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.
"Novel orthogonal functions for solving differential equations of arbitrary order." Tbilisi Math. J. 10 (1) 31 - 55, Jan 2017. https://doi.org/10.1515/tmj-2017-0004