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

Document Type : Research Article

Authors

1 Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah, Saudi Arabia & Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

2 Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt & Faculty of Engineering, Egypt University of Informatics, Knowledge City, New Administrative Capital, Egypt

3 Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt

Abstract

Herein, we present an algorithm for handling fractional delay differential equations (FDDEs). Chebyshev polynomials (CPs) class of the seventh kind is a subclass of the generalized Gegenbauer (ultraspherical) polynomials. The members of this class make up the basis functions in this paper. Our suggested numerical algorithm is derived using new theoretical findings about these polynomials and their shifted counterparts. We will use the Tau method to convert the FDDE with the governing conditions into a linear algebraic system, which can then be solved numerically using a suitable procedure. We will give a detailed discussion of the convergence and error analysis of the shifted Chebyshev expansion. Lastly, some numerical examples are provided to verify the precision and applicability of the proposed strategy.

Keywords

Main Subjects


[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.