[1] J. Boussinesq, Essai sur la théorie des eaux courantes, [Essay on the theory of flowing waters], Mem Acadèmie des Sciences, 23 (1877) 252–260.
[2] O. Darrigol, Worlds of Flow: A History of Hydrodynamics From the Bernoullis to Prandtl, Oxford University Press, 2005.
[3] A. C. Newell, Solitons in Mathematics and Physics, Siam, Philadelphia, 1985.
[4] D. J. Korteweg and G. D. Vries, XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Lond. Edinb. Dubl. Phil. Mag. 39 (240) (1895) 422–443, https://doi.org/10.1080/14786449508620739.
[5] T. Ak, S. B. G. Karakoc and A. Biswas, Application of Petrov-Galerkin finite element method to shallow water waves model: modified Korteweg-de Vries equation, Sci. Iran. 24 (3) (2017) 1148–1159, https://doi.org/10.24200/SCI.2017.4096.
[6] Y. H. Youssri and A. G. Atta, Double Tchebyshev spectral tau algorithm for solving KdV equation, with soliton application, Solitons (2022) 451–467,
https://doi.org/10.1007/978-1- 0716-2457-9_771.
[7] W. M. Abd-Elhameed and Y. H. Youssri, Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations, Comp. Appl. Math. 37 (2018) 2897–2921, https://doi.org/10.1007/s40314-017-0488-z.
[8] W. M. Abd-Elhameed and Y. H. Youssri, Sixth-kind Chebyshev spectral approach for solving fractional differential equations, Int. J. Nonlinear Sci. Numer. Simul. 20 (2) (2019) 191–203, https://doi.org/10.1515/ijnsns-2018-0118.
[9] E. M. Abdelghany, W. M. Abd-Elhameed, G. M. Moatimid, Y. H. Youssri and A. G. Atta, A Tau approach for solving time-fractional Heat equation based on the shifted sixth-kind Chebyshev polynomials, Symmetry 15 (3) (2023) p. 594, https://doi.org/10.3390/sym15030594.
[10] W. M. Abd-Elhameed, Novel expressions for the derivatives of sixth kind Chebyshev polynomials: spectral solution of the non-linear one-dimensional Burgers’ equation, Fractal Fract. 5 (2) (2021) p. 53, https://doi.org/10.3390/fractalfract5020053.
[11] A. G. Atta, G. M. Moatimid and Y. H. Youssri, Generalized Fibonacci operational tau algorithm for fractional Bagley-Torvik equation, Prog. Fract. Differ. Appl. 6 (3) (2020) 215–224.
[12] A. Napoli and W. M. Abd-Elhameed, An innovative harmonic numbers operational matrix method for solving initial value problems, Calcolo 54 (2017) 57–76, https://doi.org/10.1007/s10092-016-0176-1.
[13] A. G. Atta, G. M. Moatimid and Y. H. Youssri, Generalized Fibonacci operational collocation approach for fractional initial value problems, Int. J. Appl. Comput. Math. 5 (2019) 1–11, https://doi.org/10.1007/s40819-018-0597-4.
[14] Y. H. Youssri, Two Fibonacci operational matrix pseudo-spectral schemes for nonlinear fractional Klein–Gordon equation, Int. J. Mod. Phys. C 33 (4) (2022) p. 2250049, https://doi.org/10.1142/S0129183122500498.
[15] Y. H. Youssri and A. G. Atta, Petrov-Galerkin Lucas polynomials procedure for the time-fractional diffusion equation, Contemp. Math. 4 (2) (2023) 230–248, https://doi.org/10.37256/cm.4220232420.
[16] Y. H. Youssri, W. M. Abd-Elhameed and A. G. Atta, Spectral Galerkin treatment of linear one-dimensional telegraph type problem via the generalized Lucas polynomials, Arab. J. Math. 11 (3) (2022) 601–615, https://doi.org/10.1007/s40065-022-00374-0.
[17] W. M. Abd-Elhameed, E. H. Doha, Y. H. Youssri and M.A. Bassuony, New Tchebyshev- Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations, Numer. Methods Partial Differ. Equ. 32 (6) (2016) 1553–1571, https://doi.org/10.1002/num.22074.
[18] A. Duangpan, R. Boonklurb and M. Juytai, Numerical solutions for systems of fractional and classical integro-differential equations via finite integration method based on shifted Chebyshev polynomials, Fractal Fract. 5 (3) (2021) p. 103, https://doi.org/10.3390/fractalfract5030103.
[19] V. Saw and S. Kumar, Second kind Chebyshev polynomials for solving space fractional advection–dispersion equation using collocation method, Iran. J. Sci. Technol. Trans. Sci. 43 (2019) 1027–1037,
https://doi.org/10.1007/s40995-018-0480-5.
[20] J. Liu, X. Li and L. Wu, An operational matrix technique for solving variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials, Adv. Math. Phys. 2016 (2016) Article ID 6345978, https://doi.org/10.1155/2016/6345978.
[21] A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid and 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.
[22] Y. Xie, L. Li and M. Wang, Adomian decomposition method with orthogonal polynomials: Laguerre polynomials and the second kind of Chebyshev polynomials, Mathematics 9 (15) (2021) p. 1796, https://doi.org/10.3390/math9151796.
[23] M. Pourbabaee and A. Saadatmandi, The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applications, Int. J. Comput. Math. 98 (11) (2021) 2310–2329,
https://doi.org/10.1080/00207160.2021.1895988.
[24] M. Pourbabaee and A. Saadatmandi, Collocation method based on Chebyshev polynomials for solving distributed order fractional differential equations, Comput. Methods Differ. Equ. 9 (3) (2021) 858–873, https://doi.org/10.22034/cmde.2020.38506.1695.
[25] 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.
[26] M. Moustafa, Y. H. Youssri and A. G. Atta, Explicit Chebyshev-Galerkin scheme for the time-fractional diffusion equation, Int. J. Mod. Phys. C 35 (2024) p. 2450002, https://doi.org/10.1142/S0129183124500025.
[27] A. G. Atta, W. M. Abd-Elhameed, G. M. Moatimid and Y. H. Youssri, A fast galerkin approach for solving the fractional Rayleigh–Stokes problem via sixth-kind Chebyshev polynomials, Mathematics 10 (11) (2022) p. 1843.
[28] J. F. Traub, Analytic Computational Complexity, Academic Press, 2014.
[29] N. J. Ford, J. Xiao and Y. Yan, A finite element method for time fractional partial differential equations, Fract. Calc. Appl. Anal. 14 (2011) 454–474,
https://doi.org/10.2478/s13540- 011-0028-2.
[30] Y. Lin and C. Xu, Finite difference/spectral approximations for the timefractional diffusion equation, J. Comput. Phys. 225 (2) (2007) 1533–1552, https://doi.org/10.1016/j.jcp.2007.02.001.
[31] A. A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys. 280 (2015) 424–438, https://doi.org/10.1016/j.jcp.2014.09.031.
[32] H. Chen and T. Sun, A Petrov–Galerkin spectral method for the linearized time fractional KdV equation, Int. J. Comput. Math. 95 (6-7) (2018) 1292–1307, https://doi.org/10.1080/00207160.2017.1410544.
[33] W. M. Abd-Elhameed and Y. H. Youssri, Spectral Tau solution of the linearized time-fractional KdV-type equations, AIMS Math. 7 (8) (2022) 15138–15158,
https://doi.org/10.3934/math.2022830.
[34] Y. Zhang, Formulation and solution to time-fractional generalized Korteweg-de Vries equation via variational methods, Adv. Differ. Equ. 2014 (1) (2014) 1–12, https://doi.org/10.1186/1687-1847-2014-65.
[35] Q. Wang, Homotopy perturbation method for fractional KdV equation, Appl. Math. Comput. 190 (2) (2007) 1795–1802, https://doi.org/10.1016/j.amc.2007.02.065.
[36] S. Sabermahani and Y. Ordokhani, A numerical technique for solving fractional Benjamin– Bona–Mahony–Burgers equations with bibliometric analysis, Fractional Order Systems and Applications in Engineering, Academic Press, (2023) 93–108.
[37] P. Rahimkhani, Y. Ordokhani and S. Sabermahani, Hahn hybrid functions for solving distributed order fractional Black–Scholes European option pricing problem arising in financial market, Math. Methods Appl. Sci. 46 (6) (2023) 6558–6577,
https://doi.org/10.1002/mma.8924.
[38] F. Nourian, M. Lakestani, S. Sabermahani and Y. Ordokhani, Touchard wavelet technique for solving time-fractional Black–Scholes model, Comp. Appl. Math. 41 (4) (2022) p. 150, https://doi.org/10.1007/s40314-022-01853-y.
[39] S. Sabermahani, Y. Ordokhani and P. Rahimkhani, Application of two-dimensional Fibonacci wavelets in fractional partial differential equations arising in the financial market, Int. J. Appl. Comput. Math. 8 (3) (2022) P. 129, https://doi.org/10.1007/s40819-022-01329-x.
[40] S. Sabermahani, Y. Ordokhani and S. A. Yousefi, Two-dimensional Müntz–Legendre hybrid functions: theory and applications for solving fractional-order partial differential equations, Comput. Appl. Math. 39 (2) (2020) p. 111, https://doi.org/10.1007/s40314-020-1137-5.
[41] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 198 Academic Press, 1998.
[42] W. M. Abd-Elhameed and Y. H. Youssri, Explicit shifted second-kind chebyshev spectral treatment for fractional Riccati differential equation, Comput. Model. Eng. Sci. 121 (3) (2019) 1029–1049, https://doi.org/10.32604/cmes.2019.08378.
[43] R. Askey, Orthogonal Polynomials and Special Functions, SIAM, 1975.
[44] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, New York, 2003.
[45] D. Cen, Z. Wang and Y. Mo, Second order difference schemes for time-fractional KDV–Burgers’ equation with initial singularity, Appl. Math. Lett. 112 (2021) p. 106829, https://doi.org/10.1016/j.aml.2020.106829.
[46] A. H. Bhrawy and M. A. Zaky, A method based on the Jacobi Tau approximation for solving multi-term time–space fractional partial differential equations, J. Comput. Phys. 281 (2015) 876–895, https://doi.org/10.1016/j.jcp.2014.10.060.
[47] Y. H. Youssri and A. G. Atta, Spectral collocation approach via normalized shifted Jacobi polynomials for the nonlinear Lane-Emden equation with fractal-fractional derivative, Fractal Fract. 7 (2) (2023) p. 133, https://doi.org/10.3390/fractalfract7020133.
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