Complex order fractional derivatives in viscoelasticity

TL;DR

This paper introduces complex order fractional derivatives in viscoelastic models, establishing compatibility and physical constraints, and explores their mathematical properties and applications through numerical examples.

Contribution

It presents the first real-valued compatibility constraints for complex order derivatives and develops a new form of such derivatives for viscoelastic modeling.

Findings

Derived real-valued compatibility constraints.

Introduced a new form of complex order fractional derivative.

Numerical examples demonstrate stress relaxation and creep results.

Abstract

We introduce complex order fractional derivatives in models that describe viscoelastic materials. This can not be carried out unrestrictedly, and therefore we derive, for the first time, real valued compatibility constraints, as well as physical constraints that lead to acceptable models. As a result, we introduce a new form of complex order fractional derivative. Also, we consider a fractional differential equation with complex derivatives, and study its solvability. Results obtained for stress relaxation and creep are illustrated by several numerical examples.