Pell coefficient polynomials for solving linear hyperbolic first-order partial differential equations via the Tau approach

Waleed Mohamed Abd-Elhameed; Mohamed A. Abdelkawy; Omar Mazen Alqubori; Ahmed Gamal Atta; Waleed Mohamed Abd-Elhameed; Mohamed A. Abdelkawy; Omar Mazen Alqubori; Ahmed Gamal Atta

doi:10.3934/era.2025267

Electronic Research Archive

2025, Volume 33, Issue 10: 6012-6035. doi: 10.3934/era.2025267

Previous Article Next Article

Research article Special Issues

Pell coefficient polynomials for solving linear hyperbolic first-order partial differential equations via the Tau approach

1.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11623, Saudi Arabia
3.
Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 23831, Saudi Arabia
4.
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11341, Cairo, Egypt

Received: 21 July 2025 Revised: 24 September 2025 Accepted: 09 October 2025 Published: 16 October 2025

The main aims of this study are the introduction of Pell coefficient polynomials and their numerical treatment of the first-order hyperbolic partial differential equations. Our suggested numerical algorithm will be derived from the utilization of some novel formulas of the Pell coefficient polynomials, along with the application of the spectral tau method. For the proposed expansion, we investigate the convergence and error estimations in detail. The presented numerical results indicate that the suggested numerical method is accurate, converges exponentially, and is computationally efficient.
- Pell numbers,
- special polynomials,
- spectral methods,
- matrix system,
- convergence analysis
Citation: Waleed Mohamed Abd-Elhameed, Mohamed A. Abdelkawy, Omar Mazen Alqubori, Ahmed Gamal Atta. Pell coefficient polynomials for solving linear hyperbolic first-order partial differential equations via the Tau approach[J]. Electronic Research Archive, 2025, 33(10): 6012-6035. doi: 10.3934/era.2025267

Related Papers:

Abstract

The main aims of this study are the introduction of Pell coefficient polynomials and their numerical treatment of the first-order hyperbolic partial differential equations. Our suggested numerical algorithm will be derived from the utilization of some novel formulas of the Pell coefficient polynomials, along with the application of the spectral tau method. For the proposed expansion, we investigate the convergence and error estimations in detail. The presented numerical results indicate that the suggested numerical method is accurate, converges exponentially, and is computationally efficient.

References

[1]	J. Ockendon, S. Howison, A. Lacey, A. Movchan, Applied Partial Differential Equations, Oxford University Press, Oxford, 2003. https://doi.org/10.1093/oso/9780198527701.001.0001
[2]	T. Roubíček, Nonlinear Partial Differential Equations with Applications, 2013.
[3]	A. Sarma, T. W. Watts, M. Moosa, Y. Liu, P. L. McMahon, Quantum variational solving of nonlinear and multidimensional partial differential equations, Phys. Rev. A, 109 (2024), 062616. https://doi.org/10.1103/PhysRevA.109.062616 doi: 10.1103/PhysRevA.109.062616
[4]	B. Bhaumik, S. De, S. Changdar, Deep learning based solution of nonlinear partial differential equations arising in the process of arterial blood flow, Math. Comput. Simul., 217 (2024), 21–36. https://doi.org/10.1016/j.matcom.2023.10.011 doi: 10.1016/j.matcom.2023.10.011
[5]	S. Tang, X. Feng, W. Wu, H. Xu, Physics-informed neural networks combined with polynomial interpolation to solve nonlinear partial differential equations, Comput. Math. Appl., 132 (2023), 48–62. https://doi.org/10.1016/j.camwa.2022.12.008 doi: 10.1016/j.camwa.2022.12.008
[6]	L. Zada, R. Nawaz, K. S. Nisar, M. Tahir, M. Yavuz, M. K. A. Kaabar, et al., New approximate-analytical solutions to partial differential equations via auxiliary function method, Partial Differ. Equations Appl. Math., 4 (2021), 100045. https://doi.org/10.1016/j.padiff.2021.100045 doi: 10.1016/j.padiff.2021.100045
[7]	G. Sarma, Vezhopalu, Analytical and numerical investigation of nonlinear differential equations: A comprehensive study, J. Comput. Anal. Appl., 34 (2025), 213–222.
[8]	D. Doehring, G. J. Gassner, M. Torrilhon, Many-stage optimal stabilized Runge–Kutta methods for hyperbolic partial differential equations, J. Sci. Comput., 99 (2024), 28. https://doi.org/10.1007/s10915-024-02478-5 doi: 10.1007/s10915-024-02478-5
[9]	S. Sharma, A. Bala, S. Aeri, R. Kumar, K. S. Nisar, Vieta–Lucas matrix approach for the numeric estimation of hyperbolic partial differential equations, Partial Differ. Equations Appl. Math., 11 (2024), 100770. https://doi.org/10.1016/j.padiff.2024.100770 doi: 10.1016/j.padiff.2024.100770
[10]	S. Malik, S. T. Ejaz, A. Akgül, Subdivision collocation method: a new numerical technique for solving hyperbolic partial differential equation in non-uniform medium, Bol. Soc. Mat. Mex., 31 (2025), 50. https://doi.org/10.1007/s40590-025-00731-x doi: 10.1007/s40590-025-00731-x
[11]	S. Karthick, V. Subburayan, R. P. Agarwal, Solving a system of one-dimensional hyperbolic delay differential equations using the method of lines and Runge–Kutta methods, Computation, 12 (2024), 64. https://doi.org/10.3390/computation12040064 doi: 10.3390/computation12040064
[12]	Y. H. Youssri, R. M. Hafez, Exponential Jacobi spectral method for hyperbolic partial differential equations, Math. Sci., 13 (2019), 347–354. https://doi.org/10.1007/s40096-019-00304-w doi: 10.1007/s40096-019-00304-w
[13]	J. P. Boyd. Chebyshev & Fourier Spectral Methods, Springer-Verlag Berlin, 2001.
[14]	J. Shen, T. Tang, L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, 2011. https://doi.org/10.1007/978-3-540-71041-7
[15]	M. Kashif, M. Singh, T. Som, E. M. Craciun, Numerical study of variable order model arising in chemical processes using operational matrix and collocation method, J. Comput. Sci., 80 (2024), 102339. https://doi.org/10.1016/j.jocs.2024.102339 doi: 10.1016/j.jocs.2024.102339
[16]	G. Manohara, S. Kumbinarasaiah, An innovative Fibonacci wavelet collocation method for the numerical approximation of Emden–Fowler equations, Appl. Numer. Math., 201 (2024), 347–369. https://doi.org/10.1016/j.apnum.2024.03.016 doi: 10.1016/j.apnum.2024.03.016
[17]	E. Gebril, M. S. El-Azab, M. Sameeh, Chebyshev collocation method for fractional Newell–Whitehead–Segel equation, Alexandria Eng. J., 87 (2024), 39–46. https://doi.org/10.1016/j.aej.2023.12.025 doi: 10.1016/j.aej.2023.12.025
[18]	W. M. Abd-Elhameed, M. M. Alsuyuti, New spectral algorithm for fractional delay pantograph equation using certain orthogonal generalized Chebyshev polynomials, Commun. Nonlinear Sci. Numer. Simul., 141 (2025), 108479. https://doi.org/10.1016/j.cnsns.2024.108479 doi: 10.1016/j.cnsns.2024.108479
[19]	W. M. Abd-Elhameed, A. M. Al-Sady, O. M. Alqubori, A. G. Atta, Numerical treatment of the fractional Rayleigh–Stokes problem using some orthogonal combinations of Chebyshev polynomials, AIMS Math., 9 (2024), 25457–25481. https://doi.org/10.3934/math.20241243 doi: 10.3934/math.20241243
[20]	O. E. Hepson, Numerical simulations of Kuramoto–Sivashinsky equation in reaction-diffusion via Galerkin method, Math. Sci., 15 (2021), 199–206. https://doi.org/10.1007/s40096-021-00402-8 doi: 10.1007/s40096-021-00402-8
[21]	A. A. El-Sayed, S. Boulaaras, N. H. Sweilam, Numerical solution of the fractional-order logistic equation via the first-kind Dickson polynomials and spectral tau method, Math. Methods Appl. Sci., 46 (2023), 8004–8017. https://doi.org/10.1002/mma.7345 doi: 10.1002/mma.7345
[22]	W. M. Abd-Elhameed, O. M. Alqubori, A. K. Al-Harbi, M. H. Alharbi, A. G. Atta, Generalized third-kind Chebyshev tau approach for treating the time fractional cable problem, Electron. Res. Arch., 32 (2024), 6200–6224. https://doi.org/10.3934/era.2024288 doi: 10.3934/era.2024288
[23]	W. M. Abd-Elhameed, A. F. Abu Sunayh, M. H. Alharbi, A. G. Atta, Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation, AIMS Math., 9 (2024), 34567–34587. https://doi.org/10.3934/math.20241646 doi: 10.3934/math.20241646
[24]	T. Koshy, Fibonacci and Lucas Numbers With Applications, John Wiley & Sons, 2019. https://doi.org/10.1002/9781118742297
[25]	R. M. Hafez, H. M. Ahmed, O. M. Alqubori, A. K. Amin, W. M. Abd-Elhameed, Efficient spectral Galerkin and collocation approaches using telephone polynomials for solving some models of differential equations with convergence analysis, Mathematics, 13 (2025), 918. https://doi.org/10.3390/math13060918 doi: 10.3390/math13060918
[26]	W. M. Abd-Elhameed, O. M. Alqubori, A. G. Atta, A collocation procedure for treating the time-fractional FitzHugh–Nagumo differential equation using shifted Lucas polynomials, Mathematics, 12 (2024), 3672. https://doi.org/10.3390/math12233672 doi: 10.3390/math12233672
[27]	M. H. Heydari, Z. Avazzadeh, Fibonacci polynomials for the numerical solution of variable-order space-time fractional Burgers–Huxley equation, Math. Methods Appl. Sci., 44 (2021), 6774–6786. https://doi.org/10.1002/mma.7222 doi: 10.1002/mma.7222
[28]	W. M. Abd-Elhameed, O. M. Alqubori, A. G. Atta, A collocation approach for the nonlinear fifth-order KdV equations using certain shifted Horadam polynomials, Mathematics, 13 (2025), 300. https://doi.org/10.3390/math13020300 doi: 10.3390/math13020300
[29]	E. Aourir, H. L. Dastjerdi, M. Oudani, K. Shah, T. Abdeljawad, Numerical technique based on Bernstein polynomials approach for solving auto-convolution VIEs and the initial value problem of auto-convolution VIDEs, J. Appl. Math. Comput., 71 (2025), 4697–4727. https://doi.org/10.1007/s12190-025-02400-8 doi: 10.1007/s12190-025-02400-8
[30]	Ö. Oruç, A new numerical treatment based on Lucas polynomials for 1D and 2D sinh–Gordon equation, Commun. Nonlinear Sci. Numer. Simul., 57 (2018), 14–25. https://doi.org/10.1016/j.cnsns.2017.09.006 doi: 10.1016/j.cnsns.2017.09.006
[31]	B. P. Moghaddam, A. Dabiri, A. M. Lopes, J. A. T. Machado, Numerical solution of mixed-type fractional functional differential equations using modified Lucas polynomials, Comput. Appl. Math., 38 (2019), 46. https://doi.org/10.1007/s40314-019-0813-9 doi: 10.1007/s40314-019-0813-9
[32]	M. Pourbabaee, A. Saadatmandi, A numerical schemes based on Vieta–Lucas polynomials for evaluating the approximate solution of some types of fractional optimal control problems, Iran. J. Sci. Technol. Trans. Electr. Eng., 49 (2025), 873–895. https://doi.org/10.1007/s40998-025-00791-9 doi: 10.1007/s40998-025-00791-9
[33]	S. Çelik, I. Durukan, E. Özkan, New recurrences on Pell numbers, Pell–Lucas numbers, Jacobsthal numbers, and Jacobsthal–Lucas numbers, Chaos, Solitons Fractals, 150 (2021), 111173. https://doi.org/10.1016/j.chaos.2021.111173 doi: 10.1016/j.chaos.2021.111173
[34]	Z. Şiar, R.Keskin, k-Generalized Pell numbers which are concatenation of two repdigits, Mediterr. J. Math., 19 (2022), 180. https://doi.org/10.1007/s00009-022-02099-y doi: 10.1007/s00009-022-02099-y
[35]	J. J. Bravo, J. L. Herrera, F. Luca, On a generalization of the Pell sequence, Math. Bohemica, 146 (2021), 199–213. https://doi.org/10.21136/MB.2020.0098-19 doi: 10.21136/MB.2020.0098-19
[36]	P. K. Singh, S. S. Ray, A numerical approach based on Pell polynomial for solving stochastic fractional differential equations, Numer. Algorithms, 97 (2024), 1513–1534. https://doi.org/10.1007/s11075-024-01760-9 doi: 10.1007/s11075-024-01760-9
[37]	H. C. Yaslan, Pell polynomial solution of the nonlinear variable order space fractional PDEs, J. Fract. Calc. Appl., 16 (2025), 1–16. https://doi.org/10.21608/jfca.2024.255438.1052 doi: 10.21608/jfca.2024.255438.1052
[38]	H. C. Yaslan,, Pell polynomial solution of the fractional differential equations in the Caputo–Fabrizio sense, Indian J. Pure Appl. Math., (2024), 1–12. https://doi.org/10.1007/s13226-024-00684-3
[39]	M. Izadi, H. M. Srivastava, K. Mamehrashi, Numerical simulations of Rosenau–Burgers equations via Crank–Nicolson spectral Pell matrix algorithm, J. Appl. Math. Comput., 71 (2025), 1009–1033. https://doi.org/10.1007/s12190-024-02273-3 doi: 10.1007/s12190-024-02273-3
[40]	T. Koshy, Pell and Pell–Lucas Numbers with Applications, Springer, 2014. https://doi.org/10.1007/978-1-4614-8489-9
[41]	G. B. Djordjevic, G. V. Milovanovic, Special Classes of Polynomials, University of Nis, Faculty of Technology Leskovac, 2014.
[42]	A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid, Y. H. Youssri, Shifted fifth-kind Chebyshev Galerkin treatment for linear hyperbolic first-order partial differential equations, Appl. Numer. Math., 167 (2021), 237–256. https://doi.org/10.1016/j.apnum.2021.05.010 doi: 10.1016/j.apnum.2021.05.010
[43]	S. Kumbinarasaiah, M. Mulimani, A novel scheme for the hyperbolic partial differential equation through Fibonacci wavelets, J. Taibah Univ. Sci., 16 (2022), 1112–1132. https://doi.org/10.1080/16583655.2022.2143636 doi: 10.1080/16583655.2022.2143636
[44]	S. Singh, V. K. Patel, V. K. Singh, Application of wavelet collocation method for hyperbolic partial differential equations via matrices, Appl. Math. Comput., 320 (2018), 407–424. https://doi.org/10.1016/j.amc.2017.09.043 doi: 10.1016/j.amc.2017.09.043
[45]	A. H. Bhrawy, R. M. Hafez, E. Alzahrani, D. Baleanu, A. A. Alzahrani, Generalized Laguerre–Gauss–Radau scheme for first order hyperbolic equations on semi-infinite domains, Rom. J. Phys., 60 (2015), 918–934. Available from: http://hdl.handle.net/20.500.12416/1554.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)