# Article

 Title: Fractional-order Bessel functions with various applications (English) Author: Dehestani, Haniye Author: Ordokhani, Yadollah Author: Razzaghi, Mohsen Language: English Journal: Applications of Mathematics ISSN: 0862-7940 (print) ISSN: 1572-9109 (online) Volume: 64 Issue: 6 Year: 2019 Pages: 637-662 Summary lang: English . Category: math . Summary: 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. (English) Keyword: fractional-order Bessel functions Keyword: fractional operational matrix Keyword: error estimation MSC: 34A08 MSC: 65L70 MSC: 65M70 idZBL: 07144731 idMR: MR4042431 DOI: 10.21136/AM.2019.0279-18 . Date available: 2019-12-09T11:47:32Z Last updated: 2020-07-02 Stable URL: http://hdl.handle.net/10338.dmlcz/147926 . Reference: [1] Agarwal, R., O'Regan, D., Hristova, S.: Stability of Caputo fractional differential equations by Lyapunov functions.Appl. Math., Praha 60 (2015), 653-676. Zbl 1374.34005, MR 3436567, 10.1007/s10492-015-0116-4 Reference: [2] Baillie, R. T.: Long memory processes and fractional integration in econometrics.J. Econom. 73 (1996), 5-59. Zbl 0854.62099, MR 1410000, 10.1016/0304-4076(95)01732-1 Reference: [3] Bhrawy, A. 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