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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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
- Primary 26A33
- Secondary 34A08, 34B30
Key Words and Phrases
- fractional calculus
- fractional ordinary differential equation
- Bessel function
- asymptotic analysis