University of Tabriz Computational Methods for Differential Equations 2345-3982 12 2 2024 03 01 Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval‎ 226 235 16638 10.22034/cmde.2023.54630.2275 EN Zahra Kavooci Faculty of Sciences, Sahand University of Technology, Tabriz, Iran. Kazem Ghanbari Faculty of Sciences, Sahand University of Technology, Tabriz, Iran. School of Mathematics and Statistics, Carleton University, Ottawa, Canada. Hanif Mirzaei Faculty of Sciences, Sahand University of Technology, Tabriz, Iran. Journal Article 2022 12 26 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.  https://cmde.tabrizu.ac.ir/article_16638_a6531dd83f47c157d2e637b528ce71bc.pdf