# Subordination Principle for Generalized Fractional Zener Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- (P1)
- The set $\mathcal{CMF}$ is a convex cone: ${\lambda}_{1}{f}_{1}+{\lambda}_{2}{f}_{2}\in \mathcal{CMF}$ for all ${f}_{1},{f}_{2}\in \mathcal{CMF}$ and ${\lambda}_{1},{\lambda}_{2}\ge 0$. Moreover, ${f}_{1}\xb7{f}_{2}\in \mathcal{CMF}$ for all ${f}_{1},{f}_{2}\in \mathcal{CMF}$. The set $\mathcal{CMF}$ is closed under pointwise limits;
- (P2)
- The sets $\mathcal{SF}$, $\mathcal{BF}$ and $\mathcal{CBF}$ are convex cones. They are closed under pointwise limits;
- (P3)
- Any Stieltjes function is completely monotone. Any complete Bernstein function is a Bernstein function;
- (P4)
- Let $\phi \in \mathcal{BF}$. Then, the function $\phi \left(s\right)/s$ is completely monotone;
- (P5)
- $\mathcal{CMF}\circ \mathcal{BF}\subset \mathcal{CMF}$, where ∘ denotes composition of functions from the corresponding sets;
- (P6)
- Every function $\phi \in \mathcal{SF}$, such that ${lim}_{s\to +\infty}\phi \left(s\right)=0$, is the Laplace transform of a function $f\left(t\right)$, which is locally integrable on ${\mathbb{R}}_{+}$ and completely monotone;
- (P7)
- $\phi \in \mathcal{SF}$ if and only if $s\phi \left(s\right)\in \mathcal{CBF}$;
- (P8)
- Let $\phi \ne 0$. Then, $\phi \left(s\right)\in \mathcal{CBF}$ if and only if ${\left(\phi \left(s\right)\right)}^{-1}\in \mathcal{SF}$;
- (P9)
- Let $\phi ,\psi \in \mathcal{CBF}$. Assume ${\mu}_{1},{\mu}_{2}\in (0,1)$ are such that ${\mu}_{1}+{\mu}_{2}\le 1$. Then,$${\phi}^{{\mu}_{1}}\left(s\right)\xb7{\psi}^{{\mu}_{2}}\left(s\right)\in \mathcal{CBF};$$
- (P10)
- Any function $\phi $ from the sets $\mathcal{CBF}$ or $\mathcal{SF}$ admits an analytic extension to the complex plane cut along the negative real axis $\mathbb{C}\backslash (-\infty ,0]$, which satisfies$$|arg\phi (z\left)\right|\le |argz|,z\in \mathbb{C}\backslash (-\infty ,0].$$

## 3. Completely Monotone Relaxation Modulus

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 4. Generalized Fractional Zener Wave Equation

#### 4.1. Propagation Function

**Proposition**

**1.**

**Proof.**

#### 4.2. Propagation Speed

**Theorem**

**2.**

## 5. Subordination Principle

#### 5.1. General Formulation

**Theorem**

**3.**

**Proof.**

#### 5.2. Applications

## 6. Numerical Results

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 7. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Propagation function $w(x,t)$ as a function of x for $a=0.2$, $b=1.0$ and kernel (64); (

**a**) $\lambda =0$, $t=0.4$ and different values of $\alpha $; marked with a thinner line—the limiting elastic case $\alpha \to 0$; (

**b**) $\lambda =1$, $\alpha =1$ and $\beta =0.5$, marked with a thinner line—the function $exp(-\kappa x)$.

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Bazhlekova, E.; Bazhlekov, I. Subordination Principle for Generalized Fractional Zener Models. *Fractal Fract.* **2023**, *7*, 298.
https://doi.org/10.3390/fractalfract7040298

**AMA Style**

Bazhlekova E, Bazhlekov I. Subordination Principle for Generalized Fractional Zener Models. *Fractal and Fractional*. 2023; 7(4):298.
https://doi.org/10.3390/fractalfract7040298

**Chicago/Turabian Style**

Bazhlekova, Emilia, and Ivan Bazhlekov. 2023. "Subordination Principle for Generalized Fractional Zener Models" *Fractal and Fractional* 7, no. 4: 298.
https://doi.org/10.3390/fractalfract7040298