## 3 Fading Memory Approach (Formalism) and Concept of Model Development
### Model Build-Up Concept  
Let us take a look at the flux definition of (67) where the relaxation (memory) function can be presented as \(M(t)=\delta(t)+R(t)\) such that (67) expressed as a convolution product  
\[j(x,t)=\left[-\frac{\partial C(x,t)}{\partial x}\right]*M(t)=\left[-\frac{ \partial C(x,t)}{\partial x}\right]*\left[A_{1}\delta(t)+A_{2}R(t)\right] \tag{72}\]  
The two main requirements for \(R(t)\) are: it _should be completely monotone_, regardless of the fact that it is singular or non-singular, and that _its first derivative_\(dR(t)/dt~{}=~{}R_{t}(t)\), should also be completely monotone.  
The two main requirements for \(R(t)\) are: it _should be completely monotone_, regardless of whether it is singular or non-singular at \(t0+\), and that _its first derivatived_\(R(t)/dt~{}=~{}R_{t}(t)\) should also be a completely monotone function.  
Then, the main step in the construction of the flux constitutive model is the use \(R_{t}(t)\) instead \(R(t)\), that is  
\[j(x,t)=\left[-\frac{\partial C(x,t)}{\partial x}\right]*M(t)=\left[-\frac{ \partial C(x,t)}{\partial x}\right]*\left[A_{1}\delta(t)+A_{2}R_{t}(t)\right] \tag{73}\]  
Let us consider the convolution term alone and use integration by parts to clarify things.  
\[\begin{split}&\int\limits_{-\infty}^{t}R_{t}(t-\tau)\frac{ \partial C(x,\tau)}{\partial x}d\tau=\int\limits_{-\infty}^{t}\frac{\partial C (x,t)}{\partial x}dR(t-\tau)=\\ &=\frac{\partial C(x,\tau)}{\partial x}dR(t-\tau)\bigg{|}_{- \infty}^{t}-\int\limits_{-\infty}^{t}R(t-\tau)\frac{d}{d\tau}\bigg{[}\frac{ \partial C(x,\tau)}{\partial x}\bigg{]}\end{split} \tag{74}\]  
The term \(=\frac{\partial C(x,t)}{\partial x}dR(t-\tau)\bigg{|}_{-\infty}^{t}\) is zero when we have: \(R(t)\) is a _causal function_ and the _initial conditions corresponding to a virgin material_ (medium) are \(C(x,0)=C(0,0)=C(\infty,t)=C(-\infty,t)\), \(C_{x}(x,0)=C_{xx}(x,0)=0\) (signaling problem). As a result, we get \[\int\limits_{-\infty}^{t}R_{t}(t-\tau)\frac{\partial C(x,\tau)}{\partial x}d\tau=- \int\limits_{-\infty}^{t}R(t-\tau)\frac{d}{d\tau}\bigg{[}\frac{\partial C(x,\tau )}{\partial x}\bigg{]}d\tau \tag{75}\]  
Denoting \(\frac{\partial}{\partial x}C(x,t)=f(x,t)\) and setting the lower terminal at zero, we get a constriction of Caputo-type fractional derivative that is  
\[{}^{C}D_{t}^{\alpha}f(x,t)=\frac{M(\alpha)}{N(\alpha)}\int\limits_{0}^{t}R(t- \tau)\frac{df(x,\tau)}{d\tau}d\tau \tag{76}\]  
where the pre-factor \(M(\alpha)/N(\alpha)\) that only takes into account fractional order, but it is determined by the type of memory kernel (see Section 2.2 and the developments in the sequel).  
The flux can therefore be expressed as  
\[j(x,t)=-A_{1}\frac{\partial C(x,t)}{\partial x}-A_{2}{}^{C}D_{t}^{\alpha} \bigg{[}\frac{\partial C(x,t)}{\partial x}\bigg{]} \tag{77}\]  
In this case, the hereditary term is generically expressed as a fractional derivative of the Caputo type without any kernel-specific information. Then, applying the continuity equation \(\frac{\partial C}{\partial t}=-\frac{\partial q}{\partial x}\) we get  
\[\frac{\partial C(x,t)}{\partial t}=A_{1}\frac{\partial C(x,t)}{\partial x}+A_ {2}{}^{C}D_{t}^{\alpha}\bigg{[}\frac{\partial C(x,t)}{\partial x}\bigg{]} \tag{78}\]  
For \(\alpha=1\), this model reduces to the local diffusion equation.  
**Remark 6**.: _Specific comment is required regarding the following exposition before moving on to the modeling method. Assuming that the kernel of the most straightforward scenario is built first, the model build-ups are demonstrated inductively. The model build-ups are illustrated deductively, which suggests that the kernel of the most straightforward scenario is constructed first. When the same approach is used to handle ever-more complex kernels, the resulting fractional operator structures gradually become more complex. The approach taken here is more instructive than the one frequently used in the literature, which starts with the most complex kernel and gradually creates simpler kernels by lowering the complexity parameters._