%0 Journal Article
%T Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval
%J Computational Methods for Differential Equations
%I University of Tabriz
%Z 2345-3982
%A Kavooci, Zahra
%A Ghanbari, Kazem
%A Mirzaei, Hanif
%D 2024
%\ 03/01/2024
%V 12
%N 2
%P 226-235
%! Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval
%K Orthogonal polynomials
%K Fractional Chebyshev differential equation
%K Riemann-Liouville and Caputo derivatives
%R 10.22034/cmde.2023.54630.2275
%X Most of fractional differential equations are considered on a fixed interval. In this paper, we consider a typical fractional differential equation on a symmetric interval $[-\alpha,\alpha]$, where $\alpha$ is the order of fractional derivative. For a positive real number α we prove that the solutions are $T_{n,\alpha}(x)=(\alpha+x)^\frac{1}{2}Q_{n,\alpha}(x)$ where $Q_{n,\alpha}(x)$ produce a family of orthogonal polynomials with respect to the weight function$w_\alpha(x)=(\frac{\alpha+x}{\alpha-x})^{\frac{1}{2}}$ on $[-\alpha,\alpha]$. For integer case $\alpha = 1 $, we show that these polynomials coincide with classical Chebyshev polynomials of the third kind. Orthogonal properties of the solutions lead to practical results in determining solutions of some fractional differential equations.
%U https://cmde.tabrizu.ac.ir/article_16638_a6531dd83f47c157d2e637b528ce71bc.pdf