# Growth Equation of the General Fractional Calculus

## TL;DR

This paper investigates the growth behavior of solutions to a generalized fractional differential equation involving a convolutional derivative, extending classical Mittag-Leffler function solutions and analyzing their asymptotic properties.

## Contribution

It introduces a general convolutional derivative and derives the growth equation for solutions, extending the classical fractional calculus framework.

## Key findings

- Derived the growth equation for solutions involving the general convolutional derivative.
- Analyzed the asymptotic behavior of solutions as time approaches infinity.
- Extended the classical Mittag-Leffler function solutions to a broader fractional calculus context.

## Abstract

We consider the Cauchy problem $(\mathbb D_{(k)} u)(t)=\lambda u(t)$, $u(0)=1$, where $\mathbb D_{(k)}$ is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory {\bf 71} (2011), 583--600), $\lambda >0$. The solution is a generalization of the function $t\mapsto E_\alpha (\lambda t^\alpha)$ where $0<\alpha <1$, $E_\alpha$ is the Mittag-Leffler function. The asymptotics of this solution, as $t\to \infty$, is studied.