Fractional Calculus in Physics: A Brief Review of Fundamental Formalisms

Abstract

Fractional calculus provides powerful tools for modeling nonlocality, dissipative systems, and, when defined in the time representation, provides an interesting memory effect in mathematical physics. In this paper, we review four standard fractional approaches: the Riemann–Liouville, Gerasimov–Caputo, Grünwald–Letnikov, and Riesz formulations. We present their definitions, basic properties, Weyl–Marchaud, and physical interpretations. We also give a brief review of related operators that have been used recently in applications but have received less attention in the physical literature: the fractional Laplacian, conformable derivatives, and the Fractional Action-Like Variational Approach (FALVA) for variational principles with fractional action weights. Our emphasis is on how these operators are, and can be, applied in physical problems rather than on exhaustive coverage of the field. This review is intended as an accessible introduction for physicists working in diverse areas interested in fractional calculus and fractional methods. For deeper technical or domain-specific treatments, readers are encouraged to consult the works in the corresponding fields, for which the bibliography suggests a starting point.

1. Introduction

Recently, fractional calculus which represents a generalization of classical calculus to noninteger orders, has emerged as an important field of mathematics with applications in the study and modelling of complex physical systems. Fractional calculus is able to accommodate more naturally a range of phenomena which the standard calculus handle with more difficulty, such as memory effects, nonlocal effects, anomalous diffusion, and so on. Although its origins can be traced back to a 1695 correspondence between Leibniz and L’Hôpital [1], it is only in recent decades that fractional calculus has increasingly attracted interest in physics, engineering, biology, economy, and applied mathematics. The original question posed by L’Hôpital: “What if $n={\displaystyle \frac{1}{2}}$?” in reference to the notation ${\displaystyle \frac{d^{n}f}{d{x}^n}}$, has evolved into an exciting field of mathematics that has proved to be important to our understanding of several natural and social phenomena in general, and of physical systems across various scales, in particular [2].

The main mathematical motivation for fractional calculus resides in the intrinsic nonlocality of the fractional derivatives that mix the differentiation and integration operations with a power-law kernel that “remembers” the entire past, and sometimes the future, of a process. This feature makes fractional derivatives suitable for describing anomalous diffusion, long-range correlations, nontrivial power-law behavior, memory effects, and fractal scaling features that integer-order models cannot properly capture [2,3,4]. This nonlocal character enables fractional calculus to connect physical models of microscopic interactions and macroscopic emergent systems in a more complex way than standard calculus.

In physics, fractional calculus has found recent applications across several fundamental fields, including classical mechanics, electromagnetism, quantum mechanics, gravity, cosmology, quantum field theory, and applied physics. The construction of models within the fractional formalisms has helped to better understand systems with fractional dimensionality, disordered materials, and processes exhibiting anomalous diffusion or relaxation. The increasing interest in fractional calculus as a tool for physics is also reflected in the increasing number of specialized publications and research projects dedicated to fractional calculus [2,3,5,6,7,8].

This paper provides a general review of the most common fractional calculus formulations that have found applications to fundamental physical phenomena. We briefly review the basic mathematical definitions and properties of fractional calculus operators, and list the fields of their applications in physics. Also, we briefly present in this review two formalisms that have emerged from fractional calculus: the Fractional Action-Like Variational Approach (FALVA) and conformable calculus. FALVA represents an implementation of non-local fractional powers in the kernel of the action functional, which, conformable calculus, is a fractional trivial deformation of the standard calculus. Despite its simplicity, it has been gaining popularity as it allows for including diffusion terms in the derivative. More importantly, its fractional parameter can be connected to fractal dimensions of fractal spaces. As a final topic, we also review some concepts of the fractional Laplacian, which, in its integral representation, is associated with the Riesz derivative, and some of its recent applications in classical mechanics and field theory.

We also discus briefly the issue of operator selection in fractional modeling by illustrating the differences in fractional models of the same system based on different formalisms. In more general cases, the main problem is that different fractional operators lead to distinct physical interpretations and applications, as the relationship between fractional orders and physical properties and parameters of various systems, from anomalous diffusion to cosmological models, depends on the fractional formalism used in the description of the physical system.

Given that the subject of fractional calculus is both vast in mathematical and physical application, the present paper does not aim at being nor an exhaustive review, neither an in depth discussion of a particular topic, but rather a presentation of some the most common fractional calculus formalisms and of the derived theories from it that have been applied recently to fundamental physics and have caught the attention of authors.

The structure of this review paper is as follows. In Section 2, we present the main formulations of fractional calculus, including Riemann–Liouville, Gerasimov–Caputo, Grünwald–Letnikov, Weyl–Marchaud, and Riesz fractional derivatives, along with the fractional Laplacian and other fractional operators. Section 3 discusses conformable derivatives and their physical interpretations. Section 4 presents three examples of phenomena modelled with Gerasimov–Caputo, Riemann–Liouville, and, in the last one, conformable derivative with the objective to show how the selection of different fractal operators leads to different results. In Section 5, we briefly review the FALVA and its applications in physics and cosmology. Finally, Section 6 provides some concluding remarks on the content covered by this work.

2. Some Formulations of Fractional Calculus and Applications in Physics

Fractional calculus is a term that denotes the generalizations of classical calculus that extend the differentiation operation $d^n/d{x}^n$ to non-integer orders $d^{\alpha }/d{x}^{\alpha }$, with the corresponding inverse integration operation. The fractional generalization has found several applications in physics mainly in describing complex phenomena like memory effects, non-locality, and anomalous dynamics which are capture more naturally than the models based on standard calculus [3]. However, the consistent generalizations of standard calculus to fractional order operators are not unique. Thus, fractional calculus has developed into several inequivalent formulations, each with specific mathematical properties and physical interpretations. Today there are more than fifteen different definitions for fractional derivatives see for instance an interesting list presented in [9], of course we do not intend to explain each one and there are others works dedicated for that. However, our goal in this section is to briefly review some of the most popular and useful fractional derivatives and integrals that are encountered more frequently in physics.

2.1. Riemann–Liouville and Gerasimov–Caputo Fractional Derivatives

The Riemann–Liouville fractional derivative represents one of the oldest and most fundamental approaches to fractional differentiation. For a function $f\left(t\right)$ defined in the interval $[a,t]$, the Riemann–Liouville fractional derivative of order $\alpha$, where $n-1<\alpha <n$, $n\in \mathbb{N}$, is defined as follows:

$$D_{a+}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\displaystyle \frac{d^n}{d{t}^n}}{\int }_a^t{(t-\tau )}^{n-\alpha -1}f\left(\tau \right)d\tau .$$

(1)

The corresponding fractional integral, known as the Riemann–Liouville fractional integral, represents the inverse operation and is defined as follows:

$$J_{a+}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{(t-\tau )}^{\alpha -1}f\left(\tau \right)d\tau .$$

(2)

The integration operators satisfy the semigroup property:

$$J_{a+}^{\alpha }{J}_{a+}^{\beta }=J_{a+}^{\alpha +\beta },$$

(3)

for $\alpha ,\beta >0$, which ensures the consistency of repeated fractional integration operations [3].

The Gerasimov–Caputo fractional derivative was developed to address limitations of the Riemann–Liouville derivative in handling initial conditions and has the following definition:

$$D_{a+}^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_a^t{(t-\tau )}^{n-\alpha -1}{f}^{\left(n\right)}\left(\tau \right)d\tau .$$

(4)

In most of the literature, this derivative is shortly named after Michele Caputo, who introduced it in 1967 [5]. It is worth noting that this operator is also referred to as the Gerasimov–Caputo fractional derivative, acknowledging the independent work of Alexey Gerasimov, who arrived at a similar definition [6]. The Caputo derivative allows one to express the initial conditions in terms of integer-order derivatives, which is an important property for modeling physical systems whose physical properties depend strongly on the initial value problems [10].

An interesting relationship connects these two formulations:

$${}_{a}D_x^{n+a}f\left(x\right)={}_a{}^{C}D^{n+a}f\left(x\right)+\sum _{k=0}^n{\displaystyle \frac{f^k\left(0\right)x^{k-n-\alpha }}{\Gamma (k-\alpha -n+1)}}.$$

(5)

Besides, Riemann–Liouville and Caputo definitions have some common properties, the most important ones for applications in physics being listed below.

• Linearity: Both fractional derivatives satisfy the linearity condition, which is a consequence of the integral operation used to define them

  $$D^{\alpha }(af+bg)=a{D}^{\alpha }f+b{D}^{\alpha }g.$$

  (6)

  This property allows one to construct models that obey the principle of superposition.
• Breakdown of Leibniz Rule: The Leibniz rule no longer holds for either Riemann–Liouville or Caputo derivatives. Instead, the product rule for fractional derivatives must be treated in its series form:

  $$D^{\alpha }\left(fg\right)=\sum _{k=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{k}}\right)D^{\alpha -k}f{D}^{k}g.$$

  (7)

  The breakdown of the Leibniz rule is a fundamental feature of fractional derivatives in general. This property has profound implications for models of nonlinear systems and for the variational principle [11,12]. Also, it implies that complex analytical and computer-assisted methods should be used in order to perform actual calculations in practice.
• Non-locality: Both formulations can be used to describe memory effects through their integrals, which are defined in terms of power-law kernels. This allows one to describe systems with long-range spatial or temporal dependencies.

These properties make both Riemann–Liouville and Caputo derivatives suited for constructing models of viscoelastic materials, or which display anomalous diffusion, and systems that exhibit power-law memory kernels [3].

2.2. Grünwald–Letnikov Fractional Derivative

An important formulation of the fractional derivative is the Grünwald–Letnikov derivative, which provides an approach to fractional calculus based on finite differences. For the history of this derivative, see e.g., [13].

The Grünwald–Letnikov fractional derivative of order $\nu >0$ is defined as follows:

$$D^{\nu }f\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\nu }}}\sum _{k=0}^{\infty }{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{k}}\right)f(x-kh),$$

(8)

where

$$\left({\displaystyle \genfrac{}{}{0pt}{}{\nu }{k}}\right)={\displaystyle \frac{\Gamma (\nu +1)}{\Gamma (k+1)\Gamma (\nu -k+1)}},$$

(9)

is the generalized binomial coefficient expressed in terms of the gamma function. The definition (8) generalizes differentiation through finite differences to fractional orders. This is implemented by considering an infinite series of function values with appropriate weights. The main justification for the definition (8) is found in the standard limit definition of integer-order derivatives. For example, the first-order derivative can be written as follows:

$$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{f\left(x\right)-f(x-h)}{h}},$$

(10)

and the second-order derivative as follows:

$$f^{″}\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{f\left(x\right)-2f(x-h)+f(x-2h)}{h^2}}.$$

(11)

The Grünwald–Letnikov derivative generalizes this pattern to arbitrary orders $\nu$ by using the generalized binomial coefficients which account for the fractional and differentiation order. For negative orders ($\nu <0$), the Grünwald–Letnikov definition generates the fractional integral. At order $\alpha$, the fractional integral of is given by the following relation:

$$D^{-\alpha }f\left(x\right)=\underset{h\to 0}{lim}{h}^{\alpha }\sum _{k=0}^{\infty }{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{-\alpha }{k}}\right)f(x-kh),$$

(12)

where the binomial coefficients for negative orders are calculated as follows:

$$\left({\displaystyle \genfrac{}{}{0pt}{}{-\alpha }{k}}\right)={(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha +k-1}{k}}\right)={(-1)}^k{\displaystyle \frac{\Gamma (\alpha +k)}{\Gamma \left(\alpha \right)\Gamma (k+1)}}.$$

(13)

This fractional integral provides a discrete approximation of the Riemann–Liouville fractional integral and is particularly useful for numerical calculations. However, the Grünwald–Letnikov derivative differs from the Riemann–Liouville and Caputo derivatives in several important aspects.

• Definition: As seen above, the Riemann–Liouville derivative is defined through fractional integrals followed by integer-order differentiation, and the Caputo derivative reverses the order of these standard calculus operations. However, the Grünwald–Letnikov derivative is defined directly through a limit of finite differences. This makes it more natural for numerical applications.
• Initial Conditions: From its definition, we can see that the Caputo derivative allows us to give standard initial conditions written in terms of integer-order derivatives. In contrast, the Grünwald–Letnikov and Riemann–Liouville derivatives require initial conditions for fractional integrals, which can be more difficult to interpret from a physical point of view.
• Regularity: The Caputo derivative requires the function to be differentiable in the classical sense, while the Grünwald–Letnikov derivative can be applied to a more general class of functions, for example, functions that are not necessarily differentiable in the integer-order sense.
• Numerical applications: The main advantage of the Grünwald–Letnikov derivative is that it is suitable for numerical computations because of its discrete nature. For example, in the Grünwald–Letnikov calculus, one can immediately generalize the finite difference method. On the other hand, the Riemann–Liouville and Caputo derivatives require more complex numerical quadrature techniques.
• Memory Effects: All three derivatives incorporate non-locality and memory effects, but the Grünwald–Letnikov derivative does so through an explicit summation over previous function values, making the memory effects more transparent. This is an advantage in discrete and numerical modelling of memory effect systems, as already mentioned above [14].
• Constant Functions: One important characteristic of both the Grünwald–Letnikov and Riemann–Liouville derivatives of a constant function is that they generate a non-zero value, which contrasts with the Caputo derivative.

Due to its specific properties, the Grünwald–Letnikov fractional derivative has applications in several fields of physics, mainly in areas where the focus is on numerical implementation and modeling of non-local phenomena, like anomalous diffusion in complex media, viscoelasticity with memory effects, and non-local response in electromagnetic phenomena [10,14,15].

2.3. Weyl–Marchaud Fractional Calculus

The Weyl–Marchaud derivative’s definition arises from the idea of implementing the contribution of a function $\varphi$ at different times. The Weyl derivative is defined on the whole real axis or on periodic functions. It can be defined in terms of Fourier transforms or as the limit of symmetric fractional integrals. The Marchaud derivative is a pointwise representation for nonlocal fraction differentiation in terms of singular integrals of difference quotients. For a review and explanation, see ref. [16].

Let us review the basic operators of the Weyl–Marchaud fractional calculus. We start by denoting by $\mathcal{S}\left(\mathbb{R}\right)$ the Schwartz class and by ${\mathcal{S}}^{\prime }\left(\mathbb{R}\right)$ its dual. The Fourier transform is denoted by the following:

$$\widehat{f}\left(\xi \right)=\mathcal{F}\left\{f\right\}\left(\xi \right)={\int }_{\mathbb{R}}{e}^{-i\xi x}f\left(x\right)dx,$$

(14)

with inverse:

$${\mathcal{F}}^{-1}\left\{g\right\}\left(x\right)={\left(2\pi \right)}^{-1}{\int }_{\mathbb{R}}{e}^{i\xi x}g\left(\xi \right)d\xi .$$

(15)

As before, we consider $0<\alpha <1$ unless otherwise stated.

The left and right Liouville–Weyl fractional integrals on the real line are defined for sufficiently smooth functions f by the following:

$$\begin{array}{cc}\hfill \left(I_{+}^{\alpha }f\right)\left(x\right)& :={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{-\infty }^x{(x-t)}^{\alpha -1}f\left(t\right)dt,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \left(I_{-}^{\alpha }f\right)\left(x\right)& :={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_x^{\infty }{(t-x)}^{\alpha -1}f\left(t\right)dt.\hfill \end{array}$$

(17)

For $f\in \mathcal{S}\left(\mathbb{R}\right)$, the integrals from Equation (17) are well defined and map $\mathcal{S}$ into itself. Their Fourier multipliers are given by the following:

$$\mathcal{F}\left\{I_{\pm }^{\alpha }f\right\}\left(\xi \right)={(\mp i\xi )}^{-\alpha }\widehat{f}\left(\xi \right),$$

(18)

where a continuous branch of the complex power is chosen.

The Marchaud fractional derivative of order $0<\alpha <1$ is the singular integral operator:

$$\left({\mathcal{M}}^{\alpha }f\right)\left(x\right):=C_{\alpha }{\int }_0^{\infty }{\displaystyle \frac{f\left(x\right)-f(x-s)}{s^{1+\alpha }}}ds,$$

(19)

with normalization constant:

$$C_{\alpha }={\displaystyle \frac{\alpha }{\Gamma (1-\alpha )}}.$$

(20)

A symmetric form for two-sided decay is defined as follows:

$${\mathcal{M}}_{\mathrm{sym}}^{\alpha }f\left(x\right)={\displaystyle \frac{C_{\alpha }}{2}}{\int }_{\mathbb{R}\setminus \left\{0\right\}}{\displaystyle \frac{f\left(x\right)-f(x-s)}{{\left|s\right|}^{1+\alpha }}}ds.$$

(21)

The Weyl fractional derivative on $\mathcal{S}\left(\mathbb{R}\right)$ is defined by the Fourier multiplier:

$$\mathcal{F}\left\{D_{\mathrm{W}}^{\alpha }f\right\}\left(\xi \right)={\left(i\xi \right)}^{\alpha }\widehat{f}\left(\xi \right).$$

(22)

For the symmetric real version with ${\left|\xi \right|}^{\alpha }$, it is given by the following:

$$\mathcal{F}\left\{D_{\mathrm{sym}}^{\alpha }f\right\}\left(\xi \right)={\left|\xi \right|}^{\alpha }\widehat{f}\left(\xi \right).$$

(23)

On appropriate function classes, the Marchaud derivative (19) admits inverse operators which coincide with Liouville–Weyl integrals up to additive polynomial terms depending on decay or boundary assumptions. In particular, we can show the following:

• If $f=I_{+}^{\alpha }g$ with g decaying sufficiently fast at $-\infty$, then

  $${\mathcal{M}}^{\alpha }f=g,$$

  (24)

  so $I_{+}^{\alpha }$ is a right-inverse of ${\mathcal{M}}^{\alpha }$ under suitable conditions.
• If $f\in \mathcal{S}\left(\mathbb{R}\right)$ and $D_{\mathrm{W}}^{\alpha }$ is defined via (22), then $I_{\pm }^{\alpha }$ is the inverse in the Fourier sense:

  $$\mathcal{F}\left\{I_{\pm }^{\alpha }\left(D_{\mathrm{W}}^{\alpha }f\right)\right\}\left(\xi \right)=\widehat{f}\left(\xi \right).$$

  (25)

The Weyl–Marchaud calculus has several important properties. Consider $0<\alpha , \beta <1$ and all equalities in the maximal classes where both sides are defined.

• Semigroup structure: Weyl integrals satisfy the semigroup property:

  $$I_{\pm }^{\alpha }{I}_{\pm }^{\beta }=I_{\pm }^{\alpha +\beta }.$$

  (26)
  If we consider the Fourier multipliers, we can write the following:

  $$D_{\mathrm{W}}^{\alpha }{D}_{\mathrm{W}}^{\beta }=D_{\mathrm{W}}^{\alpha +\beta }.$$

  (27)
• Inversion and boundary dependence: The inversion relations depend on decay at infinity or on boundary data as follows:
  • If f decays sufficiently fast as $x\to -\infty$ then ${\mathcal{M}}^{\alpha }\left(I_{+}^{\alpha }f\right)=f$ holds with no additive polynomial term, as in (24).
  • On finite intervals, inversion can produce lower-order polynomial residues analogous to Riemann–Liouville and Caputo differences.
• Pointwise representation and relation to Riemann–Liouville calculus: For smooth f with well defined one-sided decay, the Marchaud derivative equals the limit of regularized Riemann–Liouville derivatives when the lower limit $a\to -\infty$. Thus, the whole-line Marchaud/Weyl pointwise formulas can be generated in this case
• Fourier multiplier: For $f\in \mathcal{S}\left(\mathbb{R}\right)$ the symmetric Marchaud derivative (21) and the symmetric Weyl derivative (23) coincide up to normalization. This can be most easily seen from their Fourier multiplier structure with sa ymbol proportional to ${\left|\xi \right|}^{\alpha }$:

  $$\mathcal{F}\left\{{\mathcal{M}}_{\mathrm{sym}}^{\alpha }f\right\}\left(\xi \right)=k_{\alpha }{\left|\xi \right|}^{\alpha }\widehat{f}\left(\xi \right).$$

  (28)
• Connection with fractional Laplacian: The symmetric Weyl–Marchaud derivative is the one-dimensional analogue of the fractional Laplacian ${(-\Delta )}^{\alpha /2}$. This is a consequence of the fact that the fractional Laplacian in $\mathbb{R}$ has a Fourier symbol ${\left|\xi \right|}^{\alpha }$ and admits a pointwise singular integral representation similar to (21).
• Mapping properties: The basic mapping properties are given by the following relations:
  • $I_{\pm }^{\alpha }:L^p\left(\mathbb{R}\right)\to L_{\mathrm{loc}}^q\left(\mathbb{R}\right)$ under standard fractional-integration index relations.
  • ${\mathcal{M}}^{\alpha }:C^{\beta }\left(\mathbb{R}\right)\to C^{\beta -\alpha }\left(\mathbb{R}\right)$ for $\beta >\alpha$ and similar Sobolev space order reductions.

2.4. Riesz Fractional Derivative and Integral

Another important formulation of fractional calculus is based on the Riesz fractional derivative, which is a symmetric derivative with respect to its variable, in the sense that it incorporates both left and right information at the point where it is evaluated. When the derivative is in the time variable, this property takes into account both past and future information. Due to this feature, the Riesz derivative is an important tool for modeling spatial and time fractional derivatives in field theories with global memory effects. The Riesz fractional derivative of order $\alpha$ is defined as follows

$${}^{R}D_x^{\alpha }f\left(x\right)={\displaystyle \frac{d^{\alpha }}{{d\left|x\right|}^{\alpha }}}f\left(x\right):=-{\displaystyle \frac{1}{2cos(\pi \alpha /2)}}\left[-\infty D_x^{\alpha }{+}_x{D}_{\infty }^{\alpha }\right]f\left(x\right),$$

(29)

where ${}^{R}D_x^{\alpha }$ and $d^{\alpha }/d{\left|x\right|}^{\alpha }$ are two common notations for Riesz derivative, and $-\infty D_x^{\alpha }$ and ${}_{x}D_{\infty }^{\alpha }$ are the left- and right-sided Riemann–Liouville fractional derivatives, respectively [1,17].

The corresponding Riesz fractional integral is the inverse operation of the derivative. This property is implemented by the following definition

$$I^{\alpha }f\left(x\right)={\displaystyle \frac{1}{2\Gamma \left(\alpha \right)cos(\pi \alpha /2)}}{\int }_{-\infty }^{\infty }{|x-\xi |}^{\alpha -1}f\left(\xi \right)d\xi ,$$

(30)

for $0<\alpha <1$ and $\alpha \ne 1,3,5,\dots$ [17].

The Riesz derivative has specific properties that distinguish it from earlier fractional derivatives. The most important are the following:

• Symmetry: The Riesz derivative is symmetric under reflection ($x\to -x$). This makes it a fractional derivative of choice for describing isotropic systems and processes without a preferred direction in the variable.
• Two-sided memory: While most of the fractional derivatives that display memory properties are one-sided derivatives, the Riesz derivative contains information from left- and right-side of the point of evaluation. This allows one to include non-local interactions in the full variable space.
• Fourier representation: The Riesz derivative has a simple representation in Fourier space, given by the relation:

  $$\mathcal{F}\left\{{\displaystyle \frac{d^{\alpha }}{{d\left|x\right|}^{\alpha }}}f\left(x\right)\right\}\left(k\right)=-{\left|k\right|}^{\alpha }\mathcal{F}\left\{f\left(x\right)\right\}\left(k\right).$$

  (31)
  This property is useful for solving fractional partial differential equations using transform methods.

The above properties make the Riesz derivative a useful tool for studying fractional quantum mechanics and anomalous transport processes [17,18].

2.5. Fractional Laplacian and Its Connection to Riesz Derivative

An important operator in fractional calculus, related to the Riesz derivative, is the fractional Laplacian ${(-\Delta )}^{\alpha /2}$, which generalizes the classical Laplacian to fractional orders. When viewed as operator on some types of spaces like Lebesgue ${\mathcal{L}}^p$, continuous functions ${\mathcal{C}}_0$, bounded uniformly continuous functions ${\mathcal{C}}_{bu}$, Schwartz space $\mathcal{S}$, etc., the fractional Laplacian can be defined using different but equivalent mathematical concepts [19].

In ${\mathbb{R}}^n$, the singular integral representation of the fractional Laplacian is defined by the following formula:

$${(-\Delta )}^{{\displaystyle \frac{\alpha }{2}}}f\left(x\right)=C_{n,\alpha }\mathrm{P}.\mathrm{V}.{\int }_{{\mathbb{R}}^n}{\displaystyle \frac{f\left(x\right)-f\left(y\right)}{{|x-y|}^{n+\alpha }}}dy,$$

(32)

where

$$C_{n,\alpha }={\displaystyle \frac{4^{\alpha /2}\Gamma \left(\frac{n+\alpha }{2}\right)}{{\pi }^{n/2}|\Gamma (-\alpha /2)|}},$$

(33)

is a normalization constant, and P.V. denotes the Cauchy principal value [20,21].

The Riesz fractional derivative and the fractional Laplacian are related operators. To see that, we consider a one-dimensional space for simplicity of discussion. The case of n-dimensional fractional Laplacian can be discussed similarly. Then the relation between the fractional Laplacian and the Riesz fractional derivative of order $\alpha \in (0,2)$ takes the following form:

$${(-\Delta )}^{\frac{\alpha }{2}}f\left(x\right)={}^{R}D_x^{\alpha }f\left(x\right),$$

(34)

where ${(-\Delta )}^{\frac{\alpha }{2}}$ denotes the fractional Laplacian. In the Fourier domain, this relation can be written as follows:

$$\mathcal{F}\left[{(-\Delta )}^{\frac{\alpha }{2}}f\right]\left(k\right)=-{\left|k\right|}^{\alpha }\widehat{f}\left(k\right),$$

(35)

which coincides with the definition of the Riesz fractional derivative in one dimension:

$$\mathcal{F}\left[{}^{R}D_x^{\alpha }f\right]\left(k\right)=-{\left|k\right|}^{\alpha }\widehat{f}\left(k\right).$$

(36)

In n spatial dimensions, the fractional Laplacian is defined as follows:

$${(-\Delta )}^{\frac{\alpha }{2}}f\left(\mathbf{x}\right)={\mathcal{F}}^{-1}\left({\left|\mathbf{k}\right|}^{\alpha }\widehat{f}\left(\mathbf{k}\right)\right)\left(\mathbf{x}\right),$$

(37)

with $\mathbf{x}\in {\mathbb{R}}^n$ and $\mathbf{k}\in {\mathbb{R}}^n$, is related to the Riesz fractional derivative in n dimensions by the following equation:

$${(-\Delta )}^{\frac{\alpha }{2}}f\left(\mathbf{x}\right)={}^{R}D_{\mathbf{x}}^{\alpha }f\left(\mathbf{x}\right),$$

(38)

with the Fourier symbol defined as $-{\left|\mathbf{k}\right|}^{\alpha }$.

The fractional Laplacian has important properties that are carried over to the models used in applications to physical systems.

• Non-locality: Due to its singular integral kernel, the fractional Laplacian describes long-range interactions. This makes it suitable for describing physical systems with power-law spatial or temporal correlations in the kinetic term.
• Scale invariance: The Laplacian operator has the following scaling property:

  $${(-\Delta )}^{\alpha /2}f\left(\lambda x\right)={\lambda }^{\alpha }\left[{(-\Delta )}^{\alpha /2}f\right]\left(\lambda x\right).$$

  (39)
  This property is particularly important for modeling critical phenomena and systems with fractal properties or defined on fractal spaces.
• Extension problem: The fractional Laplacian can be realized as a Dirichlet-to-Neumann map for a local degenerate elliptic equation in the upper half-space:

  $$\begin{array}{cc}\hfill {\Delta }_{x}u(x,y)+{\displaystyle \frac{1-2s}{y}}{u}_y(x,y)+u_{yy}(x,y)& =0,\hfill \end{array}$$

  (40)

  $$\begin{array}{cc}\hfill u(x,0)& =f\left(x\right),\hfill \end{array}$$

  (41)

  $$\begin{array}{cc}\hfill -\underset{y\to 0^{+}}{lim}{y}^{1-2s}{u}_y(x,y)& =C_s{(-\Delta )}^{s}f\left(x\right).\hfill \end{array}$$

  (42)
  This extension property represents a powerful technique for analyzing non-local problems using local methods [22]. In fractional quantum field theory, the Caffarelli-Silvestre extension problem connects fractional field theories in n dimensions with more standard quantizable field theories in $n+1$ dimensions [23,24].

Recently, the fractional Laplacian has found applications in different fields. Without wanting to be exhaustive nor in fields, nor in citations, the fractional Laplacian generated models, such as gradient elasticity [25], anomalous diffusion [26] and Lévy processes [27]. More recently, it has been used to study fundamental physical systems that incorporate processes addressed through fractional calculus, such as electrodynamic systems [28], classical particles [29,30], cosmological models [31,32], and different quantum field theories [23,33,34,35,36,37,38]. For more details on the mathematical properties of the fractional Laplacian, we refer to [21].

2.6. Other Fractional Operators

Several other fractional operators have been developed for specific applications in mathematics and physics. In what follows, we will briefly review only two of these operators.

• Atangana–Baleanu derivative: This derivative is based on non-singular Mittag–Leffler kernels, and it is defined as follows:

  $$D^{\alpha }f\left(t\right)={\displaystyle \frac{AB\left(\alpha \right)}{1-\alpha }}{\int }_a^t{E}_{\alpha }\left(-{\displaystyle \frac{\alpha }{1-\alpha }}{(t-\tau )}^{\alpha }\right)f^{\prime }\left(\tau \right)d\tau ,$$

  (43)

  where $AB\left(\alpha \right)$ is a normalization function and $E_{\alpha }$ is the Mittag–Leffler function. The Atangana–Baleanu derivative exhibits non-singular kernel properties, making it suitable for describing physical systems with non-exponential relaxation [39].
• Hadamard fractional derivative: The Hadamard fractional derivative can be defined using logarithmic kernels as follows [40]:

  $$D^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\left(t{\displaystyle \frac{d}{dt}}\right)}^n{\int }_a^t{\left(log{\displaystyle \frac{t}{\tau }}\right)}^{n-\alpha -1}{\displaystyle \frac{f\left(\tau \right)}{\tau }}d\tau .$$

  (44)
  This operator is useful for problems with logarithmic scaling properties.

It is worth noting that there exist other generalizations of fractional derivatives for the case of variable order fractional derivatives, see for example [7]. These operators allow for describing interesting memory effects, such as heredity, that can vary in time or in space. A well-known example is the use of the Gerasimov–Caputo fractional derivative in time of a variable order $0<\alpha \left(t\right)<1$ [8], which has been applied successfully to direct and inverse problems of modeling of Radon volumetric activity.

Each of the fractional derivative operators presented here has characteristic mathematical properties that make it suitable for specific physical applications, ranging from non-local continuum mechanics and anomalous transport in complex media to deformations of quantum mechanics. The different formulations of fractional calculus reflect the rich mathematical structure that arises from generalizing the standard calculus to non-integer orders. Going from standard calculus to fractional calculus, the simplicity of calculus is traded for the adaptability of derivative and integral operators to various physical phenomena. The choice of a specific fractional operator depends on the particular physical phenomena being modeled, the mathematical properties required for analysis, and the compatibility with experimental observations.

3. Conformable or Deformed Derivative

A spin-off of the fractional calculus is the conformable calculus, which originated in the idea of constructing a fractional derivative that obeys the Leibniz axiom. However, as mentioned above, a characteristic of fractional derivatives is the violation of the Leibniz rule. As it turned out, the conformable derivative is just a deformation of the standard derivative with a fractional “conformable parameter”. Also, as formulated initially, the conformable derivative does not tend to the standard derivative, casting doubt on the legitimacy of using the term ’conformable’. Nevertheless, since it is already known as such, we will use this terminology as well. A proper conformable deformation of the standard derivative was proposed in [41], but this is beyond the scope of our paper.

Although it is a simple deformation of the standard derivative, the conformable derivative and the calculus constructed from it are interesting because they are formally connected to the fractal derivative. The relationship is obtained by interpreting the conformable parameter as a fractal parameter, usually the fractal dimension of the corresponding underlying space.

For a function $f:[0,\infty )\to \mathbb{R}$, the conformable derivative of order $\alpha \in (0,1]$ is defined as follows:

$$T_{\alpha }f\left(t\right)=\underset{\epsilon \to 0}{lim}{\displaystyle \frac{f(t+\epsilon t^{1-\alpha })-f\left(t\right)}{\epsilon }}.$$

(45)

This definition satisfies the classical properties of the first-order derivative when $\alpha =1$ [42].

The corresponding conformable integral is defined as follows:

$$I_{\alpha }f\left(t\right)=\int f\left(t\right)d_{\alpha }t=\int f\left(t\right)t^{\alpha -1}dt,$$

(46)

which reduces to the standard integral when $\alpha =1$ [42].

The conformable derivative possesses interesting properties with physical implications.

• Leibniz rule: The conformable derivative satisfies the standard Leibniz rule:

  $$T_{\alpha }\left(fg\right)=f{T}_{\alpha }\left(g\right)+g{T}_{\alpha }\left(f\right).$$

  (47)
  This property facilitates calculations in variational principles and conservation laws.
• Chain rule: The conformable derivative obeys the chain rule:

  $$T_{\alpha }(f\circ g)\left(t\right)=f^{\prime }\left(g\left(t\right)\right)T_{\alpha }g\left(t\right).$$

  (48)
  As in the standard calculus, the chain rule and the Leibniz rule simplify the computation of derivatives of composite functions.
• Locality: The conformable derivative is a local operator, which makes it suitable for describing systems without memory effects but with scale-dependent dynamics.

The physical interpretation of conformable derivatives has been the subject of ongoing discussion in the literature. The proponents of this derivative argued that conformable derivatives provide a more natural extension of classical calculus for modeling systems with scale-dependent behavior, particularly when the scaling properties are power-law. The parameter $\alpha$ can be interpreted as a measure of the departure of the system from standard behavior, with $\alpha =1$ corresponding to classical dynamics [42].

However, it was pointed out that conformable derivatives lack the non-locality that characterizes traditional fractional derivatives, which is essential for describing memory and hereditary effects, and they also lack the Leibniz rule. This has led to debates about whether conformable derivatives should be considered genuine fractional operators or simply parameterized families of integer-order operators [43].

Despite these controversies, see also [44,45], the conformable derivative has found applications in various physical contexts, including fundamental physics problems in conformable Newtonian mechanics [46,47], conformable quantum mechanics [48,49,50,51,52], and deformed special relativity [53]. Two reviews of recent interpretation and elementary applications of the conformable derivative can be found in [54,55], and an interesting interpretation in terms of fractal spaces in [56].

4. Selection of Fractional Operators and Physical Interpretation of the Order

The choice of an appropriate fractional operator and the physical interpretation of its order are important challenges in fractional modeling. As the above formulations show, upon analysis, different fractional derivatives determine distinct properties of non-locality and memory effects. Therefore, the operator selection becomes crucial for an accurate physical description. In this subsection, we discuss three cases that we consider representative of distinct fractional formulations with distinct physical interpretations.

4.1. Anomalous Diffusion in Complex Media

Two popular formalisms for modelling the anomalous diffusion processes are the Riemann–Liouville and Gerasimov–Caputo. However, employing these formulations significantly affects the physical interpretation of the models and their properties. To illustrate this point, we will discuss a simple aspect of anomalous diffusion. Consider the following fractional diffusion equation

$${\displaystyle \frac{{\partial }^{\beta }P(x,t)}{\partial t^{\beta }}}=D_{\beta }{\displaystyle \frac{{\partial }^{\alpha }P(x,t)}{{\partial \left|x\right|}^{\alpha }}},$$

(49)

where $P(x,t)$ is the probability density, $D_{\beta }$ is the fractional diffusion coefficient, and $\alpha ,\beta$ are the space and time fractional orders, respectively. When using the Gerasimov–Caputo derivative for the time component, the solution exhibits a Mittag–Leffler decay:

$$P(k,t)=E_{\beta }(-D_{\beta }{\left|k\right|}^{\alpha }{t}^{\beta }),$$

(50)

where $E_{\beta }$ is the Mittag–Leffler function and the fractional order $\beta$ relates directly to the anomaly exponent in the mean squared displacement [4]:

$$⟨x^2\left(t\right)⟩\sim t^{\beta }.$$

(51)

On the other hand, the Riemann–Liouville formulation introduces different initial conditions that affect the short-time behavior and lead to the following:

$$P(k,t)=t^{\beta -1}{E}_{\beta ,\beta }(-D_{\beta }{\left|k\right|}^{\alpha }{t}^{\beta }).$$

(52)

Equations (50) and (52) show a distinction in the probability density predicted by the two models, which becomes important when systems with power-law initial conditions are analysed or when dynamics is strongly determined by the initial state [3].

4.2. Viscoelastic Material Response

In the study of viscoelastic materials, the physical properties predicted by the model also depend on the choice of the fractional operators. For example, in the fractional Kelvin-Voigt model using Gerasimov–Caputo derivative, the stress-strain relationship is given by the following equation:

$$\sigma \left(t\right)=Eϵ\left(t\right)+\eta D_t^{\alpha }ϵ\left(t\right),$$

(53)

where $\sigma \left(t\right)$ is the stress, $ϵ\left(t\right)$ is the strain, E is the elastic modulus, and $\eta$ represents the generalized viscosity coefficient. The complex modulus in the frequency domain $G^{\ast }\left(\omega \right)$ relates stress and strain amplitudes for harmonic oscillations. For example, for a strain of the form

$$ϵ\left(t\right)=\mathrm{Re}\left\{\widehat{ϵ}{e}^{i\omega t}\right\},$$

(54)

the stress is given by the following:

$$\sigma \left(t\right)=\mathrm{Re}\left\{G^{\ast }\left(\omega \right)\widehat{ϵ}{e}^{i\omega t}\right\}.$$

(55)

In the Gerasimov–Caputo mode, the complex modulus has the general form [4]

$$G^{\ast }\left(\omega \right)=E+\eta {\left(i\omega \right)}^{\alpha },$$

(56)

where the fractional order $\alpha$ relates to the slope of the storage and loss moduli in log-log plots. When using the Riemann–Liouville derivative, the stress-strain relationship becomes the following:

$$\sigma \left(t\right)=Eϵ\left(t\right)+\eta D_t^{\alpha }[ϵ\left(t\right)-ϵ\left(0\right)].$$

(57)

While the complex modulus has the same form as in Equation (56) in the steady state (frequency domain), introducing additional terms that account for the initial strain history, which is crucial for materials with significant pre-stress conditions [2].

By comparing the two fractional formalisms, one can observe that both Gerasimov–Caputo and Riemann–Liouville operators generate the multiplicative factor ${\left(i\omega \right)}^{\alpha }$ when acting on a harmonic $e^{i\omega t}$. Therefore, both approaches give the same algebraic complex modulus in the steady-state frequency domain:

$$G_{\mathrm{C}}^{\ast }\left(\omega \right)=G_{\mathrm{RL},\mathrm{steady}}^{\ast }\left(\omega \right)=E+\eta {\left(i\omega \right)}^{\alpha }.$$

(58)

However, there are different initial-condition dependences in the two formalisms. The Gerasimov–Caputo derivative of a shifted signal $f\left(t\right)-f\left(0\right)$ removes singular constant-term derivatives, and its Laplace transform has initial-value terms in the form $s^{\alpha -1}f\left(0\right)$. This is consistent with standard integer-order initial data, and the transient contributions appear in a form that is compatible with physical initial conditions expressed through integer-order values. On the other hand, the Riemann–Liouville derivative gives rise to an explicit time-singular term when applied to a constant. Consequently, the constitutive relation of the Riemann–Liouville model contains explicit power-law-in-time transients proportional to $ϵ\left(0\right)$. An important point here is that one cannot represent the Riemann–Liouville transients by a single frequency, implying that the full time-dependent response is altered even if the steady-state harmonic amplitude remains the same. This can be seen by performing the Laplace transform

$$\begin{array}{cc}\hfill \mathcal{L}\left\{{}^{C}D_t^{\alpha }f\right\}\left(s\right)& =s^{\alpha }\tilde{f}\left(s\right)-s^{\alpha -1}f\left(0^{+}\right),\mathbf{G}\mathbf{\text{-}}\mathbf{C}\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill \mathcal{L}\left\{{}^{RL}D_t^{\alpha }f\right\}\left(s\right)& =s^{\alpha }\tilde{f}\left(s\right)-s^{\alpha -1}{\left[I^{1-\alpha }f\right]}_{t=0^{+}}.\mathbf{R}\mathbf{\text{-}}\mathbf{L}\hfill \end{array}$$

(60)

The term in brackets is, in general, a fractional integral of f at zero. This difference reflects distinct initial-history structures of the two models [2,5,57,58].

4.3. Fractional Damped Oscillators: Gerasimov–Caputo, Riemann–Liouville, and Conformable

Another example that is worth discussing is the application of fractional formalisms to a single-degree-of-freedom damped oscillator. Here, we are going to include the Gerasimov–Caputo, Riemann–Liouville, and the non-fractional conformable derivative-based actuated penduli. We are using an unified notation in what follows, where all models use the following common symbols unless otherwise noted: m is the mass parameter, although the units may vary among models and fractional order, k is the generalization of the linear stiffness, $\gamma$ is the generalization on the damping coefficient, with units depending on derivative order, $x\left(t\right)$ is the displacement as a function of time, $F\left(t\right)$ represents the generalization of the external (nonholonomic) forces, $\alpha$ is the fractional order parameter, in the same sense as in the previous discussion, typically $0<\alpha \le 1$, $D_t^{\alpha }$ or ${}^{C}D_t^{\alpha }$ or ${}^{RL}D_t^{\alpha }$ are the fractional derivative operators, $T_{\alpha }$ is the conformable derivative operator and $\omega$ is the generalization of the angular frequency.

The Gerasimov–Caputo fractional damped oscillator has the following equation of motion:

$$m\ddot{x}\left(t\right)+\gamma {}^{C}D_t^{\alpha }x\left(t\right)+kx\left(t\right)=F\left(t\right).$$

(61)

For this model, the initial conditions are the standard mechanical ones: $x\left(0\right),\dot{x}\left(0\right),\dots ,x^{(n-1)}\left(0\right)$. We can calculate the quality factor in the harmonic steady-state estimate, that is, for the harmonic forcing:

$$F\left(t\right)=F_0{e}^{i\omega t},$$

(62)

and zero initial transients. The Caputo term contributes a complex-frequency factor proportional to ${\left(i\omega \right)}^{\alpha }$ with phase $-\alpha \pi /2$. The effective complex stiffness leads to the well-known frequency-dependent estimate

$$Q_C\left(\omega \right)={\displaystyle \frac{m{\omega }^2-k}{\gamma {\omega }^{\alpha }cos\left(\frac{\pi \alpha }{2}\right)}}.$$

(63)

It is worth remarking that the Gerasimov–Caputo fractional damping models can be used to represent hereditary power-law dissipation effects and to obtain frequency-scaling like in the Equation (63). More details on the Caputo definition and its use in mechanical models can be found in [10,59].

The Riemann–Liouville-based fractional damped oscillator has formally the same equation of motion:

$$m\ddot{x}\left(t\right)+\gamma {}^{RL}D_t^{\alpha }x\left(t\right)+kx\left(t\right)=F\left(t\right).$$

(64)

The initial conditions appear in terms of fractional integrals of $x\left(t\right)$ at the lower limit and are not identical to the standard mechanical initial data for $0<\alpha <1$ [10]. The quality factor in steady-state harmonic response must be appropriately interpreted with consistent initial conditions. Here, the Riemann–Liouville derivative also contributes a factor proportional to ${\left(i\omega \right)}^{\alpha }$ in the frequency domain, which gives an analogous frequency-scaling for the quality factor:

$$Q_{RL}\left(\omega \right)\approx {\displaystyle \frac{m{\omega }^2-k}{\gamma {\omega }^{\alpha }cos\left(\frac{\pi \alpha }{2}\right)}}.$$

(65)

This formula coincides with the ones obtained in the Gerasimov–Caputo model for steady-state harmonic behavior, which corresponds to the long-term behaviour. Nevertheless, the transient and initial conditions contribute differently in the two models [59,60].

A different model for which the fractional order plays the role of a conformable parameter, but without having fractional properties, is the conformable fractional damped oscillator constructed with conformable derivatives [47]. The equation of motion, which is a local differential equation, can be written as

$$m\ddot{x}\left(t\right)+\gamma T_{\alpha }\left[x\right]\left(t\right)+kx\left(t\right)=F\left(t\right).$$

(66)

The initial conditions for this model are the same as for the standard actuated oscillator $x\left(0\right),\dot{x}\left(0\right)$ since $T_{\alpha }$ is local in time. In order to compute the quality factor of this model, we note that the conformable damping term is a time-dependent prefactor multiplying $\dot{x}$. Thus, a steady-state harmonic analysis in the usual frequency-domain sense is not immediately applicable, unless the problem is formulated in terms of new variables for which the derivatives take the standard form. However, if one wishes to maintain the conformable derivatives apparent, there are two practical approaches for calculating the conformable factor:

• The Frozen-time (quasi-stationary) approximation, in which we treat t as approximately constant over an oscillation period $2\pi /\omega$, so the effective instantaneous damping is

  $${\gamma }_{\mathrm{eff}}\left(t\right)=\gamma t^{1-\alpha }.$$

  (67)
  A local-in-time estimate of Q then reads

  $$Q_{conf}(\omega ,t)\approx {\displaystyle \frac{m{\omega }^2-k}{\gamma t^{1-\alpha }\omega }}.$$

  (68)
  The Equation (68) shows that Q scales as ${\omega }^1$ in the frozen-time picture, modulated by a decaying (or growing for $\alpha >1$) time factor $t^{\alpha -1}$.
• A full transient analysis is possible. In this case, solving (66) gives a transient and aperiodic behavior whose spectral content depends on initial time and of its forcing history. However, no simple universal power-law $Q\sim {\omega }^{2-\alpha }$ would emerge in general

To conclude, the calculation of the quality factor in the steady-state harmonic/like response case in the three different models presents the following characteristics. The Gerasimov–Caputo model has a quality factor given by Equation (63) with scaling $Q_C\sim {\omega }^{2-\alpha }$ for $0<\alpha <1$ (from denominator ${\omega }^{\alpha }$) and a phase factor from $cos(\pi \alpha /2)$ [10,59]. In the Riemann–Liouville, the steady-state harmonic response gives the same frequency-scaling form for the quality factor from Equation (65), so $Q_{RL}\sim {\omega }^{2-\alpha }$ for long-time harmonic behavior. Differences between the two fractional formulations appear in transients and in how initial conditions are taken into account by the responses [60]. In the conformable model, it is not obtained the same universal frequency power law. In the frozen-time approximation (68) one obtains a local-in-time quality factor $Q_{conf}\sim {\omega }^1$ multiplied by $t^{\alpha -1}$. In general, a global steady-state $Q\left(\omega \right)$ with $Q\sim {\omega }^{2-\alpha }$ is not obtained without further, model-specific assumptions [47].

5. FALVA and Applications

Another interesting construct that was influenced by fractional calculus philosophy is the Fractional Action-Like Variational Approach (FALVA) introduced to implement fractionality in the sense of non-locality and memory effects directly in the variational principle by replacing the usual time integral by a weighted Riemann–Liouville-like integral. The motivation is basically to construct a Lagrangian variational formalism for nonconservative systems and phenomena with intrinsic memory or nonlocality while maintaining the (modified) variational structure and derive the modified Euler–Lagrange equations and generalized conservation laws from it, and we are still able to think about field theories within this context [61,62]. In this section, we will give a brief overview of this approach.

5.1. FALVA Action: Definition and Basic Properties

The basic one-dimensional FALVA action is written by introducing two time variables: an intrinsic (integration) time $\tau$ and an observer time labeled by t which often appears in the upper limit of the action integral. For a mechanical Lagrangian $L(\tau ,q\left(\tau \right),\dot{q}\left(\tau \right))$ the fractional action of order $0<\alpha \le 1$ is given (up to a normalization $\Gamma \left(\alpha \right)$) by the following:

$$S^{\alpha }[q;t]={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^{t}L\left(\tau ,q\left(\tau \right),\dot{q}\left(\tau \right)\right){(t-\tau )}^{\alpha -1}d\tau .$$

(69)

Here ${(t-\tau )}^{\alpha -1}$ is the Riemann–Liouville weight, and $\Gamma$ is the Euler’s Gamma function. Several equivalent formulations, e.g., intrinsic vs. observer times, multi-dimensional generalizations, etc., can be found in the literature. The main point is that the integrand is non-local in time and reduces to the usual action for $\alpha =1$ [61,62]. Thus, non-locality and memory effects are introduced already at the level of the action functional, rather than through the fractional derivatives.

5.2. Fractional Euler–Lagrange Equation of FALVA

Varying (69) with fixed end points, and considering that the variations vanish at boundaries, leads to the FALVA Euler–Lagrange equation in the following form [61,62]:

$$\frac{\partial L}{\partial q}-\frac{\mathrm{d}}{\mathrm{d}\tau }\frac{\partial L}{\partial \dot{q}}={\displaystyle \frac{1-\alpha }{t-\tau }}\frac{\partial L}{\partial \dot{q}}.$$

(70)

The Equation (70) displays the standard Euler–Lagrange terms in the left-hand side corrected by a term proportional to $(1-\alpha )/(t-\tau )$ times the conjugate momentum in the right-hand side. The extra term plays the role of a nonconservative force, which is used to model dissipative or memory effects, and which is controlled by the fractional parameter $\alpha$ and by the observer’s intrinsic time separation $t-\tau$. When $\alpha \to 1$ the right-hand side vanishes and one recovers the standard variational equations. The same structure can be generalized to field actions and to multidimensional fractional integrals [61,62].

5.3. Relation with Fractional Calculus

FALVA differs from the fractional derivative substitutions discussed previously, e.g., replacing derivatives by Caputo derivatives throughout. The main distinction comes from the fact that FALVA implements fractionality at the level of the action integral via a fractional weight. This keeps the Lagrangian density local in coordinates or fields, since no explicit fractional derivative operator appears inside L, while producing nonlocal equations of motion through the weighted integral and the observer’s intrinsic time structure. For comparison and rigorous multi-dimensional extensions, see the discussion from [62] and follow-up.

5.4. FALVA in Modified Gravity

One of the most interesting implementations of FALVA is in classical gravity. There, one replaces the standard Einstein–Hilbert time integral by a fractional time-weighted integral. Schematically, for spatially homogeneous cosmologies or mini-superspace settings, the gravitational fractional action reads as follows:

$$S_{\mathrm{grav}}^{\alpha }={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{t_0}^t{(t-\tau )}^{\alpha -1}\int d^{3}x\sqrt{-g(\tau ,\mathbf{x})}\left[R(\tau ,\mathbf{x})+{\mathcal{L}}_{\mathrm{matter}}\right]d\tau .$$

(71)

Variation with respect to the metric or to mini-superspace variables gives the modified gravitational field equations in which the usual Einstein tensor is supplemented by fractional memory terms of the form $(1-\alpha )/(t-\tau )$ multiplying canonical momenta defined as time derivatives of metric variables. In the weak-field or perturbative expansion, this leads to a “perturbed gravity” interpretation and to a time-dependent, dissipative effective cosmological constant in several FALVA cosmological models [63,64]. Some recent works have applied FALVA to higher-curvature theories. For example, a fractional Einstein–Gauss–Bonnet scalar-field cosmology has been formulated, and modified Friedmann and scalar field equations with fractional corrections were obtained in [65].

5.5. FALVA and Mimetic Dark Matter

Another recent application of FALVA formalism is to the mimetic dark matter. The mimetic dark matter can be constructed by adding the mimetic constraint with a Lagrange multiplier to the fractional gravity sector. In a mini-superspace cosmological ansatz, the fractional mimetic action can be schematically written as follows:

$$\begin{array}{cc}\hfill S_{\mathrm{mimetic}}^{\alpha }=& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{t_0}^t{(t-\tau )}^{\alpha -1}\int d^{3}x\sqrt{-g}\left[{\displaystyle \frac{1}{2}}R+\lambda \left(g^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -1\right)\right]d\tau .\hfill \end{array}$$

(72)

By applying the variational principle, one obtains (i) a modified Einstein equation with fractional memory terms and a contribution from the mimetic constraint, and (ii) the mimetic constraint equation $g^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi =1$. By using this idea in a simplified framework, a fractional mimetic dark matter model in the FALVA framework was constructed, and the non-local nonlinear field equations and their solutions were obtained in several homogeneous cases in [66]. There, it was shown that the metric acquires explicit fractional dependence, and geometric tensors get modified accordingly. In this model, the fractional parameter controls departures from standard mimetic phenomenology.

5.6. FALVA in Cosmology: Friedmann Equations and Phenomenology

Another interesting type of problem to which FALVA has been applied recently concerns cosmology. Friedmann–Lemaître–Robertson–Walker models with the scale factor $a\left(\tau \right)$ were studied, and it was shown that the FALVA variational principle modifies the Friedmann equations. The general result from these studies is that the standard acceleration equation picks up some memory terms proportional to $(1-\alpha )$ and inverse powers of $(t-\tau )$. The effective energy densities and pressures acquire contributions which can be interpreted as fractional dark energy dissipative components [64,67]. These works follow the line of using fractional Friedmann equations to model late-time acceleration, reconstruct scalar field potentials, and study stability in fractional models. For higher-curvature fractional cosmologies, e.g., fractional Einstein–Gauss–Bonnet models, an extended space of possible histories and reheating scenarios is obtained [65].

The FALVA approach is more directly inspired by the fractional integral method to implement the non-locality and memory effects in the action. The FALVA has some interesting features that make it attractive.

• The FALVA preserves the general variational structure of classical mechanics, while introducing memory and non-locality through the action weight.
• This method produces modified conservation and Noether relations suitable for standard analysis. There exist non-conservative Noether theorems adapted to the FALVA.
• The FALVA is a versatile method that can be applied to a variety of fields in physics: mechanics, fields, gravity, and cosmology.

However, despite these interesting features, FALVA has some limitations and open issues. For example, the known issues are as follows: (i) the choice of fractional kernel (Riemann–Liouville vs. other kernels); (ii) the interpretation of observer vs. intrinsic times in fully covariant settings; and (iii) the physical identification of the fractionality parameter $\alpha$ for which phenomenological fits exist but a microscopical origin is debated. For a mathematical treatment of FALVA, see [62] and subsequent reviews [62,64,67].

5.7. Fractional Cosmology: Comparative Analysis of Different Formulations

In order to make the general discussion more concrete, it is interesting to illustrate how different formulations of fraction calculus have found applications in cosmology, and different formulations have given distinct interpretations of dark energy and dark matter phenomena.

The FALVA formalism modifies the Friedmann equations through fractional action principles. For a flat FLRW metric, the fractional Friedmann equation can be written as follows:

$$H^2={\displaystyle \frac{8\pi G}{3}}\rho +{\displaystyle \frac{(1-\alpha )}{t}}H,$$

(73)

where the additional term ${\displaystyle \frac{(1-\alpha )}{t}}H$ mimics the dark energy behavior. The fractional parameter $\alpha$ can be related to the effective equation of state parameter $w_{eff}$ through the following equation:

$$w_{eff}=-1+{\displaystyle \frac{2(1-\alpha )}{3}}.$$

(74)

For $\alpha =0.5$, the Equation (74) gives $w_{eff}\approx -0.67$ which is consistent with some observational constraints [31].

In the fractional derivative cosmology, one can use the time-fractional derivatives in the Einstein field equations, which modify the continuity equations. For example, in the Gerasimov–Caputo formulation, the introduction of the fractional derivative generates an equation of the following form:

$$D_t^{\beta }\left(\rho a^3\right)+p{D}_t^{\beta }\left(a^3\right)=0.$$

(75)

For $\beta <1$, the Equation (75) generates an effective dark energy component whose density parameter is given by the following:

$${\Omega }_{DE}=1-{\displaystyle \frac{{\beta }^2}{t^{2(1-\beta )}}}.$$

(76)

Thus, the fractional derivative cosmology can accommodate the transitioning dark energy models without any additional fields [68].

In the conformal derivative cosmology, one can introduce a scale-dependent dynamics through the following equation:

$$T_{\alpha }H=-{\displaystyle \frac{4\pi G}{3}}(\rho +3p).$$

(77)

Here, the conformable parameter $\alpha$ contains information about the fractal dimension of spacetime. The parameter $\alpha \ne 1$ shows that the spacetime structure is different from the standard smooth geometry. Thus, the fractional-inspired model constructed from the conformable calculus can help understand the fractal spacetime model. An interesting application to cosmologies might be of transition densities of the stochastic processes energy [69].

6. Conclusions

In this review, we have discussed the fundamental formalisms of fractional calculus and some of its applications in fundamental physics, to the best of the authors’ knowledge. We note, however, that many other formulations of fractional calculus and interesting applications exist, for which extensive reviews and books are available in the literature. We have examined the main fractional derivatives, including Riemann–Liouville, Caputo, Grünwald–Letnikov, Weyl–Marchaud, and Riesz formulations, and we have discussed their basics: mathematical definitions, fundamental properties, and physical interpretations. Hopefully, these can serve as guides to understanding their applications in physics across different scales. We have included the fractional Laplacian and its connection to the Riesz derivative, which is not usually mentioned in reviews together with other fractional formalisms, along with some of its recent applications in field theory and quantum mechanics.

A related topic, the fractionality-inspired conformable derivative, was presented as an alternative approach to local calculus that maintains the Leibniz rule while introducing scale-dependent dynamics and a fractional parameter. The conformable calculus is intimately related to the fractal calculus. Also, the Fractional Action-Like Variational Approach (FALVA) was briefly discussed as a new method for introducing non-locality and memory effects into the action principle rather than the equations of motion. FALVA has many interesting applications, from classical mechanics to cosmology and modified gravity theories. Here, we have listed only those with which we are familiar.

Another aspect of this review is the illustration of the operator selection (fractional formulation) in fractional modeling. Here, we discussed how different fractional formulations lead to different physical predictions in the case of three systems: anomalous diffusion in complex media, viscoelastic material response, and fractional damped oscillators, by calculating a concrete physical observable in each model. More consequences of the operator selection can be found in each case, but their presentation falls beyond the scope of this presentation. The comparative analysis in cosmology reveals that the FALVA, fractional-derivative, and conformable approaches each provide distinct insights into dark energy phenomena, with the fractional parameter carrying different physical meanings in each formalism. There is a phenomenological correspondence between fractional orders and measurable physical parameters.

There are several important aspects that can be emphasised and which are listed below.

• Choice of Operator: It is important to emphasize that the selection of an appropriate fractional operator depends on the specific physical problem at hand. To our understanding, there is no clear-cut procedure to choose one fractional formalism over another, but some general ideas can be followed. For example, Caputo derivatives are preferable for initial value problems, Riesz derivatives for isotropic systems, Grünwald–Letnikov for numerical implementations, and FALVA for variational approaches with memory effects. More details can be found in the standard books provided in the references.
• Physical Interpretation: While fractional operators provide powerful mathematical tools especially when used together with computer assisted methods, their physical interpretation requires a careful consideration. One example of this is fractional order, which is often related to the physical properties of the system and can be encoded in parameters such as fractal dimensions, memory decay rates, or anomalous diffusion exponents.
• Computational Aspects: The numerical implementation of fractional models presents computational challenges due to non-locality. These challenges differ from one formalism to another. For example, the Grünwald–Letnikov approach offers advantages for discrete implementations, while spectral methods work well for problems with smooth analytical solutions.
• Emerging Directions in Fundamental Physics: Recent developments include variable-order fractional derivatives applied to systems with evolving memory characteristics, applications in quantum field theory and gravity, and connections between fractional calculus and fractal geometry. This is a relatively new field of applications of fractional calculus justified by hypotheses and models of the microscopic structure of space-time and strong gravity.

It is difficult to give an exhaustive review of the fractional calculus since this field, in its multiple manifestations, continues to evolve as a rich mathematical theory in continuous construction and with a growing number of applications across all physical scales. The most common formulations discussed in this review offer, each of them, some unique advantages for modeling complex systems that are endowed mainly with memory, non-locality, and anomalous dynamics. Other physical properties can be included here, such as fractal structures. As this field is in development, further exciting discoveries will certainly be made in both mathematical theory and physical applications, particularly in quantum gravity, complex systems, and interdisciplinary areas where traditional integer-order models do not provide a sufficient description of the corresponding phenomena.

For researchers new to the field of fractional calculus, we recommend starting with the Caputo derivative for physical applications involving initial value problems. Also, we suggest being mindful of the mathematical properties and physical interpretations specific to each fractional operator. The references provided throughout this review offer starting points for deeper exploration of specific topics.