# Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Theorem**

**1**

- (P1)
- The sets $\mathcal{CMF}$, $\mathcal{SF}$, and $\mathcal{CBF}$ are convex cones, closed under pointwise limits.
- (P2)
- $\phi \in \mathcal{SF}$ if and only if $s\phi \left(s\right)\in \mathcal{CBF}$.
- (P3)
- Let $\phi \ne 0$. Then, $\phi \left(s\right)\in \mathcal{SF}$ if and only if ${\left(\phi \left(s\right)\right)}^{-1}\in \mathcal{CBF}$.
- (P4)
- Any function $\phi \in \mathcal{CBF}$ has an analytic continuation to the upper half-plane,$$\mathbb{H}=\{z\in \mathbb{C}:\Im z>0\},$$$$\phi \left(z\right)=A+Bz+{\int}_{0}^{\infty}\frac{z}{z+t}\phantom{\rule{0.166667em}{0ex}}\nu \left(\mathrm{d}t\right),$$
- (P5)
- Any function φ from the classes $\mathcal{CBF}$ or $\mathcal{SF}$ admits an analytic extension to the complex plane cut along the negative real axis $\mathbb{C}\setminus (-\infty ,0]$, such that$${\left(\phi \left(z\right)\right)}^{*}=\phi \left({z}^{*}\right),$$$$|arg\phi (z\left)\right|\le |argz|$$$$\begin{array}{ccc}& & \Im \phi \left(z\right)\xb7\Im z\ge 0\phantom{\rule{4pt}{0ex}}if\phantom{\rule{4pt}{0ex}}\phi \in \mathcal{CBF};\hfill \end{array}$$$$\begin{array}{ccc}& & \Im \phi \left(z\right)\xb7\Im z\le 0\phantom{\rule{4pt}{0ex}}if\phantom{\rule{4pt}{0ex}}\phi \in \mathcal{SF}.\hfill \end{array}$$
- (P6)
- Let φ be the analytic extension of a complete Bernstein function to $\mathbb{C}\setminus (-\infty ,0]$. Then, the limit$$\phi (0+)=\underset{(0,\infty )\ni s\to 0}{lim}\phi \left(s\right)$$
- (P7)
- If $\alpha \in [0,1]$, then$$\phantom{\rule{4pt}{0ex}}{s}^{-\alpha}\in \mathcal{SF},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{s}^{\alpha}\in \mathcal{CBF}.$$

**Theorem**

**2**

## 3. Model Formulation

## 4. First Integral Representation

**Theorem**

**3.**

**Proof.**

## 5. Examples

#### 5.1. Fractional Derivative of Uniformly Distributed Order

**Example**

**1.**

#### 5.2. Convolutional Derivative with Multinomial Mittag-Leffler-Type Kernel

**Example**

**2.**

## 6. Second Integral Representation

**Theorem**

**4**

**Theorem**

**5.**

**Corollary**

**1.**

## 7. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Relaxation modulus $G\left(t\right)$ for the model of Example 1: (

**a**) $a=0.3$ and different values of b; (

**b**) $b=3$ and different values of a.

**Figure 2.**Relaxation modulus $G\left(t\right)$ for the model of Example 2 with $m=2$, ${\lambda}_{1}={\lambda}_{2}=1$, ${\alpha}_{1}=0.5$, ${\alpha}_{2}=0.3$, $\beta =0.8$: (

**a**) $a=0.3$ and different values of b; (

**b**) $b=3$ and different values of a.

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Bazhlekova, E.; Pshenichnov, S.
Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model. *Fractal Fract.* **2023**, *7*, 636.
https://doi.org/10.3390/fractalfract7080636

**AMA Style**

Bazhlekova E, Pshenichnov S.
Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model. *Fractal and Fractional*. 2023; 7(8):636.
https://doi.org/10.3390/fractalfract7080636

**Chicago/Turabian Style**

Bazhlekova, Emilia, and Sergey Pshenichnov.
2023. "Two Integral Representations for the Relaxation Modulus of the Generalized Fractional Zener Model" *Fractal and Fractional* 7, no. 8: 636.
https://doi.org/10.3390/fractalfract7080636