## 1 Introduction
### Heat Conduction as a Transport Process with Memory
#### 1.2.2 Constitutive Heat Flux with Linear Memory [22]  
Following [PERSON] and [PERSON] [22], the internal energy \(e(x,t)\) and heat flux \(q(x,t)\), and following the fading memory principle [18, 19, 20, 21] are described by the following linearized constitutive equations (with notations used in this work)  
\[e(x,t)=e_{0}+C_{p}\theta(x,t)+\int\limits_{0}^{\infty}R_{e}(s)\theta(t-s)ds \tag{9}\]  
\[q(x,t)=-k_{0}\frac{\partial T(x,t)}{\partial x}-\int\limits_{0}^{\infty}R_{q}( s)\theta(t-s)ds \tag{10}\]  
\(\theta(x,t)=T(x,t)-T_{0}(x,t),\quad T_{0}(x,t)=T(x,0)\) is temperature variation field. Further, \(R_{e}(t)\) and \(R_{q}(t)\) are relaxation functions (fading memories), and following the arguments of [PERSON] [21], the fading memory principle is equivalent to \(R_{e}(t),R_{q}(t)\in L^{1}(\mathbb{R}^{+})\) and \(R_{q}(t)\). In this context, the heat capacity \(C_{p}(t)\), and thermal conductivity \(k(t)\) are defined as [22]  
\[C_{p}(t)=C_{p0}+\int\limits_{0}^{t}R_{e}(s)ds,\quad k(t)=k_{0}+\int\limits_{0} ^{t}R_{q}(s)ds,\quad C_{p0}>0 \tag{11}\]  
and \(C_{0}=\lim_{t\to\infty}C_{p}(t)\), and \(k_{0}=\lim_{t\to\infty}k(t)\) are _equilibrium heat capacity_, and the _equilibrium thermal conductivity_, that is when the memory integrals in (11) are vanishing. At this point, we will discontinue the discussion of the thermodynamic principles of fading memory (it will be revisited in Section 3 devoted to model development with fractional operators), but it is critical to emphasize two principles in the constitutive model formulation:1. The memory integral is an additive part (in the cases discussed here, a linear contribution) to the constitutive heat flux model. This gives us a great opportunity to apply fractional modeling through the use of adequately formulated relaxation functions (memory kernels), as we will do in the sequel to this article. 2. The energy balance equation remains unchanged since the First Law of Thermodynamics holds. This directly means that the time derivative of internal energy cannot be mechanistically replaced by fractional operators. We can see that we have two versions of constitutive heat flux formulations, but the energy balance equation remains unmodified since this is the First Law of Thermodynamics. Any changes in the balance equation, which is a very common step in fractional calculus publications, caused by replacing the time derivatives with fractional counterparts are violations of the First Law of Thermodynamics. The main task of mathematical modeling, as well as fractional modeling, is to solve correctly constructed models with physically interpretable results, which are often found in formalistic fractional models obtained by replacement. Before starting with the development of fractional operators with various Mittag-Leffler kernels, we have to say that there is no mathematics only for mathematics in this study. Fractional calculus is applied mathematics invoked by reality since the non-localities are natural phenomena in the transport processes. Because of that, the following exposition stresses the attention to the physical background of the modeled problems and their adequate interpretations. The main task of this article is threefold:  
1. To start the introduction of correctly formulated models of heat conductors with memory (actually, these are diffusion models, not diffusion models commonly modeled in heterogeneous media by application of fractional derivatives), applying the fading memory formalism. 2. To introduce different fractional operators in a logical way by using different relaxation functions (completely monotone Mittag-Leffler functions) and how these operators are related. From our point of view, this approach is of high importance for fractional modeling, where there are too many operators constituted without any physical background and then incorrectly applied in the modeling of real-world problems. 3.