A Shifted Balanced Polynomial Collocation Framework for Variable-Coefficient Multi-Term Time-Fractional Convection–Diffusion–Reaction Problems
5 affiliations
- 1Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
- 2Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
- 3Department of Mathematics, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi Arabia
- 4Department of Mathematics, Al-Lith University College, Umm Al-Qura University, Al-Lith 24382, Saudi Arabia
- 5Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt
Abstract
We develop a spectral collocation method based on shifted balanced polynomials for solving variable-coefficient multi-term time-fractional convection–diffusion–reaction equations. The proposed approach uses the Caputo fractional derivative and a global space–time approximation. First, explicit formulas are derived for expressing the shifted balanced polynomials in the power basis and for expressing monomials in terms of the shifted balanced polynomial basis. These formulas are then used to obtain a representation of the Caputo operator at the selected points. The approximate solution is expanded in a tensor-product basis, and the governing equation is enforced at selected interior collocation points together with the prescribed initial and boundary conditions. Theoretical results concerning the growth of the shifted balanced polynomials and the decay of the expansion coefficients are established. These results are used to prove the absolute convergence of the proposed expansion and to derive an explicit upper bound for the truncation error. Four numerical examples are presented to examine the accuracy and effectiveness of the proposed method for different polynomial degrees and fractional orders. In addition, an account of the extension of the proposed approach to problems involving nonlinear terms is presented. The numerical results show that accurate approximations can be obtained using relatively small polynomial degrees.
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BibTeX
@article{abdelhameed2026shifted,
title = {{A Shifted Balanced Polynomial Collocation Framework for Variable-Coefficient Multi-Term Time-Fractional Convection–Diffusion–Reaction Problems}},
author = {Waleed Mohamed Abd-Elhameed and Omar Mazen Alqubori and Naher Mohammed A. Alsafri and Amr Kamel Amin and Mohammed Atef Abdallah and Ahmed Gamal Atta},
journal = {Fractal and Fractional},
year = {2026},
month = oct,
doi = {10.3390/fractalfract10110733},
url = {https://www.mdpi.com/2504-3110/10/11/733},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - A Shifted Balanced Polynomial Collocation Framework for Variable-Coefficient Multi-Term Time-Fractional Convection–Diffusion–Reaction Problems
AU - Waleed Mohamed Abd-Elhameed
AU - Omar Mazen Alqubori
AU - Naher Mohammed A. Alsafri
AU - Amr Kamel Amin
AU - Mohammed Atef Abdallah
AU - Ahmed Gamal Atta
PY - 2026
DA - 2026/10/22
JO - Fractal and Fractional
DO - 10.3390/fractalfract10110733
UR - https://www.mdpi.com/2504-3110/10/11/733
AB - We develop a spectral collocation method based on shifted balanced polynomials for solving variable-coefficient multi-term time-fractional convection–diffusion–reaction equations. The proposed approach uses the Caputo fractional derivative and a global space–time approximation. First, explicit formulas are derived for expressing the shifted balanced polynomials in the power basis and for expressing monomials in terms of the shifted balanced polynomial basis. These formulas are then used to obtain a representation of the Caputo operator at the selected points. The approximate solution is expanded in a tensor-product basis, and the governing equation is enforced at selected interior collocation points together with the prescribed initial and boundary conditions. Theoretical results concerning the growth of the shifted balanced polynomials and the decay of the expansion coefficients are established. These results are used to prove the absolute convergence of the proposed expansion and to derive an explicit upper bound for the truncation error. Four numerical examples are presented to examine the accuracy and effectiveness of the proposed method for different polynomial degrees and fractional orders. In addition, an account of the extension of the proposed approach to problems involving nonlinear terms is presented. The numerical results show that accurate approximations can be obtained using relatively small polynomial degrees.
SN - 2504-3110
ER -