Fractional differential equations: alpha-entire solutions, regular and irregular singularities

TL;DR

This paper studies fractional differential equations of order alpha in (0,1), providing growth estimates for solutions, developing a Frobenius-like method for regular singularities, and analyzing solutions near irregular singularities with convergence properties.

Contribution

It introduces a growth estimate for alpha-entire solutions, adapts Frobenius method for fractional systems with regular singularities, and examines the convergence of formal solutions near irregular singularities.

Findings

Alpha-entire solutions have a specific growth order.

A Frobenius-type method is developed for fractional equations.

Formal solutions may converge or diverge depending on the nature of singularity and alpha.

Abstract

We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order of growth of $\alpha$-entire solutions is given. An analog of the Frobenius method for systems with regular singularity is developed. For a model example of an equation with a kind of an irregular singularity, a series for a formal solution is shown to be convergent for $t>0$ (if $\alpha$ is an irrational number poorly approximated by rational ones) but divergent in the distribution sense.