Fractional-order Bessel functions with various applications

• Published in: Applications of mathematics (Prague) Vol. 64; no. 6; pp. 637 - 662
• Main Authors: Dehestani, Haniye, Ordokhani, Yadollah, Razzaghi, Mohsen
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2019; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Bessel functions; Classical and Continuum Physics; Derivatives; Differential equations; Fractional calculus; Integrals; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Optimization; Polynomials; Theoretical; error estimation; fractional-order Bessel functions; 65L70; 65M70; fractional operational matrix
• ISSN: 0862-7940; 1572-9109
• DOI: 10.21136/AM.2019.0279-18

We introduce fractional-order Bessel functions (FBFs) to obtain an approximate solution for various kinds of differential equations. Our main aim is to consider the new functions based on Bessel polynomials to the fractional calculus. To calculate derivatives and integrals, we use Caputo fractional derivatives and Riemann-Liouville fractional integral definitions. Then, operational matrices of fractional-order derivatives and integration for FBFs are derived. Also, we discuss an error estimate between the computed approximations and the exact solution and apply it in some examples. Applications are given to three model problems to demonstrate the effectiveness of the proposed method.