## 2 On the Fractionalisation Problem  
Fractional systems can be considered as the 21 st century models for many natural and man-made phenomena [9]. However, they have already been used in many real applications with phenomena that require modelling beyond traditional tools. The well-known Curie--von Schweidler law, discovered in the 19 th century, is a good example of a fractional behaviour modelled using fractional calculus (FC) [10; 11]. [PERSON] made some contributions to the application of FC in electromagnetics [12]; very curious is [PERSON]'s use of FC in the discussion about the age of the earth [13]. In the forties of the 20 th century, [PERSON] introduced FC in rheology and viscoelasticity [14; 15]. More recently, the Schrodinger fractional equation was studied [16; 17; 18; 19], as well as the fractional Maxwell equations [20; 21; 22]. Other applications in physics can be found in [20; 23; 24; 25]. In fact, we are currently dealing with fractional phenomena such as \(1/f\) noise, long-range dependence, fractional Gaussian noise, and fractional Brownian motion (fBm), which are ubiquitous in the scientific literature [26; 27; 28]. This raises a question: why use a fractional derivative? A first answer is obvious: we have another parameter to manipulate in order to get a better fit to the solution of a given problem. However, there may be another important reason: causality. In fact, we can enforce causality through an appropriate choice of the used derivatives. This reason is often overlooked. A reading of the cited references shows a great diversity regarding the use of fractional operators, often leading to results that do not enjoy backward compatibility with classical theories. This is often the case in connection with the use of Riemann-Liouville (RL) or Caputo (C) derivatives. However, instead of replacing the derivative (D) with an appropriate one, this non-compatibility is accepted as a fatality of fractional calculus. The designation \"metaphysical derivatives\" has been coined [29]. In fact, these derivatives, which enjoy privilege among mathematicians, create several problems in scientific applications, by:  
* not distinguishing between constant and Heaviside functions;
* not giving the derivative of the \(\delta(t)\);
* the derivative of a sinusoid is not a sinusoid;
* non-validity of the index rule \(D^{\kappa}D^{\delta}\
eq D^{\alpha+\beta}\) for \(\alpha,\beta>0\)[30; 31; 32];
* together with the one-sided Laplace transform, introducing wrong initial-conditions (IC) [33; 34; 35; 36]. These problems have been addressed and solved in a step-by-step deduction of the formulae from classical derivative definitions [37; 38]. This approach makes clear the existence of different derivatives for (time) causal or (space) non-causal systems. The classic derivative suitable for defining (1) must be given by: \[Dx(t)=x^{\prime}(t)=\lim\limits_{h\to 0}\frac{x(t)-x(t-h)}{h},\] (2) since it uses the present and past values of \(x(t)\) (causality). This derivative has to be generalised, so that we are able to define a fractional Ambartsumian equation, given by \[D^{\alpha}x(t)+x(t)=\frac{1}{q}x\bigg{(}\frac{t}{q}\bigg{)}\quad t\geq 0,\ q \geq 1,\] (3) under a suitable initial condition (IC) \(x_{0}=x(0)\). Let the function \(x(t)\) to be defined on \(\mathbf{R}\) and \(\alpha\) be any real number (the complex values lead to non-hermitian systems [39]). The above derivative can be generalized for any order by: \[D^{\alpha}x(t)=\lim\limits_{h\to 0^{+}}\frac{\sum\limits_{n=0}^{\infty}\frac{(- \alpha)_{n}}{n!}x(t-nh)}{h^{\alpha}},\] (4) where \(\frac{(-\alpha)_{n}}{n! },\ n=0,1,\cdots\) represent the binomial coefficients [37] with \((a)_{k}=a(a+1)(a+2)\dots(a+k-1)\) denoting the Pochammer symbol for the raising factorial. This derivativeis called \"Grunwald-Letnikov\" (GL), although it was proposed first by [PERSON].