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
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
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
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
J.P Boyd, Chebyshev and Fourier spectral methods, Courier Corporation, 2001.
L.N. Trefethen, Spectral methods in MATLAB, SIAM, (2000).
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
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
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.
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
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
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.
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.
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
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.
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
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
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
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
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
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
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.
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.
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
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.
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
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
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.
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
J. Sabatier, O.P. Agrawal and J.A. Tenreiro Machado, Advances in fractional calculus, Springer, 2007.Google Scholar
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
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.
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Published Online: 2019-01-29
Published in Print: 2019-04-26