Distributed Order Derivatives and Relaxation Patterns

TL;DR

This paper investigates the asymptotic behavior of solutions to distributed order derivative equations, which model anomalous relaxation processes, by analyzing how the measure  ho influences the solutions' long-term dynamics.

Contribution

It provides a detailed analysis of how the properties of the measure  ho affect the asymptotic behavior of solutions to distributed order derivative equations.

Findings

Asymptotic behavior depends on the measure  ho properties.

Characterization of relaxation patterns based on  ho.

Insights into modeling anomalous relaxation processes.

Abstract

We consider equations of the form $(D_{(\rho)}u)(t)=-\lambda u(t)$, $t>0$, where $\lambda >0$, $D_{(\rho)}$ is a distributed order derivative, that is the Caputo-Dzhrbashyan fractional derivative of order $\alpha$, integrated in $\alpha\in (0,1)$ with respect to a positive measure $\rho$. Such equations are used for modeling anomalous, non-exponential relaxation processes. In this work we study asymptotic behavior of solutions of the above equation, depending on properties of the measure $\rho$.