University of TabrizComputational Methods for Differential Equations2345-398212220240301Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval2262351663810.22034/cmde.2023.54630.2275ENZahraKavoociFaculty of Sciences, Sahand University of Technology, Tabriz, Iran.KazemGhanbariFaculty of Sciences, Sahand University of Technology, Tabriz, Iran.School of Mathematics and Statistics, Carleton University, Ottawa, Canada.HanifMirzaeiFaculty of Sciences, Sahand University of Technology, Tabriz, Iran.Journal Article20221226Most 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