Tau algorithm for fractional delay differential equations utilizing seventh-kind Chebyshev polynomials

[1] W.M. Abd-Elhameed, Y.H. Youssri, Fifth-kind orthonormal Chebyshev polynomial solutions for fractional
differential equations, Comput. Appl. Math. 37 (2018) 2897–2921.
[2] W.M. Abd-Elhameed, Y.H. Youssri, Sixth-kind Chebyshev spectral approach for solving fractional differen-
tial equations, Int. J. Nonlinear Sci. Numer. Simul. 20 (2019) 191–203.
[3] W.M. Abd-Elhameed, S.O. Alkhamisi, New results of the fifth-kind orthogonal Chebyshev polynomials,
Symmetry 13 (2021) 2407.
[4] 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 (2021) 53.
[5] W.M. Abd-Elhameed, J.A.T. Machado, Y.H. Youssri, Hypergeometric fractional derivatives formula of
shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations, Int. J.
Nonlinear Sci. Numer. Simul. 23 (2022) 1253–1268

[6] W.M. Abd-Elhameed, Y.H. Youssri, A.K. Amin, A.G. Atta, Eighth-kind Chebyshev polynomials collocation
algorithm for the nonlinear time-fractional generalized Kawahara equation, Fractal Frac. 7 (2023) 652.
[7] W.M. Abd-Elhameed, H.M. Ahmed, Spectral solutions for the time-fractional heat differential equation
through a novel unified sequence of Chebyshev polynomials, Aims Math. 9 (2024) 2137–2166.
[8] A. Ali, Z. Gul, W.A. Khan, S. Ahmad, S. Zeb, Investigation of fractional order sine-Gordon equation using
Laplace Adomian decomposition method, Fractals 29 (2021) 2150121.
[9] T. Allahviranloo, H. Sahihi, Reproducing kernel method to solve fractional delay differential equations, Appl.
Math. Comput. 400 (2021) 126095.
[10] G.E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, 1999.
[11] A.G. Atta, G.M. Moatimid, Y.H. Youssri, Generalized Fibonacci operational collocation approach for frac-
tional initial value problems, Int. J. Appl. Comput. Math. 5 (2019) 9.
[12] A.G. Atta, W.M. Abd-Elhameed, G.M. Moatimid, Y.H. Youssri, Shifted fifth-kind Chebyshev Galerkin treat-
ment for linear hyperbolic first-order partial differential equations, Appl. Numer. Math. 167 (2021) 237–256.
[13] A.G. Atta, W.M. Abd-Elhameed, G.M. Moatimid, Y.H. Youssri, Advanced shifted sixth-kind Chebyshev
tau approach for solving linear one-dimensional hyperbolic telegraph type problem, Math. Sci. 17 (2022)
415–429.
[14] A. G. Atta, W.M. Abd-Elhameed, G. M. Moatimid, Y. H. Youssri, A fast Galerkin approach for solving the
fractional Rayleigh–Stokes problem via sixth-kind Chebyshev polynomials, Mathematics 10 (2022) 1843.
[15] A.G. Atta, Two spectral Gegenbauer methods for solving linear and nonlinear time fractional Cable prob-
lems, Int. J. Modern Phys. C. In Press.
[16] R. Azimi, M. Mohagheghy Nezhad, R. Pourgholi, Legendre spectral tau method for solving the fractional
integro-differential equations with a weakly singular kernel, Global Anal. Discrete Math. 7 (2022) 11–31.
[17] H. Bin Jebreen, C. Cattani, Interpolating scaling functions tau method for solving space–time fractional
partial differential equations, Symmetry 14 (2022) 2463.
[18] A Draux, M Sadik, B Moalla, Markov–Bernstein inequalities for generalized Gegenbauer weight, Appl.
Numer. Math. 61 (2011)1301–1321.
[19] E.H. Doha, W.M. Abd-Elhameed, 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.
[20] E.H. Doha, W.M. Abd-Elhameed, H.M. Ahmed, The coefficients of differentiated expansions of double and
triple Jacobi polynomials, Bull. Iran. Math. Soc. 38 (2012) 739–765.
[21] W. Gautschi, Orthogonal polynomials-constructive theory and applications, J. Comput. Appl. Math. 12
(1985) 61–76.
[22] M. Izadi, S. Y¨uzbas¸ı, W. Adel, A new Chelyshkov matrix method to solve linear and nonlinear fractional
delay differential equations with error analysis, Math. Sci. 17 (2022) 267–284.
[23] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations,
elsevier, 2006.
[24] W. Koepf, Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Iden-
tities, Vieweg, Braunschweig, Germany, 1998.

[25] Y. Kumar, V.K. Singh, Computational approach based on wavelets for financial mathematical model gov-
erned by distributed order fractional differential equation, Math. Comput. Simulation 190 (2021) 531–569.
[26] M. Li, O. Nikan, W. Qiu, D. Xu, An efficient localized meshless collocation method for the two-dimensional
Burgers-type equation arising in fluid turbulent flows, Eng. Anal. Bound. Elem. 144 (2022) 44–54.
[27] Z. Li, Q. Chen, Y. Wang, X. Li, Solving two-sided fractional super-diffusive partial differential equations
with variable coefficients in a class of new reproducing kernel spaces, Fractal Fract. 6 (2022) 492.
[28] Y.L. Luke, The Special Functions and Their Approximations, Academic press, New York, 1969.
[29] S. Maitama W. Zhao, Homotopy analysis Shehu transform method for solving fuzzy differential equations of
fractional and integer order derivatives, Comput. Appl. Math. 40 (2021) 86.
[30] F. Marcell´an, Orthogonal Polynomials and Special Functions: Computation and Applications, Number
1883. Springer Science & Business Media, 2006.
[31] M. Masjed-Jamei, Some New Classes of Orthogonal Polynomials and Special Functions: A Symmetric
Generalization of Sturm-Liouville Problems and its Consequences, PhD thesis, University of Kassel, Kassel,
Germany, 2006.
[32] J.C. Mason, D.C. Handscomb, Chebyshev Polynomials, Chapman and Hall, New York, NY, CRC, Boca
Raton, 2003.
[33] M. Nadeem, Ji-Huan He, A. Islam, The homotopy perturbation method for fractional differential equations:
part 1 mohand transform, Internat. J. Numer. Methods Heat Fluid Flow, 31 (2021) 490–3504.
[34] M. Nadeem, Ji-Huan He, The homotopy perturbation method for fractional differential equations: part 2,
two-scale transform, Internat. J. Numer. Methods Heat Fluid Flow, 31 2021 559–567.
[35] K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration
to Arbitrary Order, Elsevier, 1974.
[36] N. Peykrayegan, M. Ghovatmand, M.H. Noori Skandari, On the convergence of Jacobi-Gauss collocation
method for linear fractional delay differential equations, Math. Methods Appl. Sci. 44 (2021) 2237–2253.
[37] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Dif-
ferential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, San Diego,
1998.
[38] P Rahimkhani, Y Ordokhani, PM Lima, An improved composite collocation method for distributed-order
fractional differential equations based on fractional Chelyshkov wavelets, Appl. Numer. Math. 145 (2019)
1–27.
[39] S. Sabermahani, Y. Ordokhani, Numerical solution of fractional delay differential equations via Fibonacci
polynomials, In Proceedings of the 2nd International Conference on Combinatorics, Cryptography and Com-
putation (2017) I4C2017.
[40] R. Shah, H. Khan, M. Arif, P. Kumam, Application of Laplace–Adomian decomposition method for the
analytical solution of third-order dispersive fractional partial differential equations, Entropy 21 (2019) 335.
[41] B. Shiri, Guo-Cheng Wu, D. Baleanu, Collocation methods for terminal value problems of tempered frac-
tional differential equations, Appl. Numer. Math. 156 (2020) 385–395.
[42] A.K. Singh, M. Mehra, Wavelet collocation method based on Legendre polynomials and its application in
solving the stochastic fractional integro-differential equations, J. Comput. Sci.51 (2021) 101342

[43] H. Singh, Jacobi collocation method for the fractional advection-dispersion equation arising in porous
media, Numer. Methods Partial Differential Equations 38 (2022) 636–653.
[44] M.I. Syam, M. Sharadga, I. Hashim, A numerical method for solving fractional delay differential equations
based on the operational matrix method, Chaos Solitons Fract. 147 (2021) 110977.
[45] P.T. Toan, T.N. Vo, M. Razzaghi, Taylor wavelet method for fractional delay differential equations, Eng.
Comput. 37 (2021) 231–240.
[46] H. Tu, Y. Wang, Q. Lan, W. Liu, W. Xiao, S. Ma, A Chebyshev-tau spectral method for normal modes of
underwater sound propagation with a layered marine environment, J. Sound Vib. 492 (2021) 115784.
[47] ¨O. T¨urk, R. Codina, Chebyshev spectral collocation method approximations of the Stokes eigenvalue problem
based on penalty techniques, Appl. Numer. Math.145 (2019) 188–200.
[48] C. Wu, Z. Wang, The spectral collocation method for solving a fractional integro-differential equation, AIMS
Math. 7 (2022) 9577–9587.
[49] Y. Xu, An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials, Adv. Appl.
Math. 29 (2002) 328–343.
[50] Y.H. Youssri, A.G. Atta, Petrov-Galerkin Lucas polynomials procedure for the time-fractional diffusion
equation, Contemp. Math. 4 (2023) 230–248.
[51] B. Yuttanan, M. Razzaghi, Legendre wavelets approach for numerical solutions of distributed order fractional
differential equations, Appl. Math. Model. 70 (2019) 350–364.
[52] M.A. Zaky, An accurate spectral collocation method for nonlinear systems of fractional differential equations
and related integral equations with nonsmooth solutions, Appl. Numer. Math. 154 (2020) 205–222.