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Fractional calculus

Fractional calculus is a branch of mathematical analysis that generalizes the operations of differentiation and integration to arbitrary non-integer orders, extending the classical concepts of integer-order calculus to real or complex values, thereby providing tools to model phenomena with memory effects, non-locality, and fractal behaviors.^[1] This field enables the description of systems where traditional integer-order derivatives fail to capture intermediate dynamics, such as in anomalous diffusion or viscoelastic materials.^[2] The origins of fractional calculus trace back to 1695, when Gottfried Wilhelm Leibniz and Guillaume de l'Hôpital corresponded on the meaning of a half-derivative, marking the first formal inquiry into non-integer orders shortly after the invention of classical calculus.^[1] Systematic development began in the 19th century with contributions from mathematicians like Joseph Liouville, who introduced fractional integrals via Fourier series in the 1830s, and Bernhard Riemann, who formalized the Riemann integral for fractional orders around 1854.^[2] Later, key formulations emerged, including the Riemann-Liouville fractional derivative (developed by Bernhard Riemann and Joseph Liouville in the mid-19th century) and the Caputo derivative (proposed by Michele Caputo in 1967), which differ in their treatment of initial conditions and are widely used for physical applications due to their alignment with integer-order behaviors at integer limits.^[1] Other definitions, such as Grünwald-Letnikov and Cauchy, provide alternative discrete and continuous approaches, often equivalent under certain conditions.^[1] In modern contexts, fractional calculus has gained prominence for modeling complex systems across disciplines, including engineering control theory—where fractional-order PID controllers like PI^αD^μ offer enhanced robustness and performance with fewer parameters than integer-order counterparts—and physics, particularly in describing anomalous diffusion, viscoelasticity, and heat transfer processes that exhibit long-range memory.^[1]^[2] Applications extend to bioengineering for tissue modeling, robotics for precise motion control, and even computer vision for image enhancement and denoising, leveraging the non-local properties of fractional operators to handle fractal-like structures in data.^[1]^[3] Recent advancements, such as numerical methods for solving fractional differential equations and tools like the CRONE toolbox, have facilitated practical implementations, underscoring fractional calculus's role in addressing real-world problems beyond classical limits.^[2]

Introduction

Definition and motivation

Fractional calculus generalizes the operations of differentiation and integration from integer orders to arbitrary non-integer orders, unifying them into a single framework known as the fractional derivative or integral of order \alpha \in \mathbb{R}.^[4] In the Riemann–Liouville approach, the fractional derivative of order \alpha > 0 is constructed by applying an integer-order derivative to a fractional integral of order n - \alpha, where n = \lceil \alpha \rceil, thereby extending classical calculus to model processes requiring non-local effects.^[5] The field's motivation originated in a 1695 correspondence between Gottfried Wilhelm Leibniz and Guillaume de l'Hôpital, where Leibniz pondered the meaning of a half-derivative (d^{1/2}/dx^{1/2}) in response to l'Hôpital's query about generalizing integer-order derivatives.^[6] In modern contexts, fractional calculus addresses limitations of integer-order models by capturing non-local, memory-dependent behaviors in real-world phenomena, such as anomalous diffusion where particle spread deviates from Gaussian patterns and viscoelasticity where materials exhibit time-dependent stress-strain relations influenced by long-range memory. These properties arise because fractional operators inherently incorporate hereditary effects over the entire past history, unlike local integer derivatives.^[7] Fractional orders better represent infinite-dimensional state spaces than integer orders, as they model distributed parameter systems with an infinite continuum of relaxation times or modes, leading to power-law dynamics that integer models cannot replicate without infinite parameters. For instance, the Riemann–Liouville fractional derivative of order \alpha applied to the power function x^\beta (with \beta > -1) yields \frac{\Gamma(\beta + 1)}{\Gamma(\beta - \alpha + 1)} x^{\beta - \alpha}, which is proportional to x^{\beta - \alpha} and preserves the power-law form while adjusting the exponent non-locally.^[9] A foundational operator in this framework is the fractional integral, defined for $0 < \alpha < 1 as

_{a}I_{t}^{\alpha} f(t) = \frac{1}{\Gamma(\alpha)} \int_{a}^{t} (t - \tau)^{\alpha - 1} f(\tau) \, d\tau,

which generalizes repeated integration and serves as the basis for deriving fractional derivatives.^[5]

Basic concepts and notation

The Gamma function, denoted \Gamma(z), serves as a fundamental extension of the factorial to real and complex numbers, satisfying \Gamma(n+1) = n! for positive integers n and defined for \Re(z) > 0 by the integral \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt.^[10] In fractional calculus, the Gamma function plays a crucial role in normalizing fractional integrals and derivatives, ensuring consistency with integer-order operations by incorporating the factor $1/\Gamma(\alpha) in definitions involving non-integer orders \alpha, which generalizes the denominators in repeated integration formulas.^[11] Prerequisite concepts for fractional calculus build on classical calculus, where integer-order integration can be viewed as a convolution with power-law kernels, such as the operation I^n f(t) = \frac{1}{(n-1)!} \int_a^t (t - \tau)^{n-1} f(\tau) \, d\tau for positive integer n. Fractional integrals extend this framework and satisfy the semigroup property: for orders \alpha > 0 and \beta > 0, the composition of operators yields I^\alpha I^\beta = I^{\alpha + \beta}, reflecting the additive nature of orders in successive applications.^[12] This property holds for standard fractional integral operators on suitable function spaces, such as L^1[0, T], and underscores the algebraic structure underlying fractional operators.^[13] Standard notation in fractional calculus distinguishes between left-sided and right-sided operators to account for the direction of the operation relative to the variable t. The left-sided fractional derivative is commonly denoted as _{a}D_{t}^{\alpha} f(t), where a is the lower limit, t the upper limit, and \alpha the order, while the right-sided counterpart is _{t}D_{b}^{\alpha} f(t) with upper limit b. Positive orders \alpha > 0 typically correspond to differentiation or integration depending on the context, whereas negative orders -\alpha with \alpha > 0 invert the operation, mapping derivatives to integrals and vice versa.^[14] A key tool for series expansions in fractional calculus is the generalization of the binomial theorem, which expands (1 - x)^{-\alpha} for |x| < 1 and \alpha \in \mathbb{R} as

(1 - x)^{-\alpha} = \sum_{k=0}^{\infty} \frac{\Gamma(k + \alpha)}{\Gamma(\alpha) \, k!} x^k.

This series, derived from the Taylor expansion of the binomial function, facilitates the representation of fractional operators through infinite sums and is instrumental in deriving approximations and solutions to fractional equations.^[15]

History

Early origins and precursors

The origins of fractional calculus can be traced to a famous exchange of letters between Gottfried Wilhelm Leibniz and Guillaume de l'Hôpital in 1695. On September 30, 1695, l'Hôpital wrote to Leibniz inquiring about the meaning of the notation for a fractional derivative, specifically asking, "What will be the sense of the fraction ½ in the expression d^{½}y/dx^{½}?" Leibniz responded on October 15, 1695, acknowledging the apparent paradox but suggesting that "an apparent paradox nowadays will be tomorrow a theorem" and that it could lead to useful consequences. This correspondence marked the first explicit consideration of generalizing differentiation to non-integer orders, posing a conceptual puzzle that intrigued mathematicians for centuries.^[16] In the 18th century, precursors to fractional calculus emerged through explorations of series expansions and integral representations that implicitly handled fractional powers. Leonhard Euler, in his 1748 treatise Introductio in analysin infinitorum, developed the generalized binomial theorem for arbitrary exponents, including fractions, providing a series-based framework for functions like (1 + x)^α where α is non-integer; this work hinted at the analytical extension of differentiation and integration beyond integers without formal fractional operators. Similarly, Pierre-Simon Laplace's investigations into integral transforms during the late 1770s and early 1780s, particularly in his studies of generating functions and probability, incorporated expressions with fractional powers through gamma function-like integrals, foreshadowing the representation of fractional integrals as convolutions. These contributions, while not directly addressing fractional derivatives, expanded the toolkit for handling non-integer orders in analytical contexts.^[16] Early informal attempts at fractional differentiation appeared in Joseph Fourier's work on heat conduction between 1811 and 1822. In his prize memoir of 1811 and the subsequent Théorie analytique de la chaleur (1822), Fourier employed Fourier series expansions to solve the heat equation, where term-by-term differentiation of series implicitly invoked operations akin to fractional differentiation for modeling diffusion processes; however, these were heuristic applications without rigorous definitions or general theory. Such approaches demonstrated practical utility in physical problems but remained ad hoc. Despite these pioneering ideas, no systematic theory of fractional calculus existed until the 19th century, as the scattered insights primarily served as motivational puzzles for later formal developments.^[16]

Key developments in the 19th and 20th centuries

In 1823, Niels Henrik Abel solved a generalization of the tautochrone problem by employing fractional integration of order 1/2, thereby providing one of the earliest practical applications and a foundational framework for non-integer order operators.^[17] This work demonstrated how fractional calculus could address problems in mechanics, marking a shift from conceptual curiosity to analytical tool.^[17] Building on this, Joseph Liouville introduced systematic methods for fractional integration in 1832 by generalizing Cauchy's formula for repeated integrals to arbitrary orders, establishing a rigorous basis for handling non-integer powers in analysis.^[18] Liouville's approach emphasized the extension of classical calculus to fractional domains, influencing subsequent developments in integral equations.^[18] In 1853, Bernhard Riemann advanced the theory through his work on differintegrals, proposing definitions that integrated fractional differentiation and integration, which later contributed to the formulation of the Riemann–Liouville operator.^[19] Independently, in 1868, Aleksey Letnikov developed a discrete approximation to fractional derivatives, bridging continuous and finite difference methods for practical computation.^[20] In the 20th century, Hermann Weyl introduced a conformal fractional derivative in 1917, tailored for periodic functions via Fourier series and applied in the context of quantum mechanics to model nonlocal effects.^[21] This operator, defined for functions with zero mean, offered a weaker, integral-based alternative to earlier definitions, highlighting the versatility of fractional tools in physics.^[21] Around the same period, E. L. Post explored semigroup properties of fractional operators in the 1930s, examining their composition and continuity to unify integration and differentiation under algebraic structures.^[22] A pivotal event occurred in 1974 with the first international conference on fractional calculus, organized by Bertram Ross at the University of New Haven, which formalized the field and featured contributions from key researchers including A. C. McBride and K. B. Oldham, fostering interdisciplinary dialogue.^[23] The proceedings compiled seminal works, solidifying fractional calculus as a mature mathematical discipline.^[23]

Modern extensions and applications

In the 21st century, fractional calculus has experienced rapid advancements through the introduction of new operators designed to address limitations in classical definitions, particularly regarding singular kernels that can complicate modeling of real-world phenomena with smooth memory effects. A notable development is the Caputo–Fabrizio fractional derivative, proposed in 2015, which employs an exponential non-singular kernel to provide a more accurate representation of processes exhibiting non-local and non-singular memory, such as viscoelastic materials and anomalous diffusion. This operator mitigates the singularity issues inherent in the Riemann–Liouville and Caputo derivatives, enabling better fits to empirical data in physical systems. Building on this, the Atangana–Baleanu fractional derivative, introduced in 2016, incorporates a Mittag-Leffler function in its non-singular kernel, offering improved flexibility for capturing both local and non-local behaviors in dynamical systems like heat transfer and biological models.^[24] Variable-order fractional calculus represents another key extension, allowing the fractional order to vary with time or space, which is particularly suited for modeling systems with evolving memory effects, such as aging materials or adaptive biological processes. This approach builds on earlier foundations but provides greater adaptability for applications requiring dynamic order transitions, without delving into singular behaviors.^[25] Computational methods have also proliferated to handle the increased complexity of these modern operators, with numerical schemes like the fractional Adams–Bashforth method gaining prominence for solving fractional differential equations efficiently. Originating from predictor-corrector adaptations of classical multistep methods, this scheme approximates solutions with high accuracy for nonlinear problems, as demonstrated in early implementations for initial value problems.^[26] The development of software tools, such as the FOMCON MATLAB toolbox, has further democratized these computations by providing built-in functions for fractional-order system identification, simulation, and control design, supporting both commensurate and incommensurate orders.^[27] The surge in research activity underscores the field's maturity, with over 10,000 publications on fractional calculus appearing since 2010, reflecting exponential growth driven by interdisciplinary applications.^[28] This expansion has outpaced coverage in general references, highlighting recent innovations like the Prabhakar kernel, introduced in 2018 as a three-parameter generalization of the Mittag-Leffler function, which enables more nuanced modeling of multi-scale relaxation processes in viscoelasticity and beyond. Since 2019, further advancements have included novel non-singular kernels and applications in optimization algorithms, quantum mechanics, and machine learning, continuing to broaden the field's impact as of 2025.^[29]

Fractional Integrals

Riemann–Liouville fractional integral

The Riemann–Liouville fractional integral arises as a natural extension of the Cauchy formula for repeated integer-order integration to arbitrary positive real orders. For a positive integer n, the n-fold integral of a function f from a lower limit a to t > a is expressed as

I_a^n f(t) = \frac{1}{(n-1)!} \int_a^t (t - s)^{n-1} f(s) \, ds,

which represents the iterated application of the standard Cauchy integral operator. To generalize this to a non-integer order \alpha > 0, the factorial (n-1)! is replaced by the Gamma function \Gamma(\alpha), and the exponent n-1 by \alpha - 1, yielding the fractional analogue. This extension was first proposed by Bernhard Riemann in 1847 and later refined by Joseph Liouville, forming the foundation for modern fractional calculus operators. The standard definition of the left-sided Riemann–Liouville fractional integral of order \alpha with lower limit a is

{}_a I_t^\alpha f(t) = \frac{1}{\Gamma(\alpha)} \int_a^t (t - s)^{\alpha - 1} f(s) \, ds,

where \operatorname{Re}(\alpha) > 0, f is integrable on [a, t], and \Gamma denotes the Gamma function. This operator is well-defined for sufficiently regular functions, such as continuous or Lebesgue integrable ones, and reduces to the integer case when \alpha = n. The integral is interpreted as an improper Riemann integral when necessary, particularly near the singularity at s = t. A key property of the Riemann–Liouville fractional integral is its semigroup composition rule: for \alpha > 0 and \beta > 0,

{}_a I_t^\alpha \left( {}_a I_t^\beta f(t) \right) = {}_a I_t^{\alpha + \beta} f(t),

which mirrors the additivity of integration orders in the classical setting and facilitates the analysis of fractional differential equations. Additionally, the operator is invertible by integer-order differentiation: if m is a positive integer and \alpha = m, then the m-th ordinary derivative satisfies

\frac{d^m}{dt^m} \left( {}_a I_t^m f(t) \right) = f(t),

assuming f is sufficiently differentiable and satisfies appropriate boundary conditions at t = a. These properties underscore the integral's role as a building block for fractional calculus. Illustrative examples highlight the operator's behavior. For the constant function f(t) = c with lower limit a = 0,

{}_0 I_t^\alpha c = c \frac{t^\alpha}{\Gamma(\alpha + 1)},

demonstrating that constants are mapped to power functions scaled by the Gamma factor. Similarly, for the power function f(t) = t^\beta with \beta > -1 and a = 0,

{}_0 I_t^\alpha t^\beta = \frac{\Gamma(\beta + 1)}{\Gamma(\alpha + \beta + 1)} t^{\alpha + \beta},

which generalizes the classical integration rule \int_0^t s^\beta \, ds = \frac{t^{\beta + 1}}{\beta + 1} and reveals the operator's effect on monomials through Gamma function ratios. These computations are particularly useful in solving fractional integral equations and verifying theoretical results. Despite its foundational importance, the Riemann–Liouville fractional integral features a singular kernel (t - s)^{\alpha - 1} at s = t when \operatorname{[Re](/page/Re)}(\alpha) \leq 1, which introduces non-local dependencies and can complicate convergence or numerical evaluation for certain functions. This singularity contributes to limitations in applications, notably that constant functions yield non-zero results under subsequent fractional differentiation, diverging from classical expectations where derivatives of constants vanish.

Hadamard and other variants

The Hadamard fractional integral, introduced by Jacques Hadamard in 1892, represents a variant of fractional integration that employs a logarithmic kernel, distinguishing it from the power-law kernel of the Riemann–Liouville integral.^[30] Defined for a function f on [a, b] with $0 < a < b < \infty and order \alpha > 0, it is given by

{}_{a}I^{\alpha} f(t) = \frac{1}{\Gamma(\alpha)} \int_{a}^{t} \left( \ln \frac{t}{s} \right)^{\alpha-1} \frac{f(s)}{s} \, ds, \quad t \in [a, b].

This operator is particularly suitable for modeling multiplicative and scale-invariant processes, such as those arising in quantum mechanics involving generalized coherent states.^[31] A key property of the Hadamard integral is illustrated by its action on the constant function: for f(t) = 1 with lower limit a = 1, it yields \frac{ (\ln t )^{\alpha} }{\Gamma(\alpha + 1)}, which introduces a logarithmic factor absent in the Riemann–Liouville case. This reflects the kernel's role in capturing logarithmic scaling behaviors. In contrast to the Riemann–Liouville power-law kernel (t - s)^{\alpha - 1}, the Hadamard logarithmic kernel \left( \ln (t/s) \right)^{\alpha - 1}/s better accommodates scale-invariant phenomena, such as fractal structures or systems with multiplicative noise, where translations in logarithmic scale preserve the form of solutions.^[32]^[33] Other variants extend fractional integration to weighted or multi-dimensional settings. The Erdélyi–Kober integral serves as a weighted generalization of the Riemann–Liouville operator, incorporating parameters for flexibility in representing memory effects; for parameters \alpha > 0, \sigma > 0, \mu \in \mathbb{R}, it is defined as

I^{\alpha ; \sigma, \mu }_{0+} f(x) = \frac{ x ^{-\mu \sigma} }{\Gamma(\alpha)} \int_0^x (x^\sigma - t^\sigma )^{\alpha -1 } t^{\sigma (\mu + \alpha ) -1 } f(t ) \, dt.

^[19] This form allows for tailored weighting in applications like anomalous diffusion. For multi-dimensional cases, the Riesz potential provides a symmetric fractional integral over \mathbb{R}^n, defined for order $0 < \alpha < n as

I^{\alpha} f(\mathbf{x}) = \frac{1}{\gamma_n(\alpha)} \int_{\mathbb{R}^n} \frac{f(\mathbf{y})}{|\mathbf{x} - \mathbf{y}|^{n - \alpha}} \, d\mathbf{y},

where \gamma_n(\alpha) = 2^{\alpha} \pi^{n/2} \frac{\Gamma(\alpha/2)}{\Gamma((n - \alpha)/2)} is the normalization constant; it is essential for isotropic problems in potential theory and higher-dimensional fractional models.^[34]

Fractional Derivatives

Grünwald–Letnikov and Riemann–Liouville derivatives

The Grünwald–Letnikov fractional derivative provides a discrete approximation to the concept of fractional differentiation, extending the finite difference formula to non-integer orders. For a function f sufficiently smooth on [a, t] and order \alpha > 0, the left-sided Grünwald–Letnikov derivative is defined as

_{a}D_{t}^{\alpha} f(t) = \lim_{h \to 0^+} h^{-\alpha} \sum_{k=0}^{\lfloor (t-a)/h \rfloor} (-1)^k \binom{\alpha}{k} f(t - k h),

where the generalized binomial coefficient is given by

\binom{\alpha}{k} = \frac{\Gamma(\alpha+1)}{\Gamma(k+1) \Gamma(\alpha - k + 1)}.

This formulation arises from generalizing the backward difference operator to fractional powers, capturing the non-local nature of fractional orders through an infinite series in the limit. The definition was originally proposed by A. K. Grünwald in his 1867 work on bounded derivations and independently developed by A. V. Letnikov in 1868 as part of a theory for differentiation of arbitrary index.^[35]^[36] The Riemann–Liouville fractional derivative builds on the fractional integral to define differentiation for non-integer orders, offering an integral representation that aligns with classical calculus for integer values. For \alpha > 0, let n = \lceil \alpha \rceil; the left-sided Riemann–Liouville derivative of f is

_{a}D_{t}^{\alpha} f(t) = \left( \frac{d^n}{dt^n} \, _{a}I_{t}^{n - \alpha} f(t) \right),

where _{a}I_{t}^{\beta} f(t) denotes the fractional integral of order \beta = n - \alpha > 0,

_{a}I_{t}^{\beta} f(t) = \frac{1}{\Gamma(\beta)} \int_{a}^{t} (t - s)^{\beta - 1} f(s) \, ds.

This operator was formalized through contributions from Joseph Liouville in his 1832 studies on fractional integration and Bernhard Riemann's 1847 dissertation on the foundations of integration, with the combined form later standardized as the Riemann–Liouville derivative.^[5]^[37] A distinguishing feature of the Riemann–Liouville derivative is that it does not annihilate constants, unlike integer-order derivatives. For the constant function f(t) = 1 with lower limit a = 0 and $0 < \alpha < 1,

D_{t}^{\alpha} 1 = \frac{t^{-\alpha}}{\Gamma(1 - \alpha)}.

This non-zero result reflects the operator's singular kernel and memory-dependent structure, leading to _{a}D_{t}^{\alpha} c = c \frac{(t - a)^{-\alpha}}{\Gamma(1 - \alpha)} for any constant c.^[37]^[5] Under appropriate regularity conditions—such as f being n-times continuously differentiable on [a, t] with \alpha \in (n-1, n]—the Grünwald–Letnikov and Riemann–Liouville derivatives coincide, i.e.,

_{a}D_{t}^{\alpha} f(t) = \, _{a}D_{t}^{\alpha} f(t)

. The equivalence follows from expressing the [Grünwald–Letnikov](/page/ Fractional_calculus#Grünwald–Letnikov_derivative) sum as a Riemann sum that converges to the fractional integral in the Riemann–Liouville definition; specifically, the discrete weights (-1)^k \binom{\alpha}{k} h^{\alpha} approximate the kernel (t - s)^{-\alpha}/\Gamma(1 - \alpha) as h \to 0^+, with the integer differentiation applied afterward yielding the same limit. This connection bridges discrete and continuous formulations, enabling numerical implementations via Grünwald–Letnikov approximations for the analytically defined Riemann–Liouville operator.^[38]^[39]

Caputo and Caputo–Fabrizio derivatives

The Caputo fractional derivative, introduced by Michele Caputo in 1967 as a reformulation suited for modeling dissipation in geophysical contexts, addresses limitations of the Riemann–Liouville derivative by prioritizing physical interpretability in initial value problems. Unlike its precursor, the Riemann–Liouville operator, the Caputo derivative applies the fractional integral after the integer-order differentiation, yielding zero for constant functions and polynomials of degree less than the ceiling of the fractional order. This property ensures that initial conditions correspond directly to classical integer-order derivatives, facilitating applications in physics where traditional boundary values are prescribed. For a function f(t) sufficiently differentiable and n-1 < \alpha < n where n is a positive integer, the left-sided Caputo derivative from lower limit a is defined as

{}^{\mathrm{C}}D_a^{\alpha} f(t) = I_a^{n-\alpha} \frac{d^n f(t)}{dt^n} = \frac{1}{\Gamma(n-\alpha)} \int_a^t (t-s)^{n-\alpha-1} f^{(n)}(s) \, ds,

with I_a^{n-\alpha} denoting the Riemann–Liouville fractional integral of order n-\alpha.^[40] This form guarantees that the derivative vanishes for polynomials of degree at most n-1, as the nth derivative is zero. For instance, the Caputo derivative of the power function t^\mu with \mu \geq n-1 simplifies to

{}^{\mathrm{C}}D_a^{\alpha} t^{\mu} = \frac{\Gamma(\mu + 1)}{\Gamma(\mu - \alpha + 1)} t^{\mu - \alpha},

mirroring the behavior of the Riemann–Liouville derivative for such functions while maintaining the desired initial condition properties.^[41] A key advantage of the Caputo derivative over the Riemann–Liouville operator lies in its handling of initial value problems, particularly in viscoelasticity and diffusion models, where the fractional derivative at the initial time aligns with observable integer-order rates rather than fractional integrals of initial data.^[42] This makes it preferable for physical systems requiring straightforward specification of initial velocities or positions without additional fractional terms.^[43] The Caputo–Fabrizio derivative, proposed by Caputo and Fabrizio in 2015, extends this framework by incorporating a non-singular exponential kernel to mitigate the singularity at t = s present in power-law kernels, yielding smoother representations for time-dependent processes. For $0 < \alpha < 1, the derivative is given by

{}^{\mathrm{CF}}D_a^{\alpha} f(t) = \frac{M(\alpha)}{1 - \alpha} \int_a^t f'(s) \exp\left( -\frac{\alpha (t - s)}{1 - \alpha} \right) ds,

where M(\alpha) is a normalization function, often taken as \alpha, ensuring the kernel decays exponentially and avoids infinite values. This formulation proves advantageous for modeling dielectrics and viscoelastic materials, where the exponential decay better captures relaxation behaviors without the artifacts of singular kernels.^[40] Like the standard Caputo derivative, it supports physically meaningful initial conditions based on f'(0), enhancing its utility in non-local memory effects.

Advanced non-local derivatives with non-singular kernels and variable-order derivatives

Other non-local fractional derivatives extend traditional models by incorporating kernels that capture memory effects without the singularities present in power-law kernels, such as those in Riemann-Liouville or Caputo operators. These derivatives are particularly useful for modeling complex dynamics in systems where historical dependencies decay more smoothly, like in viscoelastic materials or anomalous diffusion processes. The Atangana–Baleanu derivative, introduced in 2016, exemplifies this approach by employing a Mittag-Leffler kernel to ensure non-locality and non-singularity.^[44] The Atangana–Baleanu derivative in the Caputo sense is defined as

{}^{\mathrm{AB}}D^{\alpha}_{a} f(t) = \frac{B(\alpha)}{1 - \alpha} \int_{a}^{t} f'(s) E_{\alpha} \left( -\frac{\alpha (t - s)^{\alpha}}{1 - \alpha} \right) \, ds,

where $0 < \alpha < 1, E_{\alpha}(\cdot) is the one-parameter Mittag-Leffler function, and B(\alpha) = 1 - \alpha + \frac{\alpha}{\Gamma(1+\alpha)} serves as a normalization constant. This operator addresses limitations of singular kernels by providing a more physically realistic description of memory in time-fractional models, such as heat transfer and chaotic systems.^[44] Variable-order fractional derivatives generalize fixed-order operators by allowing the fractional order \alpha to vary as a function of time or space, denoted \alpha = \alpha(t) or \alpha = \alpha(x), enabling adaptation to evolving system behaviors. This flexibility is crucial for applications in control systems and mechanics, where the degree of non-locality changes dynamically. The Coimbra derivative, proposed in 2003, is a Caputo-based variable-order operator particularly suited for modeling such time-varying fractional dynamics in engineering contexts like viscoelastic damping.^[45] The Riesz fractional derivative provides a symmetric, space-fractional analogue to the Laplacian, defined in Fourier space as \mathcal{F}\{ {}^{R}D^{\alpha} f(x) \} = -|k|^{\alpha} \hat{f}(k) for $0 < \alpha \leq 2, or equivalently in physical space as the operator - (-\Delta)^{\alpha/2} f. This form captures isotropic anomalous diffusion and is widely used in space-fractional quantum mechanics and porous medium equations due to its conservative and non-local properties.^[46] Post-2015 developments have further enriched non-local frameworks, including the Prabhakar derivative, which incorporates a three-parameter kernel based on the Prabhakar function E_{\rho,\beta}^{\gamma}(\lambda z) to model multi-scale relaxation phenomena in dielectrics and diffusion.

Properties and Generalizations

Fundamental properties

Fractional integrals satisfy the semigroup property, which states that the composition of two fractional integrals of orders \alpha > 0 and \beta > 0 yields a fractional integral of order \alpha + \beta: I^\alpha I^\beta f(t) = I^{\alpha + \beta} f(t).^[47] This property holds for the Riemann–Liouville fractional integral and follows from the convolution structure underlying the operator, as verified through integral representations involving the Gamma function.^[47] In contrast, fractional derivatives do not generally satisfy a full semigroup property D^\alpha D^\beta f(t) = D^{\alpha + \beta} f(t) without additional conditions, such as commutativity of the operators or specific smoothness assumptions on the function f. Partial semigroup properties can be established for families of Riemann–Liouville and Caputo derivatives under restricted orders, like $0 < \beta < \alpha < 1, but the general case requires careful handling of initial conditions and boundary effects to ensure the equality holds. The Laplace transform provides a powerful tool for analyzing fractional operators, simplifying their action in the frequency domain. For the Riemann–Liouville fractional integral starting at zero, the transform is given by

\mathcal{L}\{ _{0}I^{\alpha} f(t) \}(s) = s^{-\alpha} F(s),

where F(s) = \mathcal{L}\{f(t)\}(s) and \alpha > 0, assuming zero initial conditions.^[48] This formula arises from the convolution theorem, as the kernel (t - \tau)^{\alpha - 1}/\Gamma(\alpha) has a known Laplace transform s^{-\alpha}.^[48] For the Riemann–Liouville fractional derivative of order \alpha with m-1 < \alpha \leq m and m \in \mathbb{N}, the Laplace transform is

\mathcal{L}\{ _{0}D^{\alpha} f(t) \}(s) = s^{\alpha} F(s) - \sum_{k=0}^{m-1} s^{k} \, _{0}I^{m - \alpha} f^{(m-1-k)}(0^+),

where the sum accounts for initial values involving fractional integrals of integer derivatives at the origin.^[48] These initial terms reflect the non-local nature of the derivative, distinguishing it from integer-order cases.^[48] A key representational tool in fractional calculus is the fractional Taylor series expansion, which generalizes the classical Taylor series to non-integer powers. For a suitable function f, it takes the form

f(t) = \sum_{k=0}^{\infty} \frac{ (D^{\alpha})^k f(0) }{\Gamma(k \alpha + 1)} t^{k \alpha},

where D^\alpha denotes a fractional derivative operator of order \alpha > 0, valid under analyticity conditions in appropriate domains.^[49] This series leverages the semigroup-like composition of fractional derivatives and converges for functions analytic in a sector, providing a basis for approximation and solution methods in fractional differential equations.^[49] Existence and uniqueness of solutions to fractional differential equations are often established using fixed-point theorems in Banach spaces. For initial value problems of the form D^\alpha x(t) = g(t, x(t)) with appropriate initial conditions, the Banach contraction mapping principle guarantees a unique solution when g is Lipschitz continuous, by converting the equation to an equivalent integral equation and showing the operator is a contraction on a suitable ball.^[50] Krasnoselskii's fixed-point theorem extends this to nonlinear cases where compactness arguments ensure existence, even if uniqueness requires additional Lipschitz conditions.^[50] These results hold for orders $0 < \alpha < 1 and generalize to higher orders via iterative applications.^[50]

Erdélyi–Kober and other operators

The Erdélyi–Kober operator is a generalized form of the fractional integral that introduces weighting through additional parameters, extending the standard Riemann–Liouville framework to handle non-uniform distributions in the kernel. This operator was independently introduced by Hermann Kober and Arthur Erdélyi in 1940 as a tool for fractional integration linked to Hankel transforms and Mellin convolutions. Unlike the Riemann–Liouville integral, which uses a power-law kernel without extra weighting, the Erdélyi–Kober form incorporates density-like functions via parameters \delta > 0 and \beta, enabling applications in models requiring variable pathway weighting, such as anomalous diffusion or viscoelastic pathways. The left-sided Erdélyi–Kober fractional integral operator of order \gamma > 0 is defined as

\Omega_{0+}^{\gamma, \delta; \beta} f(t) = \frac{t^{-\gamma \delta - \beta}}{\Gamma(\gamma)} \int_{0}^{t} s^{\gamma \delta + \beta - 1} \left(\frac{t}{s}\right)^{\gamma} f(s) \frac{ds}{s},

for suitable functions f on (0, \infty), where \Gamma denotes the gamma function. Corresponding fractional derivatives are obtained by inverting the integral operator or composing with integer-order derivatives, preserving semigroup properties under composition. These operators facilitate solving generalized Abel-type integral equations of the form \Omega^{\gamma, \delta; \beta} f(t) = g(t), where uniqueness of solutions follows from the invertibility established via Mellin transform techniques, which diagonalize the convolution structure.^[51] In pathway models, such as those describing non-local transport in heterogeneous media, the weighting parameters adjust for spatial or temporal variations, providing more flexible representations than uniform kernels. Other notable operators in this category include the Marchaud fractional derivative, introduced by André Marchaud in 1927 as a right-sided counterpart to the left-sided Grünwald–Letnikov derivative, particularly useful for functions on the positive real line with tail behavior. The Marchaud derivative of order \alpha \in (0,1) is given by a principal value integral emphasizing the forward direction:

D_{t+}^\alpha f(t) = \frac{\alpha}{\Gamma(1-\alpha)} \int_0^t \frac{f(t) - f(t-u)}{u^{1+\alpha}} du + \frac{1}{\Gamma(-\alpha)} \int_t^\infty \frac{f(t) - f(u)}{(u-t)^{1+\alpha}} du,

capturing non-local effects in forward time evolution.^[52] Complementing this, the Hilfer derivative, proposed by Rudolf Hilfer in 2000, serves as a hybrid between the Riemann–Liouville and Caputo derivatives, parameterized by a type index \beta \in [0,1] to interpolate their behaviors. Defined as D^{\alpha,\beta} f(t) = I^{\beta(1-\alpha)} \, {}^C D^\alpha \, I^{(1-\beta)(1-\alpha)} f(t), where I^\mu is the Riemann–Liouville integral of order \mu and {}^C D^\alpha is the Caputo derivative, it accommodates initial conditions mixing integer-order derivatives of fractional powers, enhancing solvability in hybrid physical models. These operators collectively extend fractional calculus to weighted and directional settings, underpinning analyses in generalized Abel equations and beyond.

Functional calculus approaches

Functional calculus approaches provide abstract frameworks for defining fractional powers of linear operators in Banach spaces, particularly those that are unbounded or sectorial, extending classical notions from integer powers to non-integer exponents. These methods rely on the spectral theory and resolvent estimates rather than direct integral representations, making them suitable for operators arising in evolution equations and partial differential equations. By leveraging the holomorphic functional calculus, one can define A^\alpha for a sectorial operator A with appropriate spectrum, ensuring well-posedness in appropriate domains.^[53] A key construction is the Balakrishnan formula, which defines fractional powers for closed linear operators satisfying certain resolvent bounds. For a closed operator A in a Banach space X such that \lambda > 0 is in the resolvent set for all \lambda > 0 and \|\lambda R(\lambda, A)\| \leq M < \infty (where R(\lambda, A) = (\lambda I - A)^{-1}), the fractional power A^\alpha for $0 < \Re \alpha < 1 and x \in D(A) is given by

A^\alpha x = \frac{\sin(\pi \alpha)}{\pi} \int_0^\infty \lambda^{\alpha - 1} R(\lambda, A) (-A) x \, d\lambda.

This integral representation converges in the norm of X and yields a closed operator with domain D(A^\alpha). For higher orders, n-1 < \Re \alpha < n, it extends recursively as A^\alpha = A^{\alpha - n + 1} A^{n-1}. Such operators are sectorial, facilitating the generation of analytic semigroups.^[53] The Dunford–Schwartz functional calculus offers a more general framework for defining f(A) for holomorphic functions f on a suitable sector containing the spectrum of A, including f(s) = s^\alpha for fractional powers. For a bounded operator A on a Banach space, this is achieved via the Cauchy integral formula over a contour enclosing the spectrum: f(A) x = \frac{1}{2\pi i} \int_\Gamma f(\lambda) R(\lambda, A) x \, d\lambda, where \Gamma avoids the spectrum. This extends to unbounded sectorial operators by restricting to the holomorphic functional calculus, ensuring A^\alpha is densely defined and closed when \alpha > 0. The approach unifies polynomial and rational approximations with spectral projections, providing continuity estimates in operator norms. In the context of semigroup theory, fractional powers of generators model anomalous transport phenomena, such as subdiffusion in heterogeneous media. For a generator A of a C_0-semigroup, A^\alpha (with $0 < \alpha < 1) generates a subordinate semigroup capturing non-local effects, where the transition probabilities exhibit heavy tails leading to anomalous diffusion. This resolvent-based definition avoids explicit fractional integrals, proving essential for unbounded operators in Hilbert spaces, and has been applied to fractional Laplacians in kinetic theory.

Applications

Physical and engineering models

Fractional calculus provides a powerful framework for modeling physical and engineering systems that exhibit memory-dependent behaviors, such as hereditary effects in materials and processes where the response at a given time depends on the entire history of inputs.^[54] In mechanics and engineering, fractional derivatives capture non-local and anomalous dissipation more accurately than integer-order models, particularly for phenomena involving power-law relaxation or frequency-dependent responses. These models are widely applied in viscoelastic materials, control systems, structural dynamics, and electrochemical interfaces, enhancing predictive capabilities for real-world systems. In viscoelasticity, the fractional Kelvin-Voigt model extends the classical parallel spring-dashpot configuration by incorporating a fractional derivative to describe the viscous component, effectively capturing power-law creep and relaxation behaviors observed in polymers and biological tissues.^[54] The stress-strain relation in this model is given by

\sigma(t) = E \, D^\alpha \varepsilon(t),

where \sigma(t) is the stress, \varepsilon(t) is the strain, E is a material constant, D^\alpha denotes the Caputo fractional derivative of order \alpha (with $0 < \alpha < 1), and the Caputo form ensures compatibility with physical initial conditions by aligning with integer-order derivatives at boundaries.^[54] This formulation arises from molecular theories linking microscopic chain dynamics to macroscopic hereditary responses, providing a thermodynamic-consistent basis for non-exponential decay in viscoelastic media.^[54] The model's ability to fit experimental creep data over wide time scales with fewer parameters than multi-exponential models has made it seminal for applications in damping layers and soft matter composites.^[54] Fractional-order PID controllers generalize the classical proportional-integral-derivative (PID) structure to PI^\lambda D^\mu controllers, offering enhanced robustness and flexibility in tuning for systems with irregular dynamics, such as those in process control and robotics.^[55] The transfer function is

C(s) = K_p + \frac{K_i}{s^\lambda} + K_d s^\mu,

where K_p, K_i, and K_d are gains, and $0 < \lambda, \mu < 2 allow isodamping contours in the Nyquist plane for improved phase margin specifications.^[55] This design achieves flat-phase response over a wider frequency band compared to integer-order PID, enabling better disturbance rejection and set-point tracking in uncertain environments.^[55] Seminal analyses show that optimal tuning via magnitude and phase specifications reduces overshoot in benchmark systems like the thermal process model.^[55] In structural engineering, fractional derivative models for hysteretic damping in beams replace viscous terms with non-local operators to better predict energy dissipation in viscoelastic layers, addressing limitations of classical models that fail to capture frequency-independent loss factors across broad spectra. These models describe the damping force as proportional to the fractional derivative of displacement or velocity, enabling accurate vibration analysis for composite beams under dynamic loads where experimental loss factors exhibit power-law dependence on frequency. For instance, in Euler-Bernoulli beam theory, incorporating a fractional damper term significantly improves modal damping predictions compared to viscous models for materials like epoxy-layered structures.^[56] This approach has been implemented in finite element methods, showing improved fit to experimental data for seismic isolation designs.^[56] Electrochemical analysis benefits from fractional impedance models for porous electrodes, where the constant-phase element (CPE) arises naturally from distributed resistance-capacitance networks in rough or fractal surfaces, explaining non-ideal capacitive behavior without ad hoc parameters. The impedance is modeled as Z(\omega) \propto (j\omega)^{-\alpha}, with $0 < \alpha < 1 reflecting the electrode's porosity and the penetration depth of the AC field into cylindrical pores. This transmission-line derivation, treating the electrode as a semi-infinite ladder of resistive and capacitive elements, predicts 45° linear segments in Nyquist plots at low frequencies, matching experimental spectra for electrodes like platinum black.^[57] Such models are essential for optimizing battery and supercapacitor performance, as the CPE exponent \alpha correlates with effective surface area.^[57]

Quantum and diffusion processes

Fractional calculus provides a powerful framework for modeling anomalous diffusion and quantum processes characterized by non-local space-time behaviors, such as subdiffusion and Lévy flights, where standard integer-order derivatives fail to capture long-range memory effects. In diffusion contexts, these models describe phenomena like contaminant transport in heterogeneous media, where particles exhibit slower-than-normal spreading due to trapping or fractal structures. Quantum applications extend this to wave functions evolving under fractional Laplacians, enabling descriptions of quantum tunneling and superdiffusive paths beyond classical limits.^[58] The fractional diffusion equation, given by

\partial_t^\beta u = - (-\Delta)^{\alpha/2} u,

with $0 < \beta \leq 1 and $0 < \alpha \leq 2, models subdiffusion processes where the mean squared displacement grows sublinearly with time, as opposed to the linear growth in normal diffusion. This equation arises in systems with power-law waiting times, such as porous media or biological tissues, and its fundamental solutions are expressed using Fox H-functions, which generalize Mittag-Leffler functions for complex arguments and allow closed-form representations of probability densities. For instance, the one-dimensional solution involves the H-function H_{1,3}^{2,0} to encode the anomalous scaling, facilitating analytical studies of long-time asymptotics.^[59]^[58] In environmental applications, the time-space fractional advection-dispersion equation,

\partial_t^\beta u + v \partial_x u = D \partial_x^\alpha u,

captures the transport of groundwater contaminants in fractured aquifers, where the fractional spatial derivative \partial_x^\alpha (with $1 < \alpha < 2) accounts for heavy-tailed dispersion due to scale-invariant heterogeneity, and the Caputo time derivative \partial_t^\beta incorporates temporal memory from adsorption-desorption processes. This model, pioneered in hydrological contexts, predicts non-Gaussian plume shapes with power-law tails, improving forecasts over classical Fickian advection-dispersion equations for real-world sites like the Cape Cod aquifer. Solutions often rely on Fourier-Laplace transforms, revealing dual space-time scaling that aligns with observed breakthrough curves. The Riesz derivative formulation for the Laplacian, as discussed in fractional derivative definitions, underpins the isotropic spatial operator here.^[60]^[61] Shifting to quantum mechanics, the fractional Schrödinger equation,

i \partial_t \psi = (-\Delta)^{\alpha/2} \psi + V \psi,

with $1 < \alpha < 2, describes the evolution of wave functions for particles undergoing Lévy flights, where the fractional kinetic energy operator (-\Delta)^{\alpha/2} replaces the standard Laplacian to model superdiffusive quantum paths with infinite variance. Originating from path integral formulations over Lévy processes, this equation predicts enhanced tunneling probabilities and anomalous quantum transport in disordered potentials, as verified in optical lattice experiments simulating fractional quantum walks. The Hamiltonian's self-similarity leads to scale-invariant eigenstates, distinguishing it from integer-order quantum mechanics.^[62]^[63] Recent extensions incorporate variable-order fractional derivatives, such as \alpha(t) in the Schrödinger equation, to address time-varying potentials where the non-locality evolves dynamically, as in driven quantum systems or adaptive optical traps. These models, developed in the 2020s, use Adomian decomposition or collocation methods for numerical solutions, revealing time-dependent scaling exponents that capture transient anomalous behaviors not covered by constant-order theories. For example, in a linearly varying potential, the variable-order term \partial_t^{\beta(t)} modulates the wave packet spreading, aligning with simulations of ultrafast laser interactions.^[64]^[65]

Control systems and other uses

Fractional calculus has found significant applications in control theory, where fractional-order controllers enhance system performance by introducing non-integer derivatives and integrals that provide additional degrees of freedom for tuning. These controllers, such as the fractional proportional-integral-derivative (PID) variant denoted as PI^λD^μ, allow for more precise modeling of dynamic systems with memory effects, leading to improved robustness against disturbances and better transient response compared to integer-order counterparts. For instance, in process control applications like temperature regulation, fractional PID controllers achieve faster settling times and reduced overshoot, as demonstrated in simulations of DC motor speed control where the fractional orders λ and μ are optimized to match system dynamics.^[66] In acoustics, fractional calculus models wave propagation in complex media, particularly porous materials exhibiting attenuation due to viscoelastic damping. The fractional wave equation,

\frac{\partial^2 u}{\partial t^2} = c^2 \Delta u + \eta \, D^\alpha \frac{\partial u}{\partial t},

captures frequency-dependent dispersion and absorption, where D^α denotes the fractional derivative of order α (typically 0 < α < 1) and η is a damping coefficient. This formulation accurately describes acoustic wave behavior in layered porous structures, such as soils or foams, where traditional integer-order models fail to replicate observed power-law attenuation; numerical solutions reveal enhanced prediction of scattering operators for transient waves.^[67]^[68] Fractional calculus extends conservation laws to open systems with anomalous transport, modifying the continuity equation to account for non-local fluxes. The fractional conservation of mass equation,

\nabla \cdot \mathbf{J} + D^\alpha \rho = 0,

incorporates a fractional time derivative D^α of the density ρ (with flux \mathbf{J}), enabling the description of power-law waiting times in heterogeneous media like fractured aquifers. This approach resolves limitations of classical Fickian diffusion in small control volumes, where mass balance requires recognition of scale-dependent velocity contrasts; applications in groundwater flow demonstrate improved fitting to tracer breakthrough curves.^[69] In biological modeling, fractional derivatives capture memory-dependent dynamics in neuronal systems, as seen in the fractional Hodgkin-Huxley model for action potential generation. By replacing integer-order membrane capacitance with a fractional element of order α < 1, the model incorporates long-range correlations in voltage traces, altering spiking patterns such as increasing repetitive firing rates and adaptation times in response to constant stimuli. This framework better replicates experimental observations of power-law inter-spike intervals in cortical neurons, highlighting the role of subdiffusive ion channel kinetics in information processing.^[70]^[71] Fractional calculus also informs financial modeling, particularly through geometric fractional Brownian motion (gfBm) for asset price dynamics with long-memory effects. In option pricing, gfBm replaces standard Brownian motion in the Black-Scholes framework, yielding path-dependent solutions via Monte Carlo simulations or Fourier transforms that account for Hurst parameter H ≠ 0.5, thus capturing volatility clustering in equity markets. European call options under gfBm exhibit adjusted Greeks, with delta hedging strategies showing reduced risk for H > 0.5 in persistent markets.^[72]^[73] Other applications include image processing, where fractional gradients enhance edge detection by emphasizing multi-scale features through non-local operators. The Riesz fractional derivative, of order β (0 < β < 2), computes directional gradients that suppress noise while preserving weak edges, outperforming Sobel filters in textured images; for example, in medical imaging, fractional-order Sobel kernels improve boundary delineation in MRI scans with fractional β tuned to 1.5 for optimal contrast.^[74]^[75]

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