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Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations W. M. Abd-Elhameed1 · Y. H. Youssri1

Received: 18 December 2016 / Revised: 4 July 2017 / Accepted: 17 July 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Abstract The principal aim of the current paper is to present and analyze two new spectral algorithms for solving some types of linear and nonlinear fractional-order differential equations. The proposed algorithms are obtained by utilizing a certain kind of shifted Chebyshev polynomials called the shifted fifth-kind Chebyshev polynomials as basis functions along with the application of a modified spectral tau method. The class of fifth-kind Chebyshev polynomials is a special class of a basic class of symmetric orthogonal polynomials which are constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. An investigation for the convergence and error analysis of the proposed Chebyshev expansion is performed. For this purpose, a new connection formulae between Chebyshev polynomials of the first and fifth kinds are derived. The obtained numerical results ascertain that our two proposed algorithms are applicable, efficient and accurate. Keywords Fifth-kind Chebyshev polynomials · Spectral methods · Romberg’s quadrature formulae · Connection problem · Fractional differential equations Mathematics Subject Classification 65M70 · 34A08 · 33C45 · 11B83

1 Introduction Spectral methods are crucial for obtaining solutions of ordinary, partial and fractional differential equations (FDEs). These methods have many advantages if compared with other methods such as finite element and finite difference methods. In many physical and chemical applications, solutions with many decimal places of accuracy are needed, so it is very useful to employ various spectral methods because they yield exponential convergence of the solutions. For applications of spectral methods in different disciplines, one can consult Shizgal

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Y. H. Youssri [email protected] Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

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W. M. Abd-Elhameed, Y. H. Youssri

(2014), Hesthaven et al. (2007), Boyd (2001), Trefethen (2000) and Canuto et al. (1988). The philosophy of the application of various spectral methods is built on writing the solution of a certain problem as a suitable combination of certain polynomials which are often orthogonal. There are three popular types of spectral methods, they are tau, collocation and Galerkin methods. The philosophy of applying Galerkin method is built on choosing suitable combinations of orthogonal polynomials satisfying the underlying initial/boundary conditions, and after that enforcing the residual to be orthogonal with the selected basis functions. This method is successfully applied to linear boundary value problems (see, for example Doha and Abd-Elhameed 2014). The tau method has the advantage that it avoids some of the problems of the Galerkin methods, since we can choose