[1] J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, 2002.
[2] T. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, Wiley, 1990.
[3] W. M. Abd-Elhameed, Y. H. Youssri and A. G. Atta, Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials, J. Math. Model. 12 (2024) 277–299, https://doi.org/10.22124/jmm.2024.25844.2295.
[4] Y. H. Youssri and A. G. Atta, Modal spectral Tchebyshev Petrov–Galerkin stratagem for the time-fractional nonlinear Burgers’ equation, Iran. J. Numer. Anal. Optim. 14 (2024) 172–199, https://doi.org/ 10.22067/IJNAO.2023.83389.1292.
[5] M. Moustafa, Y. H. Youssri and A. G. Atta, Explicit Chebyshev Petrov–Galerkin scheme for time-fractional fourth-order uniform Euler–Bernoulli pinned–pinned beam equation, Nonlinear Eng. 12 (2023) #20220308.
[6] W. M. Abd-Elhameed, A. M. Al-Sady, O. M. Alqubori and 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.
[7] Y. H. Youssri, M. I. Ismail and A. G. Atta, Chebyshev Petrov–Galerkin procedure for the time-fractional heat equation with nonlocal conditions, Phys. Scr. 99 (2023) #015251, https://doi.org/10.1088/1402-4896/ad1700.
[8] C. Bernardi and Y. Maday, Spectral methods, Handbook Numer. Anal. 5 (1997) 209–485.
[9] J. Shen, T. Tang and L.-L. Wang, Spectral Methods: Algorithms, Analysis, and Applications, Springer, Berlin, 2011.
[10] A. G. Atta, Two spectral Gegenbauer methods for solving linear and nonlinear timefractional Cable problems, Int. J. Mod. Phys. C 35 (2024) #2450070.
[11] W. M. Abd-Elhameed, O. M. Alqubori, N. M. A. Alsafri, A. K. Amin and A. G. Atta, A matrix approach by convolved Fermat polynomials for solving the fractional Burgers’ equation, Mathematics 13 (2025) #1135.
[12] W. M. Abd-Elhameed, A. K. Al-Harbi, O. M. Alqubori, M. H. Alharbi and A. G. Atta, Collocation method for the time-fractional generalized Kawahara equation using a certain Lucas polynomial sequence, Axioms 14 (2025) #114.
[13] W. M. Abd-Elhameed, O. M. Alqubori and A. G. Atta, A collocation procedure for the numerical treatment of FitzHugh–Nagumo equation using a kind of Chebyshev polynomials, AIMS Math. 10 (2025) 1201–1223.
[14] H. Ashry, W. M. Abd-Elhameed, G. M. Moatimid and Y. H. Youssri, Robust shifted Jacobi–Galerkin method for solving linear hyperbolic telegraph type equation, Palest. J. Math. 11 (2022) 504–518.
[15] A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid and Y. H. Youssri, Novel spectral schemes to fractional problems with nonsmooth solutions, Math. Methods Appl. Sci. 46 (2023) 14745–14764, https://doi.org/10.1002/mma.9343.
[16] A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid and Y. H. Youssri, Modal shifted fifth-kind Chebyshev tau integral approach for solving heat conduction equation, Fractal Fract. 6 (2022) #619.
[17] N. M. Yassin, E. H. Aly and A. G. Atta, Novel approach by shifted Schröder polynomials for solving the fractional Bagley–Torvik equation, Phys. Scr. 100 (2025) #015242.
[18] W. M. Abd-Elhameed and Y. H. Youssri, New formulas of the high-order derivatives of fifth-kind Chebyshev polynomials: Spectral solution of the convection–diffusion equation, Numer. Methods Partial Differ. Equ. 40 (2024) #e22756.
[19] A. H. Bhrawy and W. M. Abd-Elhameed, New algorithm for the numerical solutions of nonlinear third-order differential equations using Jacobi–Gauss collocation method, Math. Probl. Eng. 20 (2011) #837218.
[20] R. M. Hafez, Y. H. Youssri and A. G. Atta, Jacobi rational operational approach for time-fractional sub-diffusion equation on a semi-infinite domain, Contemp. Math. 4 (2023) 853–876.
[21] M. Abdelhakem, D. Abdelhamied, M. El-Kady and Y. H. Youssri, Two modified shifted Chebyshev–Galerkin operational matrix methods for even-order partial boundary value problems, Boundary Value Probl. 2025 (2025) #34.
[22] S. M. Sayed, A. S. Mohamed, E. M. Abo-Eldahab and Y. H. Youssri, A compact combination of second-kind Chebyshev polynomials for Robin boundary value problems and Bratu-type equations, J. Umm Al-Qura Univ. Appl. Sci. 11 (2025) 766–783.
[23] Y. E. Aghdam, H. Safdari, Y. Azari, H. Jafari and D. Baleanu, Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme, Discrete Contin. Dyn. Syst. - S. 14 (2021) 2025-2039.
[24] W. M. Abd-Elhameed, E. H. Doha and Y. H. Youssri, New wavelets collocation method for solving second-order multipoint boundary value problems using Chebyshev polynomials of third and fourth kinds, Abstr. Appl. Anal. 2013 (2013) #542839.
[25] W. M. Abd-Elhameed, Y. H. Youssri, A. K. Amin and A. G. Atta, Eighth-kind Chebyshev polynomials collocation algorithm for the nonlinear time-fractional generalized Kawahara equation, Fractal Fract. 7 (2023) #652.
[26] S. E. Fadugba, A. F. Olanrewaju and F. E. Duke-Umanah, Numerical solution of secondorder Fredholm integro-differential equations using the Chebyshev collocation method, Proc. Int. Conf. Sci. Eng. Bus. Driving Sustain. Dev. Goals (SEB4SDG) (2024) 1–10.
[27] M. Mulimani and K. Srinivasa, A novel approach for Benjamin–Bona–Mahony equation via ultraspherical wavelets collocation method, Int. J. Math. Comput. Eng. 2 (2024) 179–192.
[28] A. Saadatmandi and M. R. Azizi, Chebyshev finite difference method for a two-point boundary value problems with applications to chemical reactor theory, Iranian J. Math. Chem. 3 (2012) 1–7,
https://doi.org/10.22052/IJMC.2012.5197.
[29] A. Eftekhari, Spectral poly-sinc collocation method for solving a singular nonlinear BVP of reaction–diffusion with Michaelis–Menten kinetics in a catalyst/biocatalyst, Iranian J. Math. Chem. 14 (2023) 77–96, https://doi.org/10.22052/IJMC.2023.246752.1648.
[30] A. Eftekhari and A. Saadatmandi, DE Sinc–collocation method for solving a class of second-order nonlinear BVPs, Math. Interdiscip. Res. 6 (2021) 11–22, https://doi.org/10.22052/MIR.2020.220050.1195.
[31] R. Rani, G. Arora and S. Mishra, A numerical study of exponential modified cubic Bspline for solving nonlinear Fisher’s equation, Adv. Comput. Methods Model. Sci. Eng. (2025) 65–71, https://doi.org/10.1016/B978-0-44-330012-7.00012-6.
[32] K.-J. Wang and B.-R. Zou, On new abundant solutions of the complex nonlinear Fokas–Lenells equation in optical fiber, Math. Methods Appl. Sci. 44 (2021) 13881–13893.
[33] B. Bhaumik, S. De and 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.
[34] H. Zheng, Y. Huang, Z. Huang, W. Hao and G. Lin, HomPINNs: Homotopy physicsinformed neural networks for solving the inverse problems of nonlinear differential equations with multiple solutions, J. Comput. Phys. 500 (2024) #112751.
[35] M. Sohaib and S. Haq, An efficient wavelet-based method for numerical solution of nonlinear integral and integro-differential equations, Math. Methods Appl. Sci. 47 (2024) 10702–10716.
[36] N. Anwar, I. Ahmad, A. K. Kiani, M. Shoaib and M. A. Z. Raja, Novel neuro-stochastic adaptive supervised learning for numerical treatment of nonlinear epidemic delay differential system with impact of double diseases, Int. J. Model. Simul. 45 (2025) 1852–1874.
[37] A. G. Atta, Approximate Petrov–Galerkin solution for the time fractional diffusion wave equation, Math. Methods Appl. Sci. 48 (2025) 11670–11685.
[38] N. M. Yassin, A. G. Atta and E. H. Aly, Numerical solutions for nonlinear ordinary and fractional Newell–Whitehead–Segel equation using shifted Schröder polynomials, Bound. Value Probl. 2025 (2025) #57.
[39] A. G. Atta, Spectral collocation approach with shifted Chebyshev third-kind series approximation for nonlinear generalized fractional Riccati equation, Int. J. Appl. Comput. Math.10 (2024) #59.
[40] A. Verma, P. Pandey and V. Verma, Some identities involving Chebyshev polynomials of the third kind, Lucas numbers, and fourth kind, Fibonacci numbers, Int. J. Appl. Math. 53 (2023) 1–8.
[41] D. N. Edith, T. Oyedepo and A. E. Adenipekun, Fourth-kind Chebyshev computational approach for integro-differential equations, Math. Comput. Sci. 5 (2024) 55–64.
[42] W. M. Abd-Elhameed and M. A. Bassuony, On the coefficients of differentiated expansions and derivatives of Chebyshev polynomials of the third and fourth kinds, Acta Math. Sci. 35 (2015) 326–338.
[43] H. M. Ahmed, Highly accurate method for boundary value problems with Robin boundary conditions, J. Nonlinear Math. Phys. 30 (2023) 1239–1263.
[44] N. Setia and R. K. Mohanty, A third-order finite difference method on a quasi-variable mesh for nonlinear two-point boundary value problems with Robin boundary conditions, Soft Comput. 25 (2021) 12775–12788.
[45] K. Sadri and H. Aminikhah, Chebyshev polynomials of sixth kind for solving nonlinear fractional PDEs with proportional delay and its convergence analysis, J. Funct. Spaces 2022 (2022) #9512048.
[46] J. Malele, P. Dlamini and S. Simelane, Highly accurate compact finite difference schemes for two-point boundary value problems with Robin boundary conditions, Symmetry 14 (2022) #1720.
[47] N. M. Nasir, Z. A. Majid, F. Ismail and N. Bachok, Diagonal block method for solving two-point boundary value problems with Robin boundary conditions, Math. Probl. Eng. 2018 (2018) #2056834.
[48] P. Roul, V. M. K. P. Goura and R. Agarwal, A compact finite difference method for a general class of nonlinear singular boundary value problems with Neumann and Robin boundary conditions, Appl. Math. Comput. 350 (2019) 283–304.
[49] Z. A. Majid, N. M. Nasir, F. Ismail and N. Bachok, Two-point diagonally block method for solving boundary value problems with Robin boundary conditions, Malays. J. Math. Sci. 13 (2019) 1–14.
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