## 3 Fading Memory Approach (Formalism) and Concept of Model Development  
### Fading Memory Concept  
The Boltzmann linear superposition functional [16] for hereditary viscoelasticity with a time-dependent memory kernel (correlation function) \(R(t,\tau)\) is  
\[\varphi(x,t)=A_{1}[v_{x}(x,t)]+A_{2}\int\limits_{0}^{t}R(x,t-\tau)\,v_{x}( \tau)d\tau \tag{66}\]  
The fading memory concept relating the flux to its gradient, for simple materials [67; 68; 69] is modeled by the following integrodifferential equation  
\[j(x,t)=-A_{1}\frac{\partial C(x,t)}{\partial x}-A_{2}\int\limits_{-\infty}^{t }R(t-\tau)\frac{\partial C(x,\tau)}{\partial x}d\tau \tag{67}\]  
as a manifestation of the Boltzmann linear superposition functional 66). In (67) the transport coefficients \(A_{1}\) and \(A_{2}\) are diffusivities. The history value problem for (67) addresses the following integral [69]  
\[d(t)=\int\limits_{-\infty}^{0}R(t-\tau)\frac{\partial C(x,\tau)}{\partial x}d\tau \tag{68}\]  
allowing to give a function \(C(x,t)\) on \((-\infty<t\leq 0)\) From (67) and (68) it follows that  
\[\frac{\partial}{\partial x}j(x,t)=-A_{1}\frac{\partial^{2}C(x,t)}{\partial x^ {2}}-A_{2}\int\limits_{0}^{t}R(t-\tau)\frac{\partial^{2}C(x,\tau)}{\partial x ^{2}}d\tau+\frac{\partial}{\partial x}d(t) \tag{69}\]  
Since \(C(x,t)\) is a causal function (vanishing for \(t<0\)) and considered only for \(0<t<\infty\) we accept \(d(t)=0\Rightarrow\frac{\partial}{\partial x}d(t)=0\), and therefore (69) can be rewritten as  
\[\frac{\partial}{\partial x}j(x,t)=-A_{1}\frac{\partial C(x,t)}{\partial x}-A_ {2}\int\limits_{0}^{t}R(t-\tau)\frac{\partial C(x,\tau)}{\partial x}d\tau \tag{70}\]  
Reducing (69) and setting the lower terminal of the memory integral to zero have deep physical meaning when applying hereditary integrals. The essence is well expressed by [PERSON] in [70] that in fractional operators, the time _is not the chronological time_ (_instant time_) but _the intrinsic time_ of the process (the _time of duration_), _starting at the point accepted as \(t=0\)_. Hence, in the context of the reduction of (69) to (70), _there is no process before \(t=0\)_.  
The deep thermodynamic sense of the fading memory formulation is that the non-locality represented by the convolution term works for short times, while for long times, we get the first term in (70), i.e., the instant reaction of the system. Moreover, models constructed with the fading memory principle obey: the _causality principle_ (through the convolution term) [26; 27; 28], thermodynamic consistency [71; 72], and _model observability_ (objectivity) [73; 74; 75].  
Further, if \(A_{1}=0\), that is, the process modeled has no long-time asymptotically stable state, the result is a model close to the Continuum Time Random Walk (CTRW) without a stationary state [76; 77], and applying the continuity equation to such flux, we get  
\[\frac{\partial C(x,t)}{\partial t}=\frac{\partial}{\partial x}j(x,t)\Rightarrow \frac{\partial C(x,t)}{\partial t}=A_{2}\int\limits_{0}^{t}R(t-\tau)\frac{ \partial C(x,\tau)}{\partial x}d\tau \tag{71}\]  
To a greater extent, the model (71) mimics the master equation [77; 78]; this is only a remark, and we will not elaborate on it further in this study but refer readers to [79; 80; 81] where this issue was developed in detail regarding Prabhakar-type memory.  
**Remark 5**: _Fading memory concept in the case of heat conduction is well applicable to the so-called simple materials [21; 82] where the flux is proportional to the temperature gradient (the term was coined after the work of [PERSON] [82]), as in all examples studied here._