# Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation

^{*}

## Abstract

**:**

## 1. Introduction and Model Formulation

## 2. Preliminaries

- (A)
- The class $\mathcal{CMF}$ is closed under point-wise multiplication;
- (B)
- If $\phi \in \mathcal{CMF}$ and $\psi \in \mathcal{BF}$ then the composite function $\phi \left(\psi \right)\in \mathcal{CMF}$;
- (C)
- $\mathcal{SF}\subset \mathcal{CMF}$ and $\mathcal{CBF}\subset \mathcal{BF}$;
- (D)
- $\phi \in \mathcal{CBF}$ if and only if $1/\phi \in \mathcal{SF}$;
- (E)
- If $\phi ,\psi \in \mathcal{CBF}$ then $\sqrt{\phi .\psi}\in \mathcal{CBF}$;
- (F)
- If $\alpha \in [0,1]$ then ${\lambda}^{\alpha}\in \mathcal{CBF}$ and ${\lambda}^{-\alpha}\in \mathcal{SF}$, $\lambda >0$;
- (G)
- If $0<\alpha \le 1$ and $a>0$ then ${E}_{\alpha}(-a{\lambda}^{\alpha})\in \mathcal{CMF}$ and ${\lambda}^{\alpha -1}{E}_{\alpha ,\alpha}(-a{\lambda}^{\alpha})\in \mathcal{CMF}$ as functions of $\lambda $.

## 3. Representations as Generalized Diffusion-Wave Equations

**Proposition**

**1.**

**Proof.**

#### 3.1. Generalized Diffusion Equation (${\tau}_{q}<{\tau}_{T}$)

#### 3.2. Generalized Wave Equation (${\tau}_{q}>{\tau}_{T}$)

## 4. Fundamental Solution

**Proposition**

**2.**

**Proof.**

#### 4.1. One-Dimensional Solution

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

#### 4.2. Mean Squared Displacement

## 5. Numerical Examples

## 6. Subordination Principles and Multi-Dimensional Fundamental Solutions

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 7. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plots of the the fundamental solution ${\mathcal{G}}_{1}(x,t)$ versus x ($x>0$) for different values of t; (

**a**) diffusion regime (${\tau}_{q}<{\tau}_{T}$); (

**b**) propagation regime (${\tau}_{q}>{\tau}_{T}$).

**Figure 2.**Plots of the the fundamental solution ${\mathcal{G}}_{1}(x,t)$ versus x ($x>0$) for fixed t and different values of $\alpha $, $\alpha =0.05,0.25,0.5,0.75,0.95$, compared to $\alpha =1$ (dashed line); (

**a**) diffusion regime (${\tau}_{q}<{\tau}_{T}$); (

**b**) propagation regime (${\tau}_{q}>{\tau}_{T}$).

**Figure 3.**Plots of the fundamental solution ${\mathcal{G}}_{1}(x,t)$ versus x ($x>0$) for different values of the relaxation times ${\tau}_{q}$ and ${\tau}_{T}$: (

**a**) ${\tau}_{q}/{\tau}_{T}=0.1$—diffusion regime; (

**b**) ${\tau}_{T}/{\tau}_{q}=0.1$—propagation regime.

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**MDPI and ACS Style**

Bazhlekova, E.; Bazhlekov, I.
Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation. *Fractal Fract.* **2020**, *4*, 32.
https://doi.org/10.3390/fractalfract4030032

**AMA Style**

Bazhlekova E, Bazhlekov I.
Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation. *Fractal and Fractional*. 2020; 4(3):32.
https://doi.org/10.3390/fractalfract4030032

**Chicago/Turabian Style**

Bazhlekova, Emilia, and Ivan Bazhlekov.
2020. "Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation" *Fractal and Fractional* 4, no. 3: 32.
https://doi.org/10.3390/fractalfract4030032