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A note on fractional Bessel equation and its asymptotics

Fractional Calculus and Applied Analysis volume 16, pages 559–572 (2013)Cite this article

Abstract

In this note we propose a fractional generalization of the classical modified Bessel equation. Instead of the integer-order derivatives we use the Riemann-Liouville version. Next, we solve the fractional modified Bessel equation in terms of the power series and provide an asymptotic analysis of its solution for large arguments. We find a leading-order term of the asymptotic formula for the solution to the considered equation. This behavior is verified numerically and shows high accuracy and fast convergence. Our results reduce to the classical formulas when the order of the fractional derivative goes to integer values.

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Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Wroclaw University of Technology, ul. Janiszewskiego 14a, 50-372, Wroclaw, Poland

    Wojciech Okrasiński & Łukasz Płociniczak

Authors
  1. Wojciech Okrasiński
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  2. Łukasz Płociniczak
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Correspondence to Wojciech Okrasiński.

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Okrasiński, W., Płociniczak, Ł. A note on fractional Bessel equation and its asymptotics. fcaa 16, 559–572 (2013). https://doi.org/10.2478/s13540-013-0036-5

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  • DOI: https://doi.org/10.2478/s13540-013-0036-5

MSC 2010

  • Primary 26A33
  • Secondary 34A08, 34B30

Key Words and Phrases

  • fractional calculus
  • fractional ordinary differential equation
  • Bessel function
  • asymptotic analysis