# Algorithms for Fractional Dynamical Behaviors Modelling Using Non-Singular Rational Kernels
## 2 Approximation of a Pure Power Law Behavior by Non-Singular Rational Kernels  
This section shows the efficiency of non-singular rational kernels for the modelling of fractional behaviors. The fractional behavior considered is first the pure power law function:  
\[\eta_{\
u}(t)=Kt^{-\
u} \tag{1}\]  
This function indeed falls within the definition of a fractional integration order operator defined in the Riemann-Liouville sense by [15; 16]  
\[I^{\gamma}[u(t)]=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}\frac{1}{(t-\tau)^{1- \gamma}}u(\tau)d\tau=\eta_{PL}(t)*u(t), \tag{2}\]  
with \(\eta_{PL}(t)=t^{\gamma-1}/\Gamma(\gamma).\) The idea is to approximate the kernel \(\eta_{PL}(t)\) by a non-singular rational kernel \(\eta(t)\), to avoid the limitations cited in the introduction, while allowing the model  
\[y(t)=\int_{0}^{t}\eta(t-\tau)u(\tau)d\tau=\eta(t)*u(t), \tag{3}\]  
to capture fractional behaviors. For the approximation of \(\eta_{\
u}(t)\), the following rational kernel is considered:  
\[\eta(t)=C_{0}\frac{\prod_{j=1}^{N}\left(\frac{t}{t_{j}^{j}}+1\right)}{\prod_{ j=1}^{N}\left(\frac{t}{t_{j}}+1\right)}. \tag{4}\]  
The decimal logarithm of \(\eta(t)\) is  
\[\log(\eta(t))=\log(C_{0})+\sum_{j=1}^{N}\log(\frac{t}{t_{j}^{j}}+1)-\sum_{j=1 }^{N}\log(\frac{t}{t_{j}}+1) \tag{5}\]Decimal logarithm applied to the function \(\eta_{v}(t)\) gives:  
\[\log(\eta_{v}(t))=\log(K)-\
u\;\log(t). \tag{6}\]  
At \(t=1\), from (1),  
\[\eta_{v}(1)=K \tag{7}\]  
and using relation (4),  
\[\eta(1)=C_{0}\frac{\prod_{j=1}^{N}\left(\frac{1}{t_{j}}+1\right)}{\prod_{j=1}^{ N}\left(\frac{1}{t_{j}}+1\right)}. \tag{8}\]  
Thus, to ensure that \(\eta(1)=\eta_{v}(1)\), it is imposed that  
\[C_{0}=K\frac{\prod_{j=1}^{N}\left(\frac{1}{t_{j}}+1\right)}{\prod_{j=1}^{N} \left(\frac{1}{t_{j}}+1\right)}. \tag{9}\]  
The functions \(\eta_{z}(t)=\log\left(\frac{t}{t_{j}}+1\right)\) and \(\eta_{p}(t)=-\log\left(\frac{t}{t_{j}}+1\right)\) of relation (5) are represented in Figure 1 for \(t_{j}^{\prime}=t_{j}=1\), as a function of \(\log(t)\). For \(t>\left(t_{j}^{\prime}=1\right)\) and \(t>\left(t_{j}=1\right)\), the functions \(\eta_{z}(t)\) and \(\eta_{p}(t)\) behave, respectively, as lines of slope \(1\) and \(-1\). The function given by relation (2) has a slope of \(-\
u\) on a logarithmic scale. Thus, taking inspiration from the work which aimed to approximate fractional integrator operators in the frequency domain [27; 28; 29; 30; 31; 32], and adapting them to the time domain, the function \(\eta_{v}(t)\) can be approximated by choosing the appropriate values of \(t_{j}^{\prime}\) and \(t_{j}\) in an alternation of function \(\eta_{p}(t)\) and \(\eta_{z}(t)\). Figure 2 describes how the functions \(\eta_{z}(t)\) and \(\eta_{p}(t)\) are used to produce the desired slope. On this figure, it is assumed that an approximation of relatio  
Based on this analysis we can say that  
\[t^{-V}=\underset{N\rightarrow\infty}{\lim}\underset{j=-N}{\prod}\frac{\left( \frac{t}{t_{j}^{\prime}}+1\right)}{\left(\frac{t}{t_{j}^{\prime}}+1\right)} \tag{16}\]  
in which the poles and zeros are linked by relations (10) to (12). It must be noticed that the kernel \(\eta_{\
u}(t)\) is singular at time \(t=0\). With the method we propose, we can get as close as necessary to 0 with the approximation method developed as it is done on a defined time interval. The accuracy of the approximation then depends on the number of poles and zeros used in the approximation. But the problem is not whether it is possible or not to fit a pure power law, because in practice it is not necessary to do so. As mentioned in the first paragraph of the introduction, if a pure power law is used in a convolution product as in relation (3), the resulting model has infinitely large and infinitely small time constants. In modelling context, these infinitely large and infinitely small time constants are not required to capture the behaviour of a real system that operates on finite spatial and time scales.