TY - JOUR
ID - 16638
TI - Fractional Chebyshev differential equation on symmetric $\alpha$ dependent interval
JO - Computational Methods for Differential Equations
JA - CMDE
LA - en
SN - 2345-3982
AU - Kavooci, Zahra
AU - Ghanbari, Kazem
AU - Mirzaei, Hanif
AD - Faculty of Sciences, Sahand University of Technology, Tabriz, Iran.
Y1 - 2024
PY - 2024
VL - 12
IS - 2
SP - 226
EP - 235
KW - Orthogonal polynomials
KW - Fractional Chebyshev differential equation
KW - Riemann-Liouville and Caputo derivatives
DO - 10.22034/cmde.2023.54630.2275
N2 - 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.
UR - https://cmde.tabrizu.ac.ir/article_16638.html
L1 - https://cmde.tabrizu.ac.ir/article_16638_a6531dd83f47c157d2e637b528ce71bc.pdf
ER -