*2.3. Interpretation*

We have seen that nonlocality (as considered above) plus Hamilton's principle implies that the equations are not causal, in the sense that the evolution at a given time *t* depends on values from all times and not just from the present (or the past).

It would indicate that if we want evolution equations that involve values just from the past, as in all the heuristic models, we cannot use the framework of the calculus of variations: the motion is no longer along paths that make an action stationary. This is irrelevant for both mathematicians and engineers, but it is irksome for physicists (such a pretty theory. . . ).

But the same can be seen to happen in classical mechanics: as long as the solution is unique for any initial problem at a given time, the solution can be ran backwards or forwards in time. We can express our solution using any time we like: from *t* = 0 we may predict the values for any *t* > 0, but from *t* = *t*max we can give the values for any *t* < *t*max.

But, in that case, we can always use only values from the past to ge<sup>t</sup> our solution while in this framework this is no longer possible.

We may change our idea of causality (at least inside the "ideal" world described by mechanics) and allow that, as long as the solution can be determined for all times, it is irrelevant what values are involved: the behaviour of the system is given. This whenever we can establish existence and unicity of the solution which is, in general, still an open problem for many fractional equations.

#### **3. New Mathematical Scenarios: New Families of Functions and Equations**

As mathematical tool, fractional operators establish important relations between transform integrals and special functions. So, the combined use of integral representations, exponential operational rules and special polynomials provides a powerful tool in the formalism of fractional calculus ([20,21]). Furthermore, fractional operators allow to elude singularities and reduce linear ordinary equations with variable coefficients. As a consequence, an extension of the classical integral representation of the related special functions can be obtained by using fractional operators ([22,23]).

From a applied point of view, fractional calculus offers a modeling scenario where fundamental mathematical questions converge and appropriate numerical algorithms can de developed. For these reasons, fractional calculus has many applications in different areas [24].

The main contribution of the fractional calculus is the consideration of intermediate-order dimensions through integrals and derivatives of arbitrary order ([25,26]). This has allowed to ge<sup>t</sup> a better modeling in different applications, for instance to model biomedical and biological phenomena ([27]). A large number of models considering long-range dependence and systems with memory are constructed with integro-differential and fractional equations.

In classical physics, many fundamental equations are based on similar laws:


As an interpolation of these equations, a fractional approach gives the possibility to look for intermediate or mixed behaviours:

$$F(t) = k \frac{d^\alpha x}{dt^\alpha}(t) \tag{89}$$

Some other contexts are the diffusion processes associated to the basic diffusion equation:

$$\frac{\partial \mu}{\partial t} = \frac{\partial^2 \mu}{\partial x^2} \tag{60}$$

as we show in Table 1:

**Table 1.** Contexts of diffusion.


The diffusion equation can be generalized through the fractional operators that allow to make a natural interpolation among equations, starting with the first order wave equation and ending with the second order wave equation:

$$\text{First order wave equation (hyperbola): } \frac{\partial u}{\partial t} = \frac{\partial u}{\partial x} \tag{91}$$

$$\text{Interpolation: } \frac{\partial u}{\partial t} = \frac{\partial^a u}{\partial x^a} \tag{92}$$

$$\text{Diffusion equation (parabolic): } \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} \tag{93}$$

$$\text{Interpolation: } \frac{\partial^a u}{\partial t^a} = \frac{\partial^2 u}{\partial x^2} \tag{94}$$

$$\text{Wave equation (hyperbola): } \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} \tag{95}$$

Another fractional approach associated to the previous one is the use of Dirac-type fractional equations [28–30].

*A ∂ψ∂t* + *B∂ψ∂x* = 0 *ψ* = *ϕξ A ∂αψ ∂tα* + *B∂ψ∂x* = 0 --*A ∂*1/2*ψ ∂t*1/2 + *B∂ψ∂x* = 0 *∂*2*u ∂t*<sup>2</sup> − *∂*2*u ∂x*<sup>2</sup> = 0 Pauli *A*<sup>2</sup> = *I* algebra *B*<sup>2</sup> = *I* {*<sup>A</sup>*, *B*} = 0 *γ* = 2*α ∂γu ∂tγ* − *∂*2*u ∂x*<sup>2</sup> = 0 -- *∂u ∂t* − *∂*2*u ∂x*<sup>2</sup> = 0 

In this way, the following equation,

$$A\frac{\partial^{1/2}\Psi}{\partial t^{1/2}} + B\frac{\partial\Psi}{\partial x} = 0,\tag{96}$$

describes two coupled diffusion processes or a diffusion process with internal degrees of freedom. Depending on the chosen representation of the Pauli algebra, that *A* and *B* must verify, we obtain a system of equations coupled or decoupled:

$$A\_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad B\_1 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \quad \implies \quad \left\{ \begin{array}{ll} \partial\_t^a \varphi = \varphi \\ \partial\_t^a \tilde{\varphi} = -\tilde{\varphi} \end{array} \right. \tag{97}$$

$$A\_2 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \quad B\_2 = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \quad \implies \quad \left\{ \begin{array}{ll} \partial\_t^u \varphi = \xi \\ \partial\_t^u \tilde{\xi} = \varphi \end{array} \right. \tag{98}$$

$$A\frac{\partial^{\alpha}\psi}{\partial t^{\alpha}} + B\frac{\partial\psi}{\partial x} = 0 \xrightarrow{\gamma = 2\alpha} \frac{\partial^{\gamma}u}{\partial t^{\gamma}} - \frac{\partial^{2}u}{\partial x^{2}} = 0$$

In the study of the temporal inversion (*t* → −*<sup>t</sup>*), we have the invariance of the fractional Dirac equation for the values 0 < *α* < 1:

> *α*= 1 3, 2 3, 1 5, 2 5, 3 5, 4 5, 1 7, 2 7,..., 6 7, 1 9,...

Some other fractional differential equations are obtained by considering the root 1/3 of both the wave and diffusion equations:

$$\text{Wave equation: } P\partial\_t^{2/3}\varphi + Q\partial\_x^{2/3}\varphi = 0 \tag{99}$$

$$\text{Diffusion equation: } P\partial\_t^{1/3}\varphi + Q\partial\_x^{2/3}\varphi = 0 \tag{100}$$

where

$$P^3 = I \qquad Q^3 = -I \qquad PPQ + PQP + QPP = 0 \qquad QQP + QPQ + PQQ = 0 \tag{101}$$

A possible realization is in terms of the 3 × 3 matrices associated to the Silvester algebra:

$$P = \begin{pmatrix} 0 & 0 & 1 \\ \omega^2 & 0 & 0 \\ 0 & \omega & 0 \end{pmatrix}, \quad Q = \Omega \begin{pmatrix} 0 & 0 & 1 \\ \omega & 0 & 0 \\ 0 & \omega^2 & 0 \end{pmatrix}, \tag{102}$$

with *ω* a cubic root of the unity and Ω a cubic root of the negative unity. In this case, *ϕ* has three components.

#### **4. Nonlocal Phenomena in Space and/or Time. Applications**

We use the term non-locality if what happens in a spatial point or at a given time depends on an average over an interval that contains that value. Thus, the non-local effects in space correspond to long-range interactions (many spatial scales), while the non-local effects in time suppose memory or delay effects (many temporal scales).

These phenomena are associated to integral or integro-differential equations, which appear in multiple contexts:


These different phenomena can be described by fractional differential equations, and it sets out two fundamental questions:


As example to answer the first question, it is interesting to remark that, for instance, the fractional diffusion equation with some kind of time fractional derivative,

$$\text{Interpolation: } \frac{\partial^a u}{\partial t^a} = \frac{\partial^2 u}{\partial \mathbf{x}^2}, \tag{103}$$

verifies the second law of thermodynamics only if the following the generalized Fourier law is satisfied:

$$\frac{\partial^{\alpha-1}\mu}{\partial x^{\alpha-1}}\frac{\partial \mu}{\partial x} > 0. \tag{104}$$

Not all fractional operators satisfy the condition aforementioned, so it might be the key to choose the convenient fractional derivative or to apply restrictions to the initial and boundary conditions of the problem. Let us define *ρ* = *α* − 1. When *ρ* = 1, the condition Equation (104) is trivial; but when 0 < *ρ* < 1, this issue is a complex problem with different solutions according to the selected fractional operator and conditions [31].

#### *4.1. Application of Fractional Calculus to Model Atmospheric Effects of Absorption*

The time-fractional Cauchy problem is well-known:

$$
\lambda\_0^C D\_t^a u(t, \mathbf{x}) - \lambda\_+^L D\_x^\beta u(t, \mathbf{x}) = 0, \quad t > 0, \mathbf{x} \in \mathbb{R}, 0 < a \le 1, \emptyset > 0 \tag{105}
$$

$$\lim\_{x \to \pm \infty} u(t, x) = 0, \quad u(0+, x) = \operatorname{g}(x). \tag{106}$$

and its solution in the space of functions with Laplace and Fourier transforms, *LF* = *<sup>L</sup>*(*R*+)x*<sup>F</sup>*(*R*), is defined by

$$u(t,x) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} G(k) E\_{\mathfrak{a}}(\lambda(-ik)^{\mathfrak{f}}t^{\mathfrak{a}}) e^{-ikx} dk,\tag{107}$$

where the Mittag-Leffler function is evaluated on the complex plane and *G*(*k*) represents the Fourier transform of *g*(*x*).

For instance, for *β* = 1 and *g*(*x*) = *<sup>e</sup>*<sup>−</sup>*μ*|*x*|, *μ* > 0:

$$u(t, \mathbf{x}) = e^{-\mu|\mathbf{x}|} E\_{\mathbf{a}}(-\mu\lambda t^{\mathbf{a}}) \tag{108}$$

The fundamental solution of the problem is obtained for *g*(*x*) = *<sup>δ</sup>*(*x*), *G*(*k*) = 1 and, in this case, the moments for *β* = 1 have the following expression:

$$\mathbf{x} < \mathbf{x}^n > = \int\_{-\infty}^{\infty} \mathbf{x}^n \boldsymbol{u}(t, \mathbf{x}) d\mathbf{x} = (-\lambda t^n)^n \frac{\Gamma(n+1)}{\Gamma(an+1)}, \qquad n = 0, 1, 2, \dots \tag{109}$$

When replacing *t* by the wave-length of the radiation, *λ*, the moment for *n* = 1 and an appropriate constant *β* returns the Angstrom law,

$$
\pi = \frac{\beta}{\lambda^{a\_{\prime}}} \tag{110}
$$

that is used to model the coefficient of molecular scattering *τ* for the absorption of the incoming energy inside the Martian atmosphere due to the dust [32,33]. The parameters *α* and *β* would be fixed in function of the Martian dust features. So, this relation shows a possible application of fractional calculus to model the dynamic of the Martian atmosphere. Deep studies, by using cloud computing, on this issue have been developed in [34–36].

In the context of the spatial exploration, other new original application of the fractional calculus analysis is the prediction and identification of dust devils and correlations between wind and seismic signals in a Martian meteorological payload packet. We recently started this project on the basis of our previous experience in the missions to Mars and to apply in ExoMars22 (initially ExoMars20 but now delayed due to the Covid-19) [37–40].

#### *4.2. Chaos in a Fractional Duffing'S Equation*

Duffing's equation has been a model for many studies on chaotic systems. It considers a simple but complex system where chaos can appear depending on the values of the parameters. The mathematical model is a time-forced, dissipative, second order nonlinear differential equation that can be viewed as a perturbed Hamiltonian or Lagrangian system [41]. Different possible potentials can be considered but the fundamental equation corresponds to a model for a long and slender vibrating beam set between two permanent magnets, subjected to an external sinusoidal force.

Duffing's equation shows many paradigmatic features of chaotic systems, in a somewhat simple frame, in the Theory of Dynamical Systems. The fractional counterpart we have chosen may possess similar relevant behaviours of other more general fractional models. This is the basis for this study. Besides, although the presence of chaotic behaviour in fractional Duffing's equations has been documented, many questions remain open.

It is possible to extend Duffing's equation into a fractional one in many ways, either as a second order differential equation, or as a system of simultaneous two first order equations, with some or all of these derivatives replaced by fractional ones. Different authors have, thus, considered different fractional equations, playing also with the fractional order of derivation [42,43]. In our case, we have chosen to replace only the first order derivative by a fractional Caputo derivative. The equation we consider is

$$
\ddot{\mathbf{x}} + \gamma D\_t^\mathbf{u} \mathbf{x} - \mathbf{x} + \mathbf{x}^3 = f\_0 \cos(\omega t),
\tag{111}
$$

where *α* ∈ (0, 1) and *Dα t x* stands for the Caputo fractional derivative with lower limit 0 of order *α*. This equation has the advantage of a regular solution (at least *C*2) whose existence can be ensured [44]. This is not merely a mathematical model since it can be viewed as the same mechanical device represented by Duffing's equation but immersed in a viscous medium. For some values of the parameters, a strange attractor is obtained, quite similar to the one that appears in the classical (i.e., non fractional) Duffing's equation.

We have studied the controlability of the chaotic regime in the presence of both harmonic and nonharmonic external perturbations, considering geometrical resonances for the second case. Using resonant Jacobi functions, we obtain conditions for external, additional, drivings that ensure chaos-free responses in our model [44].

We have also characterized the chaotic behaviour in our fractional model computing the maximum but, also, all the other Lyapunov characteristic exponents. We have used, as a reference, the fiduciary orbit technique and we have built a perturbative approach with a local equation that allows to estimate all the exponents [45].

The results show that a chaotic regime exists for the fractional Duffing's equation but also a regular regime with a long transient time. This regular regime, in practice, can be assimilated in many cases to a chaotic regime for quite long transient times, although the solutions for even longer times tend to regular, almost-periodic limiting curves.
