Wiman-Valiron method for fractional derivatives and sharp growth estimates of $\alpha$-analytic solutions for linear fractional differential equations

TL;DR

This paper extends classical growth estimates to solutions of linear fractional differential equations using a generalized Wiman-Valiron theory, establishing existence, uniqueness, and sharp growth bounds for $ ext{α}$-analytic solutions.

Contribution

It generalizes the Wiman-Valiron method to fractional derivatives and provides sharp growth estimates for solutions of fractional differential equations with $ ext{α}$-analytic coefficients.

Findings

Proves existence and uniqueness of solutions in $ ext{α}$-analytic class.

Establishes exact growth order for solutions with polynomial coefficients.

Demonstrates sharpness and generalizes Kochubei's results.

Abstract

We consider a fractional linear differential equation with successive derivatives given by $ \mathbb{D}_\alpha^{n}y+ p_{n-1}(x) \mathbb{D}_\alpha^{n-1}y+ \dots +p_{1}(x)\mathbb{D}_\alpha y+p_0(x)y=0$, where $\mathbb{D}_\alpha^{j}$ is the $j$th iteration of the Caputo-Djrbashian fractional derivative of order $\alpha>0$, $p_j$ are $\alpha$-analytic functions for $0<x^\alpha <R$. Generalizing a result of Kilbas, Rivero Rodr\'iguez-Germ\'a and Trujillo, we prove the existence and uniqueness of the corresponding Cauchy problem in the class of $\alpha$-analytic functions. We establish an exact growth order for the solution when $p_j(x)=P_j(x^\alpha)$, where $P_j$ are polynomials, and $p_0$ dominates in some sense. This is the full counterpart of the classical case of ordinary differential equations. In particular, we demonstrate the sharpness of Kochubei's result and generalize it. To achieve this, we extend the Wiman-Valiron theory to analytic functions and the Djrbashian-Gelfond-Leontiev generalized fractional derivatives.