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Fractional Chebyshev Differential Equation and New Family of Orthogonal polynomials

Article

Last updated: 29 Dec 2024

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Tags

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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.

DOI

10.21608/jfca.2023.208544.1017

Keywords

Fractional Chebyshev equation, Fractional Sturm-Liouville operator, Riemann-Liouville and Caputo derivatives

Authors

First Name

Kazem

Last Name

Ghanbari

MiddleName

-

Affiliation

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

Email

kazemghanbari@math.carleton.ca

City

Ottawa

Orcid

-

First Name

Hanif

Last Name

Mirzaei

MiddleName

-

Affiliation

Department of Mathematics Sahand University of Technology

Email

h_mirzaei@sut.ac.ir

City

-

Orcid

-

First Name

Zahra

Last Name

Kavooci

MiddleName

-

Affiliation

Department of Mathematics Sahand University of Technology

Email

z_kavousi98@sut.ac.ir

City

-

Orcid

-

Volume

14

Article Issue

2

Related Issue

39573

Issue Date

2023-07-01

Receive Date

2023-05-02

Publish Date

2023-07-01

Page Start

1

Page End

13

Print ISSN

2090-584X

Online ISSN

2090-5858

Link

https://jfca.journals.ekb.eg/article_311727.html

Detail API

https://jfca.journals.ekb.eg/service?article_code=311727

Order

311,727

Type

Regular research papers

Type Code

2,646

Publication Type

Journal

Publication Title

Journal of Fractional Calculus and Applications

Publication Link

https://jfca.journals.ekb.eg/

MainTitle

Fractional Chebyshev Differential Equation and New Family of Orthogonal polynomials

Details

Type

Article

Created At

18 Dec 2024