A note on a modified fractional Maxwell model

TL;DR

This paper introduces a modified fractional Maxwell model using Hadamard-type derivatives, capturing simultaneous memory effects and time-dependent viscosity, resulting in an ultra-slow relaxation described by a Mittag-Leffler function.

Contribution

It presents a novel fractional Maxwell model incorporating Hadamard derivatives to account for complex viscoelastic behaviors with explicit relaxation solutions.

Findings

The model exhibits ultra-slow relaxation behavior.

Explicit relaxation response is given by Mittag-Leffler function with logarithmic argument.

Graphical analysis confirms the model's main properties and asymptotic behavior.

Abstract

In this paper we consider a modified fractional Maxwell model based on the application of Hadamard-type fractional derivatives. The model is physically motivated by the fact that we can take into account at the same time memory effects and the time-dependence of the viscosity coefficient. We obtain an ultra-slow relaxation response whose explicit analytic form is given by the Mittag-Leffler function with a logarithmic argument. We show graphically the main properties of this relaxation response, also with the asymptotic behaviour.