# Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

^{*}

## Abstract

**:**

## 1. Introduction

_{F}is the Fourier thermal diffusivity, $\tau ={k}_{F}/{C}^{2}$ is the relaxation time of the heat waves propagation, which relates the moments of the temperature change and of the respective heat flux change. Cattaneo-Vernotte constitution implies a phase lag between the heat flux vector and the temperature gradient; in addition to heat diffusion the temperature perturbation propagates in matter like damped sound-wave at finite speed $C=\sqrt{{k}_{F}/\tau}$. The relaxation time τ is associated with the phonon–phonon interaction time; at normal conditions it is very small: τ ≈ 10

^{−13}s. The Cattaneo Equation (1) is the particular case of the telegrapher’s equation (TE).

_{b}, is the ballistic type heat conductivity. The parameters in the above Equations (3) and (4) have the following dimensions: $[\tau ]=s$, $[{k}_{F}]=\frac{{m}^{2}}{s}$, $[{D}_{b}]={m}^{2}$, $[\mu ]=\frac{1}{s}$, $\left[\kappa \right]=\frac{1}{{s}^{2}}$, $[\alpha ]=\frac{{m}^{2}}{{s}^{2}}$, $[\delta ]=\frac{{m}^{2}}{s}$, and $[\epsilon ]=\frac{1}{s}$.

## 2. Ballistic Heat Transport Equations with Substantial Derivative

_{F}= 5/3, D

_{b}= 3/2, τ = 1/2, μ = −1/2 in (4). We assume dimensionless equations here and in what follows. In the weak ballistic case, Kn = 0.1, we get α = 0.03333, δ=0.03, ε = 2, κ = −1 in (3) and k

_{F}= 0.01666, k

_{b}= 0.015, τ = 1/2, μ = −1/2 in (4). In the distinct ballistic case, Kn = 1, all heat transport terms in GK-type equation contribute more or less equally, in the weak ballistic case, Kn = 0.1, the Cattaneo wave-term prevails. The inhomogeneous system of PDE (5) and (6) for ballistic heat transport thin film was studied numerically in [30]; based on the periodic analytical solutions to GK-type equation [72] some solutions to (5) and (6) were obtained in [73,74].

## 3. Operational Approach to Transport Equations

## 4. Exact Periodic Solutions to GK-Type Equation with Substantial Derivative

## 5. Exact Periodic Solutions to Ballistic Heat Transport in Thin Films

_{d}= Kn

_{b}= 1 and v = 10, are presented in Figure 2, for v = −10 in Figure 3.

_{b}= Kn

_{d}= 0.1 ${\theta}_{b}\left(x,t\right)$ and ${\theta}_{d}\left(x,t\right)$ constituents behave differently (see Figure 4). The diffusive constituent slowly grows as the ballistic constituent shows gradual amplitude decrease, as shown in Figure 5. Moreover, for v = 10 the complete quasi-temperature $\theta \left(x,t\right)={\theta}_{b}\left(x,t\right)+{\theta}_{d}\left(x,t\right)$ monotonously increases, following the behavior of the diffusive constituent as shown in Figure 5. The maximum principle, established though for parabolic equations, is violated for the diffusive component and complete quasi-temperature with Cauchy conditions for non-Fourier ballistic heat transport in thin films. In all of the cases, the evolution of ${\theta}_{b}\left(x,t\right)$ occurs faster than that of ${\theta}_{d}\left(x,t\right)$; the two-speed heat propagation process can be seen in every set of plots in Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5.

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Evolution of quasi-temperature components in a thin stationary film with Kn

_{b}= Kn

_{d}= 1 for v = 0: ${\theta}_{b}\left(x,t\right)$—top left plot, ${\theta}_{d}\left(x,t\right)$—top right plot and $\theta \left(x,t\right)={\theta}_{b}\left(x,t\right)+{\theta}_{d}\left(x,t\right)$—bottom plot. The Cauchy conditions for PDEs (9) and (10) system are ${\theta}_{b}\left(x,0\right)={\theta}_{d}\left(x,0\right)={e}^{inx}$, ${\partial}_{t}{\theta}_{b}\left(x,0\right)={\partial}_{t}{\theta}_{d}\left(x,0\right)=0$.

**Figure 2.**Evolution of quasi-temperature components in a thin film with Kn

_{b}= Kn

_{d}= 1 for v = +10: ${\theta}_{b}\left(x,t\right)$—top left plot, ${\theta}_{d}\left(x,t\right)$—top right plot and $\theta \left(x,t\right)={\theta}_{b}\left(x,t\right)+{\theta}_{d}\left(x,t\right)$—bottom plot. The Cauchy conditions for PDEs (9) and (10) system are ${\theta}_{b}\left(x,0\right)={\theta}_{d}\left(x,0\right)={e}^{inx}$, ${\partial}_{t}{\theta}_{b}\left(x,0\right)={\partial}_{t}{\theta}_{d}\left(x,0\right)=0$.

**Figure 3.**Evolution of quasi-temperature components in a thin film with Kn

_{b}= Kn

_{d}= 1 for v = −10: ${\theta}_{b}\left(x,t\right)$—top left plot, ${\theta}_{d}\left(x,t\right)$—top right plot and $\theta \left(x,t\right)={\theta}_{b}\left(x,t\right)+{\theta}_{d}\left(x,t\right)$—bottom plot. The Cauchy conditions for PDEs (9) and (10) system are ${\theta}_{b}\left(x,0\right)={\theta}_{d}\left(x,0\right)={e}^{inx}$, ${\partial}_{t}{\theta}_{b}\left(x,0\right)={\partial}_{t}{\theta}_{d}\left(x,0\right)=0$.

**Figure 4.**Evolution of quasi-temperature components in a thin film with Kn

_{b}= Kn

_{d}= 0.1 for v = +10: ${\theta}_{b}\left(x,t\right)$—top left plot, ${\theta}_{d}\left(x,t\right)$—top right plot and $\theta \left(x,t\right)={\theta}_{b}\left(x,t\right)+{\theta}_{d}\left(x,t\right)$—bottom plot. The Cauchy conditions for PDEs (9) and (10) system are ${\theta}_{b}\left(x,0\right)={\theta}_{d}\left(x,0\right)={e}^{inx}$, ${\partial}_{t}{\theta}_{b}\left(x,0\right)={\partial}_{t}{\theta}_{d}\left(x,0\right)=0$.

**Figure 5.**Evolution of quasi-temperature components in a stationary thin film with Kn

_{b}= Kn

_{d}= 0.1 for v = 0: ${\theta}_{b}\left(x,t\right)$—left plot, ${\theta}_{d}\left(x,t\right)$—right plot. The Cauchy conditions for PDEs (9) and (10) system are ${\theta}_{b}\left(x,0\right)={\theta}_{d}\left(x,0\right)={e}^{inx}$, ${\partial}_{t}{\theta}_{b}\left(x,0\right)={\partial}_{t}{\theta}_{d}\left(x,0\right)=0$.

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Zhukovsky, K.; Oskolkov, D.; Gubina, N.
Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative. *Axioms* **2018**, *7*, 48.
https://doi.org/10.3390/axioms7030048

**AMA Style**

Zhukovsky K, Oskolkov D, Gubina N.
Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative. *Axioms*. 2018; 7(3):48.
https://doi.org/10.3390/axioms7030048

**Chicago/Turabian Style**

Zhukovsky, Konstantin, Dmitrii Oskolkov, and Nadezhda Gubina.
2018. "Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative" *Axioms* 7, no. 3: 48.
https://doi.org/10.3390/axioms7030048