Fractional Chebyshev Differential Equation and New Family of Orthogonal polynomials

Document Type : Regular research papers

Authors

1 Adjunct Research Professor of Mathematics School of Mathematics and Statistics, Carleton University

2 Department 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.

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