AU  - Amr Kamel Amin
AU  - Ahmed Gamal Atta
PY  - 2026
DA  - 2026/10/01
JO  - Fractal and Fractional
DO  - 10.3390/fractalfract10100687
UR  - https://www.mdpi.com/2504-3110/10/10/687
AB  - A compact operational-matrix collocation algorithm is introduced and analyzed for the numerical treatment of the time-fractional supergeneralized viscous Burgers equation. The approximation space is generated by the shifted one-parameter Horadam polynomials Hjc, for a positive real number c, and the algebraic properties required by the algorithm are derived explicitly. In particular, we obtain a power form representation, a monomial inversion relation, and closed-form operational matrices for integer and Caputo fractional derivatives. These relations allow the nonlinear fractional equation to be discretized directly in a tensor-product polynomial space and converted into a finite nonlinear system for the expansion coefficients. The resulting system is treated by the standard Newton iteration method. The convergence discussion and the error estimates support the proposed approximation, while the numerical experiments show that accurate solutions are obtained with a relatively small number of collocation points. Comparisons with existing methods indicate the efficiency of the present approach for Burgers-type fractional models.
SN  - 2504-3110
ER  -