## 5 Experiments with Other Known Kernels
### Rabotnov Kernel  
The Rabotnov function is defined originally as [89; 95]  
\[M_{Rab}(\alpha,\lambda t)=\frac{t^{\alpha-1}}{\left(\frac{1}{\lambda}\right) ^{\alpha}}\sum\limits_{k=0}^{\infty}\frac{(-1)^{k}(\lambda t)^{\alpha k}}{ \Gamma[\alpha(k+1)]},\quad\lambda=\frac{1}{\tau_{k}},-1<\alpha<0 \tag{143}\]  
and can be presented alternatively as [31]  
\[M_{Rab}(\alpha,\lambda)=t^{\alpha}\sum\limits_{k=0}^{\infty}\frac{\lambda^{k} t^{k(\alpha+1)}}{\Gamma[(k+1)(\alpha+1)]},\quad-1<\alpha<0 \tag{144}\]  
or in terms of two-parameter Mittag-Leffler functions as [31]  
\[M_{Rab}(\alpha,\lambda)=t^{\alpha}E_{\alpha+1,\alpha+1}\Big{(}\lambda t^{ \alpha+1}\Big{)},\quad M_{Rab}(\alpha,\lambda)=t^{\alpha-1}E_{\alpha,\alpha} (\lambda t^{\alpha}) \tag{145}\]  
For \(\alpha=1\) the function \(M_{Rab}(1,\lambda)\) reduces to the exponential function. However, when \(\alpha\to 0\) it goes to \(\frac{1}{2}\delta(t)\)[89]. Its time derivative, applying (25) to the second version of (145), can be expressed as  
\[\frac{d}{dt}[M_{Rab}(\alpha,\lambda)]=t^{\alpha-2}E_{\alpha,\alpha}(\lambda t ^{\alpha}) \tag{146}\]  
The original Rabotnov function (143) is a growing in time function, conceived to model creep (deformation, extension) of viscoelastic and elastoplastic materials [89], and in such a case, the kernel in a hereditary integral should be  
\[M_{Rab}(-\alpha,\lambda)=(t-\tau)^{-\alpha}\sum\limits_{k=0}^{\infty}\frac{(- \lambda)^{k}(t-\tau)^{k(1-\alpha)}}{\Gamma[(k+1)(1-\alpha)]} \tag{147}\]  
A decaying (monotonic) version was used in a new fractional operator known as the Yang-Abdel-Aty-Cattani (YAC) derivative [92; 96; 97] (see also [98; 99; 100])  
\[M_{YAC}(-\lambda,t^{\alpha})=\sum\limits_{k=0}^{\infty}\frac{(-\lambda)^{k}t^ {(k+1)(\alpha+1)-1}}{\Gamma[(k+1)(\alpha+1)]} \tag{148}\]  
Its Laplace transform is [98; 99]  
\[\mathcal{L}[M_{Rab}(\lambda t^{\alpha})]=\frac{1}{s^{\alpha+1}}\frac{1}{1- \frac{\lambda}{s^{\alpha+1}}},\quad\left|\frac{\lambda}{s^{\alpha+1}}\right|<1 \tag{149}\]  
The time derivative of \(M_{Rab}(\lambda t^{\alpha})\) through the Laplace transform is  
\[\mathcal{L}\bigg{[}\frac{d}{dt}M_{Rab}(\lambda t^{\alpha})\bigg{]}=s\bigg{(} \frac{1}{s^{\alpha+1}}\frac{1}{1-\frac{\lambda}{s^{\alpha+1}}}\bigg{)}=\frac{1 }{s^{\alpha}}\frac{1}{1-\frac{\lambda}{s^{\alpha+1}}},\quad\left|\frac{ \lambda}{s^{\alpha+1}}\right|<1 \tag{150}\]Now, constructing the heat flux constitutive equation, we get  
\[q_{YAC}(x,t)=-k_{1}\frac{\partial T(x,t)}{\partial x}-k_{2}\int\limits_{0}^{t} \left[\frac{d}{dt}M_{Rab}(\lambda t^{\alpha})\right]\frac{\partial T(x,\tau)}{ \partial x}d\tau \tag{151}\]  
Then, applying the Laplace transform to both sides of (151) and taking into account (149), we have  
\[q_{YAC}(x,s)=-k_{1}\frac{\partial T(x,s)}{\partial x}-k_{2}\int\limits_{0}^{t} \left[\left(\frac{1}{s^{\alpha+1}}\frac{1}{1-\frac{\lambda}{s^{\alpha+1}}} \right)\right]\left[s\frac{\partial T(x,s)}{\partial x}\right]ds \tag{152}\]  
The inverse Laplace transform yields  
\[M_{Rab}(\alpha,\lambda)=t^{\alpha}\sum\limits_{k=0}^{\infty}\frac{\lambda^{k} t^{k(\alpha+1)}}{\Gamma[(k+1)(\alpha+1)]},\quad-1<\alpha<0 \tag{144}\]0  
Consequently, the application of the energy balance equation results in  
\[M_{Rab}(\alpha,\lambda)=t^{\alpha}\sum\limits_{k=0}^{\infty}\frac{\lambda^{k} t^{k(\alpha+1)}}{\Gamma[(k+1)(\alpha+1)]},\quad-1<\alpha<0 \tag{144}\]1  
We may suggest again that \(\lambda=-\frac{\alpha}{1-\alpha}\) and normalization function \(N(\alpha)=1-\alpha\).