Linear and nonlinear fractional Voigt models

TL;DR

This paper explores fractional generalizations of the Voigt model for creep phenomena, providing explicit solutions for the linear case and proving existence of solutions for the nonlinear case, with an illustrative example.

Contribution

It introduces both linear and nonlinear fractional Voigt models, deriving explicit solutions for the linear case and establishing existence results for the nonlinear case.

Findings

Explicit Volterra solution involving Mittag-Leffler functions

Existence of solutions for nonlinear fractional Voigt model

Illustrative nonlinear example provided

Abstract

We consider fractional generalizations of the ordinary differential equation that governs the creep phenomenon. Precisely, two Caputo fractional Voigt models are considered: a rheological linear model and a nonlinear one. In the linear case, an explicit Volterra representation of the solution is found, involving the generalized Mittag-Leffler function in the kernel. For the nonlinear fractional Voigt model, an existence result is obtained through a fixed point theorem. A nonlinear example, illustrating the obtained existence result, is given.