Fractional Chebyshev Differential Equation and New Family of Orthogonal polynomials | ||||
| Journal of Fractional Calculus and Applications | ||||
| Volume 14, Issue 2, July 2023, Page 1-13 PDF (483.72 K) | ||||
| Document Type: Regular research papers | ||||
| DOI: 10.21608/jfca.2023.208544.1017 | ||||
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| Authors | ||||
Kazem Ghanbari  1; Hanif Mirzaei2; Zahra Kavooci2
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| 1Adjunct Research Professor of Mathematics School of Mathematics and Statistics, Carleton University | ||||
| 2Department of Mathematics Sahand University of Technology | ||||
| Abstract | ||||
| In the recent years, it has been proved that in some applications modeling by fractional derivatives generate more accurate solutions than modeling by integer order derivatives. Developing classical integer order differential equations to the fractional order has been beneficial in many applications. For example it has been shown that the classical integer order oscillator circuits are only a special case of the more general fractional oscillators . In this paper we consider a typical Fractional Chebyshev Differential Equation (FCDE) and we investigate the solutions, their properties and applications. For a positive real number $\alpha$ we prove that FCDE has solutions of the form $T_{n,\alpha}(x)=(1+x)^\frac{\alpha}{2}P_{n,\alpha}(x)$, where $P_{n,\alpha}(x)$ produce a family of orthogonal polynomials with respect to weight function $w_\alpha(x)=(\frac{1+x}{1-x})^{\frac{\alpha}{2}}$ on $[-1,1]$. For integer case $\alpha=1$ we show that these polynomials coincide with classical Chebyshev polynomials of third kind. Finally, we give some applications of $T_{n,\alpha}(x)$ in determining the solutions of some fractional order differential equations by defining a suitable integral transform. | ||||
| Keywords | ||||
| Fractional Chebyshev equation; Fractional Sturm-Liouville operator; Riemann-Liouville and Caputo derivatives | ||||
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