Study of a Fractional Creep Problem with Multiple Delays in Terms of Boltzmann's Superposition Principle

TL;DR

This paper investigates nonlinear fractional differential equations with multiple delays in viscoelasticity, establishing existence, stability, and uniqueness of solutions using Boltzmann's superposition principle and Banach's contraction, with illustrative examples.

Contribution

It introduces a novel analysis of nonlinear fractional creep models with delays, applying Boltzmann's superposition principle to prove solution properties.

Findings

Existence and uniqueness of solutions established.

Solutions depend continuously on initial data.

Model stability analyzed via Ulam stability.

Abstract

We study a class of nonlinear fractional differential equations with multiple delays, which is represented by the Voigt creep fractional model of viscoelasticity. We discuss two Voigt models, the first being linear and the second being nonlinear. The linear Voigt model give us the physical interpretation and is associated with important results since the creep function characterizes the viscoelastic behavior of stress and strain. For the nonlinear model of Voigt, our theoretical study and analysis provides existence and stability, where time delays are expressed in terms of Boltzmann's superposition principle. By means of the Banach contraction principle, we prove existence of a unique solution and investigate its continuous dependence upon the initial data as well as Ulam stability. The results are illustrated with an example.