## 4 Heat Conduction Models with Mittag-Leffler-Type Memories: Examples
### Heat Flux with Prabhakar Memory Kernel  
After the two prior examples, which used relatively simple kernels, we now attempt to create a fractional derivative using the Prabhakar kernel and the fading memory formalism. Therefore, applying (36) and its first derivative (43) we get for the heat flux with relaxation function defined as  
\[R_{PB}=k_{1}\delta(t)+k_{2}\frac{d}{dt}\Bigl{[}e_{\alpha,\beta}^{\gamma}(- \lambda t^{\alpha})\Bigr{]}=k_{1}\delta(t)+k_{2}\Bigl{[}t^{\beta-2}E_{\alpha, \beta-1}^{\gamma}(-\lambda t^{\alpha})\Bigr{]} \tag{105}\]  
Then, the heat flux can be formulated as  
\[q(x,t)=-k_{1}\frac{\partial T(x,t)}{\partial x}-k_{2}\int\limits_{-\infty}^{t }\frac{\partial T(x,u)}{\partial x}\frac{d}{du}\Bigl{[}e_{\alpha,\beta}^{ \gamma}(t,-\lambda u^{\alpha})\Bigr{]}du \tag{106}\]  
Assuming for the sake of simplicity \(\lambda=1\), and applying integration by parts in the second term of (106), and what _mutatis mutandis_, we get  
\[q(x,t)=-k_{1}\frac{\partial T(x,t)}{\partial x}-k_{2}\int\limits_{0}^{t}e_{ \alpha,\beta}^{\gamma}(t,-u^{\alpha})\frac{d}{du}\biggl{[}\frac{\partial T(x, u)}{\partial x}\biggr{]}du \tag{107}\]In general, the second term in (107) mimics the fractional derivative of the Caputo type. Alternately, by using the Laplace transform on (106) and accounting for the convolution product in the second term, we obtain (see (38))  
\[\mathcal{L}[q(x,t)]=\mathcal{L}\bigg{[}-k_{1}\frac{\partial T(x,t)}{\partial x} \bigg{]}-k_{2}\mathcal{L}\bigg{[}\frac{d}{dt}e^{\gamma}_{\alpha,\beta}(- \lambda t^{\alpha})\bigg{]}\times\mathcal{L}\bigg{[}\frac{\partial T(x,t)}{ \partial x}\bigg{]} \tag{108}\]  
that is  
\[ \begin{split}& q(x,s)=-k_{1}\frac{\partial T(x,s)}{\partial x}-k_{2 }\bigg{[}\frac{s^{\alpha\gamma-\beta+1}}{(s+\lambda)^{\gamma}}\bigg{]}\times \frac{\partial T(x,s)}{\partial x}=\\ &=-k_{1}\frac{\partial T(x,s)}{\partial x}-k_{2}\bigg{[}\frac{s ^{\alpha\gamma-\beta}}{(s+\lambda)^{\gamma}}\bigg{]}\times s\frac{\partial T (x,s)}{\partial x}\end{split}  \tag{109}\]  
The inverse transformation \(\mathcal{L}^{-1}[q(x,s)]\) yields  
\[q(x,t)=-k_{1}\frac{\partial T(x,t)}{\partial x}-k_{2}\int\limits_{0}^{t}e^{ \gamma}_{\alpha,\beta}(t,-\lambda u^{\alpha})\frac{d}{du}\bigg{[}\frac{ \partial T(x,u)}{\partial x}\bigg{]}du \tag{110}\]  
The second term in (110) has the construction of Caputo-type with the Prabhakar memory kernel (see (49)). Assuming a normalization function \(N(\alpha)=(1-\alpha)\) and setting\(\lambda=\alpha/(1-\alpha)\) (110) can be reformulated as follows without loss of generality  
\[ \begin{split}& q(x,t)=-k_{1}\frac{\partial T(x,t)}{\partial x}-k_{2 }(1-\alpha)\left\{\frac{1}{1-\alpha}\int\limits_{0}^{t}e^{\gamma}_{\alpha, \beta}(t,-\lambda u^{\alpha})\frac{d}{du}\bigg{[}\frac{\partial T(x,u)}{ \partial x}\bigg{]}du\right\}\\ &\lambda=\frac{\alpha}{1-\alpha},\quad t>0\end{split}  \tag{111}\]  
Now, applying the energy balance equation, we have  
\[ \begin{split}&\frac{\partial T(x,t)}{\partial t}=a_{1}\frac{ \partial T(x,t)}{\partial x}+a_{2}(1-\alpha)\left\{\frac{1}{1-\alpha}\int \limits_{0}^{t}e^{\gamma}_{\alpha,\beta}(t,-\lambda u^{\alpha})\frac{d}{du} \bigg{[}\frac{\partial T(x,u)}{\partial x}\bigg{]}du\right\},\\ &\lambda=\frac{\alpha}{1-\alpha},\quad t>0\end{split}  \tag{112}\]  
For \(\alpha=1\), the non-locality in (112) is absent, and it reduces to the Fourier model. The construction  
\[{}_{H}^{CP}\mathfrak{D}^{\gamma}_{\alpha,\beta}f(t)=\frac{1}{1-\alpha}\int \limits_{0}^{t}e^{\gamma}_{\alpha,\beta}(t,-\lambda u^{\alpha})\frac{d}{du}f(u )du,\quad\lambda=\frac{\alpha}{1-\alpha},\quad 0<\alpha<1,\quad t>0 \tag{113}\]  
is a fractional operator (derivative) that uses a base point zero but does not exactly match the definition (49). The differences result from the fact that (49) is formulated constitutively, mathematically correct (but with no physics behind it), and similar to the traditional Caputo derivative. They also result from the definition of the Prabhakar fractional integral (45) (in a Riemann-Liouville sense) [38; 41; 43; 45; 49].