## 4 Heat Conduction Models with Mittag-Leffler-Type Memories: Examples
### Heat Flux with Exponential Memory Kernel  
Consider a virgin semi-infinite material (medium) with initial and boundary conditions (signaling problem)  
\[T(x,0)=T(\infty,t)=T(-\infty,t)=0,\quad T_{x}(x,0)=T_{xx}(x,0)=0,\quad T(0,t)=T _{s} \tag{83}\]  
These conditions will be valid for all examples developed in the sequel. We start with the simplest case when the kernel is  
\[e_{1,1}^{1}(-\lambda t)=\exp(-\lambda t),\quad\frac{d}{dt}\left[e_{1,1}^{1}(- \lambda t)\right]=-\lambda\exp(-\lambda t) \tag{84}\]  
which will allow an easier understanding of the technology of the solution. This development follows the results of [66]. Hence, we consider an exponential memory (Jeffrey's kernel) \(R(t)\)[83] with a finite relaxation time \(\tau\) (and \(\lambda=1/\tau\) )  
\[R(t-s)=\exp\biggl{(}-\frac{t-s}{\tau}\biggr{)},\quad R_{t}(t)=-\frac{1}{\tau }\exp\biggl{(}-\frac{(t-s)}{\tau}\biggr{)} \tag{85}\]  
If we now apply the energy balance (79) (with \(\rho C_{p}=const.\)), the result is the Cattaneo equations [23] modeling only the relaxation part of the heat conduction (the RHS of (86))  
\[\frac{\partial T(x,t)}{\partial t}=-\frac{a_{2}}{\tau}\int\limits_{0}^{t}\exp \biggl{[}-\biggl{(}\frac{t-s}{\tau}\biggr{)}\biggr{]}\frac{\partial T(x,s)}{ \partial x}ds,\quad a_{2}=\frac{k_{2}}{\rho C_{p}} \tag{86}\]  
The Fourier law is the Cattaneo equation's limit \(\tau\to 0\) resulting in (82)  
Now, let us accept the construction of the relaxation kernel (73), namely  
\[M(t)=k_{1}\delta(t)+k_{2}R_{t}(t)\Rightarrow M(t-s)=k_{1}\delta(s)+k_{2}\frac {1}{\tau}\exp\biggl{(}-\frac{(t-s)}{\tau}\biggr{)} \tag{87}\]  
In (87) \(\delta(s)\) is the Dirac delta function, while \(a_{1}\) and \(a_{2}\) are the _effective thermal conductivity_ and the _elastic conductivity_. Then, the heat flux can be expressed as  
\[q(x,t)=-k_{1}\frac{\partial T(x,t)}{\partial x}-\frac{k_{2}}{\tau}\int\limits_ {-\infty}^{t}e^{-\left(\frac{t-s}{\tau}\right)}\frac{\partial T(x,s)}{ \partial x}ds \tag{88}\]  
The energy balance (79) yields  
\[\frac{\partial T(x,t)}{\partial t}=a_{1}\frac{\partial^{2}T(x,t)}{\partial x^ {2}}+a_{2}\lambda\int\limits_{-\infty}^{t}e^{-\lambda(t-s)}\frac{\partial^{2} T(x,s)}{\partial x^{2}}ds,\quad\lambda=\frac{1}{\tau} \tag{89}\]  
Here \(a_{1}=k_{1}/\rho C_{p}\), \(a_{2}=k_{2}/\rho C_{p}\) are thermal diffusivities: they have equal dimensions \(\left[m^{2}/s\right]\) because the exponential second term in (87) is dimensionless. Denoting \(F(x,t)=\partial^{2}T(x,t)/\partial x^{2}\), for the sake of the simplicity of the expressions, and after integration by parts in the second term in (89) we get [66]. \[\lambda\int\limits_{-\infty}^{t}e^{-\lambda(t-s)}F(x,s)ds=\left.e^{- \lambda(t-s)}[F(x,s)-F(x,t)]\right|_{-\infty}^{t}+ \tag{90}\] \[+\lambda\int\limits_{-\infty}^{t}e^{-\lambda(t-s)}[F(x,t)-F(x,s)]ds\]  
The first term in the RHS of (90), is zero, but the second one matches the definition of the Caputo-Fabrizio fractional derivative [53] (see also the general construction (75) of a Caputo-type derivative). Precisely, we have the formulation [53]  
\[{}^{CF}D_{t}^{\alpha}=\frac{\alpha}{\left(1-\alpha\right)^{2}}\int\limits_{- \infty}^{t}[f(t)-f_{a}(s)]\exp\left[-\frac{\alpha}{1-\alpha}(t-s)\right]ds, \quad t>0 \tag{91}\]  
This can be considered as a _pro-Caputo_ (non-normalized) derivative denoted as \({}_{PC}D_{t}^{\lambda}\). It can be expressed in two equivalent forms (following the notations used in [53] namely [66]  
\[e_{1,1}^{1}(-\lambda t)=\exp(-\lambda t),\quad\frac{d}{dt}\left[e_{1,1}^{1}(- \lambda t)\right]=-\lambda\exp(-\lambda t) \tag{84}\]0 \[e_{1,1}^{1}(-\lambda t)=\exp(-\lambda t),\quad\frac{d}{dt}\left[e_{1,1}^{1}(- \lambda t)\right]=-\lambda\exp(-\lambda t) \tag{84}\]1  
where the rate constant \(\lambda\in(0,\infty)\) controls the kernel.