## 6 Heat Conduction Model Analysis
### Heat Conduction and Diffusion Models with Mittag-Leffler Kernels: An Analysis  
[PERSON]'s Non-Linear Heat Conduction Model  
In the general context of this study, we address a model of a non-linear heat equation formulated as (section 4 in [43])  
\[D_{\alpha,\beta,\lambda}^{\gamma}T(x,t)=K(T)\frac{\partial^{2}T(x,t)}{ \partial x^{2}}-\beta T,\quad K(T)=k_{0}T^{\xi},\quad\xi>0,\quad t>0,\quad x\in\mathbb{R} \tag{168}\]  
The derivative \(D_{\alpha,\beta,\lambda}^{\gamma}\) was applied in two versions: a non-regularized (Riemann-Liouville type) derivative (169), and a regularized (Caputo-type) derivative (170), defined as (in the original notations established in [101]) (see also the definitions in Section 2.2.2),  
\[(_{0}D_{t}^{\alpha}+\lambda)^{\gamma}f(t)\equiv\frac{d^{m}}{dt^{m}}\int\limits _{0}^{t}e_{\alpha,\tau\gamma}^{\gamma}(t-u;-\lambda)f(u)du \tag{169}\]  
\[{}^{C}(_{0}D_{t}^{\alpha}+\lambda)^{\gamma}f(t)\equiv\int\limits_{0}^{t}e_{ \alpha,\tau\gamma}^{\gamma}(t-u;-\lambda)f^{(m)}(u)du \tag{170}\]  
with a Prabhakar integral (see also the definition (45))  
\[(_{0}I_{t}^{\alpha}+\lambda)^{\gamma}f(t)\equiv e_{\alpha,\tau\gamma}^{\gamma} (t;\lambda)*f(t)=\int\limits_{0}^{t}e_{\alpha,\tau\gamma}^{\gamma}(t-u;- \lambda)f(u)du \tag{171}\]In general, this type of non-linearity in the diffusion term, when the equation is local in time, makes the heat conduction model a degenerate parabolic equation. It is well-known, since the first solutions in the 50s of the last century, that such degenerate parabolic problems have solutions moving with finite speeds, in contrast to the case with \(\zeta=0\) where the flux speed is infinite (see [102] and the reference therein). Now, for the sake of the coherence of the exposition, we return to the common symbol used here (let the definitions (169)-(171) remain, demonstrating that different versions of the symbols used are also possible; in general, there are various notations of these operators, so the reader should carefully understand the meaning in each particular case). If we like to construct (derive) such a heat conduction model, we may define the flux-gradient relationship as  
\[ \begin{split}& q(x,t)=-\int\limits_{0}^{t}e_{\alpha,\beta}^{ \gamma}(\lambda,t-\tau)\bigg{[}k_{0}T^{\zeta}\frac{\partial T(x,t)}{\partial x }\bigg{]}=\\ &=-\frac{k_{0}}{1+\zeta}\int\limits_{0}^{t}e_{\alpha,\beta}^{ \gamma}(\lambda,t-\tau)\bigg{[}\frac{\partial T^{\zeta+1}(x,t)}{\partial x} \bigg{]}\end{split}  \tag{172}\]  
If the energy balance equation is (1) is applied (neglecting the sink term for the sake of simplicity), then the heat conduction model would be  
\[\frac{\partial T(x,t)}{\partial t}=\frac{k_{0}}{1+\xi}\int\limits_{0}^{t}e_{ \alpha,\beta}^{\gamma}(\lambda,t-\tau)\bigg{[}\frac{\partial^{2}T^{\zeta+1}( x,\tau)}{\partial x^{2}}\bigg{]} \tag{173}\]  
or in a more compact form as  
\[\frac{\partial T(x,t)}{\partial t}=\frac{k_{0}}{1+\xi}{}^{p}I_{\alpha,\beta}^ {\gamma}\bigg{[}\frac{\partial^{2}T^{\zeta+1}(x,t)}{\partial x^{2}}\bigg{]} \tag{174}\]  
In the local case when memory is \(\delta(t)\) it will be  
\[\frac{\partial T(x,t)}{\partial t}=\frac{k_{0}}{1+\xi}\frac{\partial^{2}T^{ \zeta+1}(x,t)}{\partial x^{2}} \tag{175}\]  
and the solutions are mainly approximate (see [102]) with convex temperature profiles (with almost infinite gradient close to the solution front for high values of \(\zeta\)), in contrast to the case for \(\zeta=0\) with concave temperature distributions. Now, the principle question is: how to transform (173) to (168)? Taking into account that the regularized Caputo-type derivative \({}^{C}D_{\alpha,\beta,\lambda}^{\gamma}\) acts as left-inverse of the Prabhakar fractional integral [41; 49], as well as that \({}^{C}D_{\alpha,\beta,\lambda}^{\gamma}f(t)={}^{RLC}D_{\alpha,\beta,\lambda}^ {\gamma}f(t)\) at zero initial conditions (see eq. 5.11 in [41]), we may apply \({}^{C}D_{\alpha,\beta,\lambda}^{\gamma}\) to both sides of (173) or (174).