Tbilisi Mathematical Journal

Novel orthogonal functions for solving differential equations of arbitrary order

Kourosh Parand, Mehdi Delkhosh, and Mehran Nikarya

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Fractional calculus and the fractional differential equations have appeared in many physical and engineering processes. Therefore, an efficient and suitable method to solve them is very important. In this paper, novel numerical methods are introduced based on the fractional order of the Chebyshev orthogonal functions (FCF) with Tau and collocation methods to solve differential equations of the arbitrary (integer or fractional) order. The FCFs are obtained from the classical Chebyshev polynomials of the first kind. Also, the operational matrices of the fractional derivative and the product for the FCFs have been constructed. To show the efficiency and capability of these methods we have solved some well-known problems: the momentum, the Bagley-Torvik, and the Lane-Emden differential equations, then have compared our results with the famous methods in other papers.

Article information

Tbilisi Math. J., Volume 10, Issue 1 (2017), 31-55.

Received: 28 April 2015
Accepted: 13 June 2016
First available in Project Euclid: 26 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34A08: Fractional differential equations
Secondary: 34A34: Nonlinear equations and systems, general 33F05: Numerical approximation and evaluation [See also 65D20] 34K07: Theoretical approximation of solutions 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Fractional order of Chebyshev functions Operational matrix Tau method Bagley-Torvik equation Lane-Emden equation momentum equation


Parand, Kourosh; Delkhosh, Mehdi; Nikarya, Mehran. Novel orthogonal functions for solving differential equations of arbitrary order. Tbilisi Math. J. 10 (2017), no. 1, 31--55. doi:10.1515/tmj-2017-0004. https://projecteuclid.org/euclid.tbilisi/1527300016

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