@article {
author = {Kavooci, Zahra and Ghanbari, Kazem and Mirzaei, Hanif},
title = {Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval},
journal = {Computational Methods for Differential Equations},
volume = {12},
number = {2},
pages = {226-235},
year = {2024},
publisher = {University of Tabriz},
issn = {2345-3982},
eissn = {2383-2533},
doi = {10.22034/cmde.2023.54630.2275},
abstract = {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. },
keywords = {Orthogonal polynomials,Fractional Chebyshev differential equation,Riemann-Liouville and Caputo derivatives},
url = {https://cmde.tabrizu.ac.ir/article_16638.html},
eprint = {https://cmde.tabrizu.ac.ir/article_16638_a6531dd83f47c157d2e637b528ce71bc.pdf}
}