Robust numerical techniques for solving nonlinear ordinary and fractional partial differential problems

Waleed Mohamed Abd-Elhameed; Hussam Salah Alahmadi; Omar Mazen Alqubori; Amr Kamel Amin; Ahmed Gamal Atta; Waleed Mohamed Abd-Elhameed; Hussam Salah Alahmadi; Omar Mazen Alqubori; Amr Kamel Amin; Ahmed Gamal Atta

doi:10.3934/era.2026346

Electronic Research Archive

2026, Volume 34, Issue 11: 8119-8139. doi: 10.3934/era.2026346

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Robust numerical techniques for solving nonlinear ordinary and fractional partial differential problems

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, Adham University College, Umm Al-Qura University, Makkah 28653, Saudi Arabia; akgadelrab@uqu.edu.sa
4.
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo 11341, Egypt

Received: 29 July 2026 Revised: 29 August 2026 Accepted: 11 September 2026 Published: 22 September 2026

This manuscript presents novel spectral techniques based on shifted orthonormal normalized Jacobi polynomials (SONJPs) for solving two types of fractional differential equations. The first method uses the tau technique to solve the nonlinear boundary value problem (NBVP), and the second method uses the collocation approach to solve the time-fractional generalized Fisher problem (TFGFP) that has importance in mathematical biology, population dynamics, and economic modeling. We derive new relations that form the foundation for the proposed numerical techniques, thereby reducing the problem and its governing conditions to a nonlinear algebraic system that can be solved efficiently using Newton's method or another suitable iterative technique. We present numerical results with comparisons to illustrate the suggested algorithm's high accuracy and applicability.
- Jacobi polynomials,
- spectral methods,
- nonlinear boundary value problems,
- time-fractional generalized Fisher equation
Citation: Waleed Mohamed Abd-Elhameed, Hussam Salah Alahmadi, Omar Mazen Alqubori, Amr Kamel Amin, Ahmed Gamal Atta. Robust numerical techniques for solving nonlinear ordinary and fractional partial differential problems[J]. Electronic Research Archive, 2026, 34(11): 8119-8139. doi: 10.3934/era.2026346

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Abstract

This manuscript presents novel spectral techniques based on shifted orthonormal normalized Jacobi polynomials (SONJPs) for solving two types of fractional differential equations. The first method uses the tau technique to solve the nonlinear boundary value problem (NBVP), and the second method uses the collocation approach to solve the time-fractional generalized Fisher problem (TFGFP) that has importance in mathematical biology, population dynamics, and economic modeling. We derive new relations that form the foundation for the proposed numerical techniques, thereby reducing the problem and its governing conditions to a nonlinear algebraic system that can be solved efficiently using Newton's method or another suitable iterative technique. We present numerical results with comparisons to illustrate the suggested algorithm's high accuracy and applicability.

References

[1]	R. Magin, Fractional calculus in bioengineering, part 1, Crit. Rev. Biomed. Eng., 32 (2004). https://doi.org/10.1615/CritRevBiomedEng.v32.i1.10
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, in North-Holland Mathematics Studies, Elsevier, 204 (2006).
[3]	M. Harker, Fractional Differential Equations: Numerical Methods for Applications, Springer Cham, 2026.
[4]	F. Wang, S. Y. Wen, Q. Fang, P. Wang, Application of Adomian decomposition method to fractional order partial differential equations, Therm. Sci., 29 (2025), 1375–1381. https://doi.org/10.2298/TSCI2502375W doi: 10.2298/TSCI2502375W
[5]	C. V. D. Kumar, D. G. Prakasha, P. Veeresha, M. Kapoor, A homotopy-based computational scheme for two-dimensional fractional cable equation, Mod. Phys. Lett. B, 38 (2024), 2450292. https://doi.org/10.1142/S0217984924502920 doi: 10.1142/S0217984924502920
[6]	M. Shams, A. Alalyani, High performance adaptive step size fractional numerical scheme for solving fractional differential equations, Sci. Rep., 15 (2025), 13006. https://doi.org/10.1038/s41598-025-95613-7 doi: 10.1038/s41598-025-95613-7
[7]	A. García, M. Negreanu, F. Ureña, A. M. Vargas, On the numerical solution to space fractional differential equations using meshless finite differences, J. Comput. Appl. Math., 457 (2025), 116322. https://doi.org/10.1016/j.cam.2024.116322 doi: 10.1016/j.cam.2024.116322
[8]	Z. Odibat, A finite difference-based adams-type approach for numerical solution of nonlinear fractional differential equations: a fractional Lotka-Volterra model as a case study, J. Comput. Nonlinear Dyn., 20 (2025), 011004. https://doi.org/10.1115/1.4066885 doi: 10.1115/1.4066885
[9]	H. Jafari, S. Salati, M. Matinfar, V. T. Nguyen, A numerical scheme for a class of nonlinear multi-order fractional differential equations, Fractals, 33 (2025), 2540132. https://doi.org/10.1142/S0218348X25401322 doi: 10.1142/S0218348X25401322
[10]	S. M. Sivalingam, P. Kumar, H. Trinh, V. Govindaraj, A novel L1-predictor-corrector method for the numerical solution of the generalized-Caputo type fractional differential equations, Math. Comput. Simul., 220 (2024), 462–480. https://doi.org/10.1016/j.matcom.2024.01.017 doi: 10.1016/j.matcom.2024.01.017
[11]	J. Singh, J. Kumar, D. Kumar, D. Baleanu, A reliable numerical algorithm based on an operational matrix method for treatment of a fractional order computer virus model, AIMS Math., 9 (2024), 3195–3210. https://doi.org/10.3934/math.2024155 doi: 10.3934/math.2024155
[12]	R. Chaudhary, S. Aeri, A. Bala, R. Kumar, D. Baleanu, Solving system of fractional differential equations via Vieta-Lucas operational matrix method, Int. J. Appl. Comput. Math., 10 (2024), 14. https://doi.org/10.1007/s40819-023-01656-7 doi: 10.1007/s40819-023-01656-7
[13]	X. Zhang, Y. Feng, Z. Luo, J. Liu, A spatial sixth-order numerical scheme for solving fractional partial differential equation, Appl. Math. Lett., 159 (2025), 109265. https://doi.org/10.1016/j.aml.2024.109265 doi: 10.1016/j.aml.2024.109265
[14]	A. S. Bekela, A. T. Deresse, An efficient numerical method for nonlinear time fractional hyperbolic partial differential equations based on fractional Shehu transform iterative method, J. Appl. Math., 2025 (2025), 7007124. https://doi.org/10.1155/jama/7007124 doi: 10.1155/jama/7007124
[15]	W. M. Abd-Elhameed, H. M. Alshehri, M. H. Alharbi, O. M. Alqubori, A. K. Amin, A. G. Atta, Shifted Pell-Lucas polynomial-based algorithms for fractional and classical Rosenau-Hyman models, Fractal Fract., 10 (2026), 359. https://doi.org/10.3390/fractalfract10060359 doi: 10.3390/fractalfract10060359
[16]	M. A. Abdelkawy, W. M. Abd-Elhameed, S. S. Alzahrani, A. G. Atta, A. Biswas, Numerical solutions via shifted Pell polynomials for third-order Rosenau-Hyman and Gilson-Pickering equations, Mathematics, 14 (2026), 582. https://doi.org/10.3390/math14030582 doi: 10.3390/math14030582
[17]	O. M. Alqubori, S. A. Alsulami, A. G. Atta, A. K. Amin, W. M. Abd-Elhameed, Numerical scheme for solving the time-fractional Huxley equation using shifted Dickson polynomials, AIMS Math., 11 (2026), 13744–13766. https://doi.org/10.3934/math.2026566 doi: 10.3934/math.2026566
[18]	P. Senfiazad, M. H. Heydari, M. Bayram, D. Baleanu, A numerical method based on the shifted Jacobi polynomials for a class of tempered fractional quadratic integro-differential equations, Results Appl. Math., 27 (2025), 100601. https://doi.org/10.1016/j.rinam.2025.100601 doi: 10.1016/j.rinam.2025.100601
[19]	K. Sadri, D. Amilo, E. Hinçal, A combination of classical and shifted Jacobi polynomials for two-dimensional time-fractional diffusion-wave equations, Chaos, Solitons Fractals, 198 (2025), 116569. https://doi.org/10.1016/j.chaos.2025.116569 doi: 10.1016/j.chaos.2025.116569
[20]	S. S. Mirshojaei, E. Shivanian, B. Ghayebi, S. J. Hosseini Ghoncheh, A computational method based on shifted Jacobi polynomials for the fractional differential equations, J. Nonlinear Math. Phys., 32 (2025), 15. https://doi.org/10.1007/s44198-025-00273-w doi: 10.1007/s44198-025-00273-w
[21]	M. H. Heydari, S. Zhagharian, M. Razzaghi, Jacobi polynomials for the numerical solution of multi-dimensional stochastic multi-order time fractional diffusion-wave equations, Comput. Math. Appl., 152 (2023), 91–101. https://doi.org/10.1016/j.camwa.2023.10.010 doi: 10.1016/j.camwa.2023.10.010
[22]	M. B. Melad, H. F. Ahmed, Romanovski-Jacobi polynomials approach for fractional mobile-immobile advection-dispersion model with Atangana-Baleanu-Caputo derivative, Phys. Scr., 100 (2025), 095229. https://doi.org/10.1088/1402-4896/adfea0 doi: 10.1088/1402-4896/adfea0
[23]	M. H. Heydari, M. Razzaghi, Jacobi polynomials method for a coupled system of Hadamard fractional Klein-Gordon-Schrödinger equations, Alexandria Eng. J., 107 (2024), 73–86. https://doi.org/10.1016/j.aej.2024.07.016 doi: 10.1016/j.aej.2024.07.016
[24]	Y. Zhou, Y. Jiao, Spectral method for one dimensional Benjamin-Bona-Mahony-Burgers equation using the transformed generalized Jacobi polynomial, Math. Model. Anal., 29 (2024), 509–524. https://doi.org/10.3846/mma.2024.18595 doi: 10.3846/mma.2024.18595
[25]	H. Singh, R. K. Pandey, H. M. Srivastava, Solving non-linear fractional variational problems using Jacobi polynomials, Mathematics, 7 (2019), 224. https://doi.org/10.3390/math7030224 doi: 10.3390/math7030224
[26]	S. C. Shiralashetti, V. R. Pala, S. I. Hanaji, Jacobi wavelet integration operational matrix method for the numerical solution of laminar nanofluid flow problem, Int. J. Ambient Energy, 46 (2025), 2458178. https://doi.org/10.1080/01430750.2025.2458178 doi: 10.1080/01430750.2025.2458178
[27]	M. M. Alsuyuti, E. H. Doha, S. S. Ezz-Eldien, Numerical simulation for classes of one- and two-dimensional multi-term time-fractional diffusion and diffusion-wave equation based on shifted Jacobi Galerkin scheme, Math. Methods Appl. Sci., 48 (2025), 8217–8244. https://doi.org/10.1002/mma.9659 doi: 10.1002/mma.9659
[28]	Y. Jiao, T. Li, Z. Zhang, Jacobi spectral collocation method of space-fractional Navier-Stokes equations, Appl. Math. Comput., 488 (2025), 129111. https://doi.org/10.1016/j.amc.2024.129111 doi: 10.1016/j.amc.2024.129111
[29]	W. M. Abd-Elhameed, A. H. Al-Mehmadi, N. M. A. Alsafri, O. M. Alqubori, M. Adel, A. G. Atta, Spectral collocation approach for solving the time-fractional Kuramoto-Sivashinsky equation using the Fibonacci coefficient polynomials, AIMS Math., 10 (2025), 18070–18093. https://doi.org/10.3934/math.2025805 doi: 10.3934/math.2025805
[30]	A. A. E. 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
[31]	A. G. Atta, M. A. Abdelkawy, A. Napoli, W. M. Abd-Elhameed, Galerkin approach by certain shifted Jacobi polynomials for solving the time-fractional Black-Scholes equation, Boundary Value Probl., 2025 (2025), 138. https://doi.org/10.1186/s13661-025-02131-6 doi: 10.1186/s13661-025-02131-6
[32]	P. Rahimkhani, Y. Ordokhani, E. Babolian, A new operational matrix based on Bernoulli wavelets for solving fractional delay differential equations, Numer. Algorithms, 74 (2017), 223–245. https://doi.org/10.1007/s11075-016-0146-3 doi: 10.1007/s11075-016-0146-3
[33]	H. M. Ahmed, Enhanced shifted Jacobi operational matrices of integrals: Spectral algorithm for solving some types of ordinary and fractional differential equations, Boundary Value Probl., 2024 (2024), 75. https://doi.org/10.1186/s13661-024-01880-0 doi: 10.1186/s13661-024-01880-0
[34]	R. S. V. Kumar, G. Sowmya, S. K. Abhilasha, B. C. Prasannakumara, Solution by operational matrix based Vieta-Lucas collocation method to analyze the thermal performance of the convex spine fin applicable in air conditioning systems, Int. Commun. Heat Mass Transfer, 161 (2025), 108397. https://doi.org/10.1016/j.icheatmasstransfer.2024.108397 doi: 10.1016/j.icheatmasstransfer.2024.108397
[35]	D. Zaidi, I. Talib, M. B. Riaz, P. Agarwal, Novel derivative operational matrix in Caputo sense with applications, J. Taibah Univ. Sci., 18 (2024), 2333061. https://doi.org/10.1080/16583655.2024.2333061 doi: 10.1080/16583655.2024.2333061
[36]	W. M. Abd-Elhameed, A. H. Al-Mehmadi, N. M. A. Alsafri, O. M. Alqubori, A. K. Amin, A. G. Atta, Numerical solution for gas dynamics equation involving Caputo-time fractional derivative using a family of shifted Chebyshev polynomials, Fractal Fract., 10 (2026), 299. https://doi.org/10.3390/fractalfract10050299 doi: 10.3390/fractalfract10050299
[37]	A. E. Owoyemi, C. Phang, Y. T. Toh, An efficient numerical scheme for solving multiorder tempered fractional differential equations via operational matrix, J. Math., 2022 (2022), 7628592. https://doi.org/10.1155/2022/7628592 doi: 10.1155/2022/7628592
[38]	I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, 198 (1998).
[39]	W. M. Abd-Elhameed, A. Ali, Y. H. Youssri, Newfangled linearization formula of certain nonsymmetric Jacobi polynomials: numerical treatment of nonlinear Fisher's equation, J. Funct. Spaces, 2023 (2023), 6833404. https://doi.org/10.1155/2023/6833404 doi: 10.1155/2023/6833404
[40]	W. M. Abd-Elhameed, E. H. Doha, Y. H. Youssri, Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials, Quaestiones Math., 36 (2013), 15–38. https://doi.org/10.2989/16073606.2013.779945 doi: 10.2989/16073606.2013.779945
[41]	Y. L. Luke, The Special Functions and Their Approximations, Academic Press, New York, 1969.
[42]	E. H. Doha, A. H. Bhrawy, S. S. Ezz-Eldien, A new Jacobi operational matrix: an application for solving fractional differential equations, Appl. Math. Modell., 36 (2012), 4931–4943. https://doi.org/10.1016/j.apm.2011.12.031 doi: 10.1016/j.apm.2011.12.031
[43]	L. Chai, L. Wu, X. Yang, A fast parallel difference method for solving the time-fractional generalized Fisher equation, J. Appl. Anal. Comput., 15 (2025), 1216–1240. https://doi.org/10.11948/20240159 doi: 10.11948/20240159
[44]	Y. H. Youssri, A. G. Atta, Spectral collocation approach via normalized shifted Jacobi polynomials for the nonlinear Lane-Emden equation with fractal-fractional derivative, Fractal Fract., 7 (2023), 133. https://doi.org/10.3390/fractalfract7020133 doi: 10.3390/fractalfract7020133
[45]	H. Li, W. Zhao, F. Chen, Stability-preserving reduced-order modeling for caputo-type time-fractional nonlinear PDEs via L1-Crank-Nicolson POD framework, Commun. Nonlinear Sci. Numer. Simul., 161 (2026), 110119. https://doi.org/10.1016/j.cnsns.2026.110119 doi: 10.1016/j.cnsns.2026.110119
[46]	H. Li, W. Zhao, X. Zhang, Stability and convergence analysis of reduced-order finite difference schemes for a class of nonlinear time-fractional partial differential equations, J. Math. Anal. Appl., 559 (2026), 130465. https://doi.org/10.1016/j.jmaa.2026.130465 doi: 10.1016/j.jmaa.2026.130465
[47]	P. Roul, V. Rohil, A high-order numerical technique and its analysis for nonlinear generalized Fisher's equation, J. Comput. Appl. Math., 406 (2022), 114047. https://doi.org/10.1016/j.cam.2021.114047 doi: 10.1016/j.cam.2021.114047

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