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

Igor Chyzhykov

TL;DR

This work addresses the growth and analyticity of solutions to sequential fractional linear differential equations driven by Caputo-Djrbashian derivatives. By extending Wiman-Valiron theory to the Djrbashian-Gelfond-Leontiev operator, it obtains precise asymptotics for fractional derivatives of entire functions and uses a majorant-series approach to prove existence and uniqueness of $α$-analytic Cauchy data in the presence of $α$-analytic coefficients. When the coefficients are polynomials in $x^α$, all $α$-analytic solutions are shown to have the form $y(x)=v(x^α)$ with $v$ entire of finite order, and sharp bounds on the order are established, generalizing Kochubei’s one-term results to higher-order equations. These results provide a complete fractional-ODE analogue of classical growth theory, linking exact growth rates to coefficient degrees and extending analytical techniques for fractional differential equations. The work thus broadens the mathematical toolkit for fractional dynamics by unifying existence theory with sharp asymptotic growth estimates in the $α$-analytic setting.

Abstract

We consider a fractional linear differential equation with successive derivatives given by $ \mathbb{D}_α^{n}y+ p_{n-1}(x) \mathbb{D}_α^{n-1}y+ \dots +p_{1}(x)\mathbb{D}_αy+p_0(x)y=0$, where $\mathbb{D}_α^{j}$ is the $j$th iteration of the Caputo-Djrbashian fractional derivative of order $α>0$, $p_j$ are $α$-analytic functions for $0<x^α<R$. Generalizing a result of Kilbas, Rivero Rodríguez-Germá and Trujillo, we prove the existence and uniqueness of the corresponding Cauchy problem in the class of $α$-analytic functions. We establish an exact growth order for the solution when $p_j(x)=P_j(x^α)$, 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.

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

TL;DR

This work addresses the growth and analyticity of solutions to sequential fractional linear differential equations driven by Caputo-Djrbashian derivatives. By extending Wiman-Valiron theory to the Djrbashian-Gelfond-Leontiev operator, it obtains precise asymptotics for fractional derivatives of entire functions and uses a majorant-series approach to prove existence and uniqueness of -analytic Cauchy data in the presence of -analytic coefficients. When the coefficients are polynomials in , all -analytic solutions are shown to have the form with entire of finite order, and sharp bounds on the order are established, generalizing Kochubei’s one-term results to higher-order equations. These results provide a complete fractional-ODE analogue of classical growth theory, linking exact growth rates to coefficient degrees and extending analytical techniques for fractional differential equations. The work thus broadens the mathematical toolkit for fractional dynamics by unifying existence theory with sharp asymptotic growth estimates in the -analytic setting.

Abstract

We consider a fractional linear differential equation with successive derivatives given by , where is the th iteration of the Caputo-Djrbashian fractional derivative of order , are -analytic functions for . Generalizing a result of Kilbas, Rivero Rodríguez-Germá and Trujillo, we prove the existence and uniqueness of the corresponding Cauchy problem in the class of -analytic functions. We establish an exact growth order for the solution when , where are polynomials, and 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.
Paper Structure (9 sections, 13 theorems, 101 equations)

This paper contains 9 sections, 13 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Fractional linear differential equations
  3. Fractional integrals and derivatives
  4. Wiman-Valiron theory
  5. Preleminaries
  6. Auxiliary results from Wiman-Valiron theory
  7. Chain rule for higher derivaitves and Bell polynomials
  8. Proof of Theorem \ref{['th:frwv']}
  9. $\alpha$-analyticity of soultions for \ref{['e:frac_lin_eq']}

Key Result

Lemma 1

If $u\in H_\alpha(R)$, then $D^\alpha$ is associative and commutative providied the conditions $u_m(0)=0$ for $m<\frac{\gamma+\beta}{2}$, $\gamma, \beta>0$, i.e. In particular, $\left(D^\alpha\right)^j =D^{j\alpha}$ if $u_m(0)=0$ for $m<j$.

Theorems & Definitions (21)

  • Example 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof : Proof of Lemma \ref{['l:commute_d_al']}
  • Theorem A
  • Theorem 1
  • Lemma 2: Lemma 3.4. Sher, cf. Lemma 2 Hay74
  • Lemma 3: Lemma 3.5. Sher, cf. Lemma 7 Hay74
  • Lemma 4: Lemma 3.6. Sher, cf. Lemma 8, Hay74
  • ...and 11 more