## 1 Introduction
### Heat Conduction as a Transport Process with Memory
#### 1.2.1 Constitutive Fractional Heat Flux with Caputo Derivative [15]  
Let us start with the energy balance (the First Law of Thermodynamics) in a rigid heat conductor  
\[\frac{\partial e(x,t)}{\partial t}=-\frac{\partial q(x,t)}{\partial x}+r \tag{1}\]  
where \(e=\rho C_{p}T\) is the internal energy, \(q(x,t)\)is the heat flux, and \(r\) is the heat generation (per unit volume). With the Fourier constitutive equation  
\[q(x,t)=-k(T(x,t))\frac{\partial T(x,t)}{\partial x},\quad k(T(x,t)) \tag{2}\]  
where \(k(T(x,t))\)is the thermal conductivity. This construction of the heat flux _has no memory_, propagates with _infinite speed_, and with the energy balance equation results in _the parabolic Fourier model_. Attempts to model heat flux propagation with a finite speed were done by [PERSON] [23; 24] and [PERSON] [25] with slight modifications of the heat flux (via a Taylor series expansion of the heat flux, in time, and as a first-order approximation), namely  
\[\frac{\partial q(x,t)}{\partial t}=q(x,t)+k(T(x,t))\frac{1}{\tau}\frac{ \partial q(x,t)}{\partial x},\quad\tau>0 \tag{3}\]  
with exponential memory ([PERSON]'s memory kernel) \(R(t)=\exp(-t/\tau\) ), and a finite relaxation time \(\tau\), named the Cattaneo equation (see more details in Section 4.2). As [PERSON] [15] demonstrated, replacing the time derivative in (2) by a fractional (Caputo) derivative, we get (in the original notations)  
\[{}^{\mathbb{C}}_{-\infty}D^{\alpha}_{t}q(x,t)=q(x,t)+k(T(x,t))\frac{\partial T (x,t)}{\partial x},\quad\alpha>0,\quad t>0 \tag{4}\]  
The thermodynamic restriction from the Second Law of Thermodynamics for a rigid heat conductor needs the following inequality to be obeyed [14]  
\[\frac{\partial q(x,t)}{\partial t}\geq\frac{\frac{\partial e(x,t)}{\partial t }}{T(x,t)}+\frac{1}{\left(T(x,t)\right)^{2}}\frac{\partial T(x,t)}{\partial x} \tag{5}\]  
where the internal energy \(e(x,t)\) and the entropy \(\eta(x,t)\) are state functions. In this framework, the following extension of (2) was proposed [15]  
\[ \begin{split}& q(x,t)={}^{\mathbb{C}}_{-\infty}D^{\alpha}_{t}\left[ \frac{\partial q(x,t)}{\partial x}\right]=-k(x)\frac{1}{\Gamma(1-\alpha)}\int \limits_{-\infty}^{t}\frac{1}{\left(t-\tau\right)^{\alpha}}\frac{\partial}{ \partial t}\left[\frac{\partial T(x,\tau)}{\partial x}\right]d\tau=\\ &=-k(x)\frac{\alpha}{\Gamma(1-\alpha)}\int\limits_{-\infty}^{t} \frac{1}{\left(t-\tau\right)^{\alpha+1}}\bigg{[}\frac{\partial T(x,\tau)}{ \partial x}-\frac{\partial T(x,\tau)}{\partial x}\bigg{]}d\tau\end{split}  \tag{6}\]The second version (in the second row) in (6) is an equivalent presentation of the Caputo derivative. Finally, with the assumption that the internal energy \(e(x,t)\) depends only on \(T(x,t)\) (i.e., the density \(\rho\) and the heat capacity \(C_{p}\) are temperature independent) and applying the energy balance equation the following model was obtained  
\[\rho C_{p}\frac{\partial T(x,t)}{\partial t}=\frac{\partial}{\partial x}k(x) \frac{\alpha}{\Gamma(1-\alpha)}\int\limits_{-\infty}^{t}\frac{1}{(t-\tau)^{ \alpha+1}}\bigg{[}\frac{\partial T(x,\tau)}{\partial x}-\frac{\partial T(x, \tau)}{\partial x}\bigg{]}d\tau+r(x,t) \tag{7}\]  
or in a more convenient form, for the presentation in this article, as  
\[\frac{\partial T(x,t)}{\partial t}=\frac{\partial}{\partial x}a(x)\frac{1}{ \Gamma(1-\alpha)}\int\limits_{-\infty}^{t}\frac{1}{(t-\tau)^{\alpha}}\frac{d} {d\tau}\bigg{[}\frac{\partial T(x,\tau)}{\partial x}\bigg{]}d\tau+r(x,t),\quad a (x)=\frac{k(x)}{\rho C_{p}},\quad t>0 \tag{8}\]  
**Remark 1**. : _It is worth noting and important for the entire discussion developed in this article that the time derivative in the energy balance equation (sometimes termed a continuity equation) is not replaced by a fractional derivative. The flux constitutive model is satisfying the causality principle [26, 27, 28], that between the cause (temperature gradient) and the result (output of the system), there is a time shift. This time shift in (7) is assured by the fractional Caputo derivative.