Complex order fractional derivatives in viscoelasticity

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.Mathematics Subject Classification (2010): Primary: 26A33; Secondary: 74D05

Wave equation for generalized Zener model containing complex order fractional derivatives

“…Following the same procedure as in [5] in obtaining (16), we conclude that the maximal and minimal values of f (τ, ϕ) and g(τ, ϕ) with respect to τ are f (τ f , ϕ) = ± cos(αϕ) cosh(βϕ) 1 + tg(αϕ) tgh(βϕ) 2 , Proof. Set s = ρe iϕ , ρ = s 2 0 + p 2 , ϕ ∈ [0, π/2].…”

“…Properties of functions f and g for ϕ = π/2 were investigated in [5]. It was shown that the extremal values of f and g are attained at points τ f and τ g respectively, where…”

“…To the best of our knowledge, so far only factional derivatives of real order have been used for describing waves in viscoelastic media. In this work, following our previous results [3,5], we study waves in a viscoelastic rod whose material is described by fractional derivatives of complex order. Our particular interest is related to waves in a specific viscoelastic material described by a generalized Zener standard model.…”

Properties of the Caputo-Fabrizio fractional derivative and its distributional settings

“…We can define CF t 0 D α t f (t) by (1.1). Since we prefer to work with real valued functions after fractional differentiation, we follow our approach presented in [9] and define combination of (CFFD) of complex order as:…”

Theory of Fractional Implicit Differential Equations With Complex Order

“…Atanackovi et al established complex order fractional derivatives in models that describe viscoelastic materials in [31]. The authors investigated existence and uniqueness by using a classical fixed point theorem.…”