## 5 Experiments with Other Known Kernels
### Miller-Ross Kernel  
The Miller-Ross function (MR) function is defined as  
\[M_{MR}(\alpha,\lambda t)=t^{\alpha}\sum\limits_{k=0}^{\infty}\frac{(\lambda t )^{k}}{\Gamma(\alpha+k+1)}=\sum\limits_{k=0}^{\infty}\frac{\lambda^{k}t^{k+ \alpha}}{\Gamma(\alpha+k+1)}t^{\alpha} \tag{130}\]  
It is defined as the \(D_{t}^{-\alpha}[e(\lambda t)]=I_{t}^{\alpha}[e(\lambda t)]\) as the \(\alpha-th\) integral of the exponential function\((\lambda t)\)  
It can be represented as the two-parameter Mittag-Leffler function by [31]  
\[M_{MR}(\alpha,\lambda t)=t^{\alpha}E_{1,\alpha+1}(\lambda t),\quad 0<\alpha<1 \tag{131}\]  
The Laplace transform of the MR function is  
\[\mathcal{L}[M_{MR}(\alpha,\lambda t^{\alpha})]=\frac{1}{s^{\alpha-1}}\bigg{(} 1-\frac{\lambda}{s}\bigg{)}^{-1},\quad\bigg{|}\frac{\lambda}{s}\bigg{|}<1 \tag{132}\]The time derivative of the MR function is  
\[\frac{d}{dt}[M_{MR}(\alpha,\lambda t)]=\frac{d}{dt}\sum_{k=0}^{\infty}\frac{ \lambda^{k}t^{k+\alpha}}{\Gamma(\alpha+k+1)}=t^{\alpha-1}\sum_{k=0}^{\infty} \frac{(k+\alpha)\lambda^{k}t^{k+\alpha}}{\Gamma(\alpha+k+1)} \tag{133}\]  
and through the Laplace transform, with the rule \(\mathcal{L}\Big{[}\frac{d}{dt}f(t)\Big{]}=s\mathcal{L}[f(t)]-f(0)=sf(s)-f(0)\)  
\[\mathcal{L}\Big{[}\frac{d}{dt}M_{MR}(\alpha,-\lambda t^{\alpha})\Big{]}=s \Bigg{[}\frac{1}{s^{\alpha-1}}\bigg{(}1+\frac{\lambda}{s}\bigg{)}^{-1}\Bigg{]} =s^{2-\alpha}\bigg{(}1-\frac{\lambda}{s}\bigg{)}^{-1},\quad\bigg{|}\frac{ \lambda}{s}\bigg{|}<1 \tag{134}\]  
because \(M_{MR}(\alpha,-\lambda t^{\alpha})(t=0)=0\). Then, if we define a relaxation function \(\frac{d}{dt}[M_{MR}(\alpha,-\lambda t)]=\frac{d}{dt}[t^{\alpha}E_{1,\alpha+1} (-\lambda t)]=R_{MR}(\alpha,-\lambda t)\) such that \(R_{MR}(\alpha,-\lambda t)\) and \(M_{MR}(\alpha,-\lambda t)\) are monotonically decaying functions, we may write the heat flux constitutive equation as  
\[q_{MR}(x,t)=-k_{0}\frac{\partial T(x,t)}{\partial x}-k_{2}\int\limits_{0}^{t} {(t-\tau)^{\alpha}E_{1,\alpha+1}\big{(}-\lambda(t-\tau)^{\alpha}\big{)}\frac{ \partial T(x,\tau)}{\partial x}d\tau} \tag{135}\]  
Then, following the integration by parts in the second term of (135), and applying all steps as in the preceding examples, as well as faster by the Laplace transform of \(q_{MR}(x,t)\)  
\[q_{MR}(x,s)=-k_{0}\frac{\partial T(x,s)}{\partial x}-k_{2}\int\limits_{0}^{t} {s\Bigg{[}\frac{1}{s^{\alpha-1}}\bigg{(}1+\frac{\lambda}{s}\bigg{)}^{-1}\Bigg{]} \frac{\partial T(x,s)}{\partial x}ds},\quad 0<\alpha<1 \tag{136}\]  
Then, a rearrangement of the second term in (136) yields  
\[\int\limits_{0}^{t}{s\Bigg{[}\frac{1}{s^{\alpha-1}}\bigg{(}1-\frac{\lambda}{s }\bigg{)}^{-1}\Bigg{]}\frac{\partial T(x,s)}{\partial x}ds}=\int\limits_{0}^{ t}{\Bigg{[}\frac{1}{s^{\alpha-1}}\bigg{(}1+\frac{\lambda}{s}\bigg{)}^{-1} \Bigg{]}\bigg{[}s\frac{\partial T(x,s)}{\partial x}\bigg{]}} \tag{137}\]  
remembering that \(\frac{\partial T(x,0)}{\partial x}=0\), as the initial condition defined at the beginning. This allows expressing the hereditary term with the MR function as a memory kernel, namely  
\[q_{MR}(x,t)=-k_{0}\frac{\partial T(x,t)}{\partial x}-k_{2}\int\limits_{0}^{t} M_{MR}\big{[}-\lambda(t-\tau)^{\alpha}\big{]}\frac{\partial T(x,t)}{ \partial x}d\tau,\quad 0<\alpha<1,\quad t>0 \tag{138}\]  
Now, without loss of generality, we may assume \(-\lambda=-\frac{\alpha}{1-\alpha}\), as well as defining the normalization function \(N(\alpha)=(1-\alpha)\). Thus, we may formulate a Miller-Ross fractional operator of Caputo-type, namely  
\[{}^{MRC}D_{t}^{\pi}f(t)=\frac{1}{1-\alpha}\int\limits_{0}^{t}{M_{MR}\bigg{[}- \frac{\alpha}{1-\alpha}(t-\tau)^{\alpha}\bigg{]}\frac{d}{d\tau}[f(\tau)]d\tau}. \tag{139}\]  
Applying the energy conservation equation with the flux expressed as (138) we get  
\[M_{MR}(\alpha,\lambda t)=t^{\alpha}E_{1,\alpha+1}(\lambda t),\quad 0<\alpha<1 \tag{131}\]0  
For \(\alpha=1\), Equation (140) reduces to the local Fourier equation.