Jump to ContentJump to Main Navigation
Show Summary Details
More options …

International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Angheluta-Bauer, Luiza / Chen, Xi / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi / Yang, Xu


IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

Online
ISSN
2191-0294
See all formats and pricing
More options …
Volume 20, Issue 2

Issues

Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations

W. M. Abd-Elhameed / Y. H. YoussriORCID iD: https://orcid.org/0000-0003-0403-8797
Published Online: 2019-01-29 | DOI: https://doi.org/10.1515/ijnsns-2018-0118

Abstract

The basic aim of this paper is to develop new numerical algorithms for solving some linear and nonlinear fractional-order differential equations. We have developed a new type of Chebyshev polynomials, namely, Chebyshev polynomials of sixth kind. This type of polynomials is a special class of symmetric orthogonal polynomials, involving four parameters that were constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. The proposed algorithms are basically built on reducing the fractional-order differential equations with their initial/boundary conditions to systems of algebraic equations which can be efficiently solved. The new proposed algorithms are supported by a detailed study of the convergence and error analysis of the sixth-kind Chebyshev expansion. New connection formulae between Chebyshev polynomials of the second and sixth kinds were established for this study. Some examples were displayed to illustrate the efficiency of the proposed algorithms compared to other methods in literature. The proposed algorithms have provided accurate results, even using few terms of the proposed expansion.

Keywords: Chebyshev polynomials of sixth kind; tau method; collocation method; connection formulae; fractional differential equations

JEL Classification: 65M70; 34A08; 33C45; 11B83

References

  • [1]

    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.Web of ScienceCrossref

  • [2]

    K. Maleknejad, K. Nouri and L. Torkzadeh. Operational matrix of fractional integration based on the shifted second kind Chebyshev polynomials for solving fractional differential equations, Mediterr. J. Math. 13 (3) (2016), 1377–1390.Web of ScienceCrossref

  • [3]

    E.H. Doha, W.M. Abd-Elhameed and M.A. Bassuony, New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods, J. Comput. Phys. 236 (2013), 563–579.CrossrefWeb of Science

  • [4]

    W.M. Abd-Elhameed, E.H. Doha 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 (2) (2015), 326–338.Web of ScienceCrossref

  • [5]

    J.P Boyd, Chebyshev and Fourier spectral methods, Courier Corporation, 2001.

  • [6]

    L.N. Trefethen, Spectral methods in MATLAB, SIAM, (2000).

  • [7]

    W.M. Abd-Elhameed and Y.H. Youssri, Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations, Comput. Appl. Math. 37 (3) (2018), 2897–2921.CrossrefWeb of ScienceGoogle Scholar

  • [8]

    M.A. Zaky, E.H. Doha and J.A.T. Machado, A spectral framework for fractional variational problems based on fractional Jacobi functions, Appl. Numer. Math. 132 (2018), 51–72.Web of ScienceCrossref

  • [9]

    A.H. Bhrawy and M.A.Zaky, Numerical simulation of multi-dimensional distributed-order generalized Schrdinger equations, Nonlinear Dynam. 89 (2) (2017), 1415–1432.Crossref

  • [10]

    M.A. Zaky, E. H. Doha and J.A.T. Machado, A spectral numerical method for solving distributed-order fractional initial value problems, J. Comput. Nonlinear Dynam. 13 (10) (2018), 101007.

  • [11]

    A.H. Bhrawy and M.A. Zaky, A method based on the Jacobi tau approximation for solving multi-term timespace fractional partial differential equations, J. Comput. Phys. 281 (2015), 876–895.Crossref

  • [12]

    M.A. Zaky, A Legendre spectral quadrature tau method for the multi-term time fractional diffusion equations, Comput. Appl. Math. 37 (3) (2018), 3525–3538.Web of ScienceCrossrefGoogle Scholar

  • [13]

    E.H. Doha, Y.H. Youssri and M.A. Zaky, Spectral solutions for differential and integral equations with varying coefficients using classical orthogonal polynomials, Bull. Iran. Math. Soc. (2018). doi:.CrossrefGoogle Scholar

  • [14]

    E.H. Doha and W.M. Abd-Elhameed, On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds, Bull. Malays. Math. Sci. Soc. 37 (2) (2014), 383–398.

  • [15]

    W.M. Abd-Elhameed, On solving linear and nonlinear sixth-order two point boundary value problems via an elegant harmonic numbers operational matrix of derivatives, CMES Comput. Model. Eng. Sci. 101 (3) (2014), 159–185.

  • [16]

    S. Esmaeili, M. Shamsi and Y. Luchko. Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials, Comput. Math. Appl. 62 (3) (2011), 918–929.Crossref

  • [17]

    E.H. Doha, W.M. Abd-Elhameed and Y.H. Youssri, Second kind Chebyshev operational matrix algorithm for solving differential equations ofLane-Emden type, New Astron. 23 (2013), 113–117.

  • [18]

    M. Masjed-Jamei, Some new classes of orthogonal polynomials and special functions: A symmetric generalization of Sturm-Liouville problems and its consequences. PhD thesis, 2006.Google Scholar

  • [19]

    W. Koepf and M. Masjed-Jamei, A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it, Integral Transforms Spec. Funct. 17 (8) (2006), 559–576.Crossref

  • [20]

    M. Masjed-Jamei, A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions, J. Math. Anal. Appl. 325 (2) (2007), 753–775.CrossrefWeb of Science

  • [21]

    W. Koepf and M. Masjed-Jamei, A generic formula for the values at the boundary points of monic classical orthogonal polynomials, J. Comput. Appl. Math. 191 (1) (2006), 98–105.Crossref

  • [22]

    M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (1) (2006), 80–90.Crossref

  • [23]

    V. Daftardar-Gejji and H. Jafari, Solving a multi-order fractional differential equation using Adomian decomposition, Appl. Math. Comput. 189 (1) (2007), 541–548.Web of Science

  • [24]

    W.M. Abd-Elhameed and Y.H. Youssri, New spectral solutions of multi-term fractional order initial value problems with error analysis, Cmes-Comp. Model. Eng. 105 (2015), 375–398.

  • [25]

    M. ur Rehman and R.A. Khan, The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 16 (11) (2011), 4163–4173.

  • [26]

    W.M. Abd-Elhameed and Y.H. Youssri, A novel operational matrix of Caputo fractional derivatives of Fibonacci polynomials: Spectral solutions of fractional differential equations, Entropy, 18 (10) (2016), 345.Web of Science

  • [27]

    W.M. Abd-Elhameed and Y.H. Youssri, Spectral solutions for fractional differential equations via a novel Lucas operational matrix of fractional derivatives, Rom. J. Phys. 61 (5–6)(2016), 795–813.

  • [28]

    A.H. Bhrawy and M.A. Zaky, Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations, Appl. Math. Model. 40 (2) (2016), 832–845.Web of Science

  • [29]

    J.A.T. Machado and B.P. Moghaddam, A robust algorithm for nonlinear variable-order fractional control systems with delay, Int. J. Nonlinear Sci. Numer. Simul. 19 (3–4) (2018), 1–8.Web of Science

  • [30]

    B.b. Hu, N. Zhang and J.B. Wang, Initial-boundary value problems for the coupled higher-order nonlinear Schrdinger equations on the half-line, Int. J. Nonlinear Sci. Numer. Simul. 19 (1) (2018), 83–92.

  • [31]

    K.B. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, volume 111. Elsevier, 1974.Google Scholar

  • [32]

    J. Sabatier, O.P. Agrawal and J.A. Tenreiro Machado, Advances in fractional calculus, Springer, 2007.Google Scholar

  • [33]

    T. Allahviranloo, Z. Gouyandeh and A. Armand, Numerical solutions for fractional differential equations by tau-collocation method, Appl. Math. Comput. 271 (2015), 979–990.Web of Science

  • [34]

    S. Irandoust-Pakchin, M. Lakestani and H. Kheiri, Numerical approach for solving a class of nonlinear fractional differential equation, Bull. Iranian Math. Soc. 42 (5) (2016), 1107–1126.

About the article

Received: 2018-05-07

Accepted: 2019-12-01

Published Online: 2019-01-29

Published in Print: 2019-04-26


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 20, Issue 2, Pages 191–203, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2018-0118.

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in