Fractional Powers of the Directional Derivative and a Maxwell–Gegenbauer Multipole Identity

Fractal and Fractional are two words referring to some characteristics and fundamental problems which arise in all fields of science and technology. [...]

We develop a rigorous framework for Gaer fractional directional calculus , providing a complex‐analytic extension of classical differentiation and integration along arbitrary directions in . Starting from Gaer's contour representation of integer‐order directional derivatives, we construct a fractional operator that extends differentiation to complex order while preserving analyticity, decay properties, and geometric invariance. The resulting operator forms an analytic family with respect to the order parameter and unifies several classical fractional constructions, including the Gaer, Weyl, and Riesz formulations, within a common analytic framework. Building on this structure, we derive a fractional extension of Maxwell's multipole expansion . By analytically continuing the differentiation order from integer to complex , we obtain a generalized Maxwell–Legendre formula in which the Legendre polynomials are replaced by Legendre functions of complex degree. Applied to the Newtonian potential, this construction produces a continuous family of fractional multipole potentials that interpolates smoothly between the classical monopole, dipole, quadrupole, and higher multipole fields. The resulting theory establishes a natural analytic bridge between fractional calculus, harmonic analysis, and potential theory, and provides new tools for the study of nonlocal field models and fractional generalizations of classical electrodynamics.

The study discussed in this article is driven by the realization that many physical processes may be understood by using applications of fractional operators and special functions. In this study, we present and examine a fractional integral operator with an I-function in its kernel. This operator is used to solve several fractional differential equations (FDEs). FDE has a set of particular cases whose solutions represent different physical phenomena. Many mathematical physics, biology, engineering, and chemistry problems are identified and solved using FDE. Specifically, a few exciting relations involving the new fractional operator with incomplete I-function (IIF) in its kernel and classical Riemann Liouville fractional integral and derivative operators, the Hilfer fractional derivative operator, and the generalized composite fractional derivate (GCFD) operator are established. The discovery and investigation of several important exceptional cases follow this.

Fractional derivatives with no-index law property: Application to chaos and statistics

The calculus of non-integer order, also known as fractional calculus, has gained prominence in recent years, both for its well-established theory and its applications for providing results more consistent with reality. From this, the problem arises: by the expressive number of definitions of integrals and fractional derivatives, knowing which best integral and fractional derivative to use to model a particular physical problem. So one way to overcome such a problem is to propose more general fractional derivatives. In this paper, we will investigate in detail the fractional derivative -Hilfer which contains as particular cases many of the usual fractional derivatives, from appropriate choices of the functions pq, f pq, the limits a, b and the parameters and . We present and demonstrate some fundamental properties and relations for the -Hilfer fractional derivative through the fractional derivatives: -Caputo and -Riemann-Liouville. Results of uniformly convergent sequences and uniformly continuous functions, using the fractional operator -Hilfer and the fractional integral operator -Riemann-Liouville, are investigated. Also, we emphasize an example through a lemma, which involves the Mittag-Leffler function. In this sense, we investigate the Leibniz I type rule and the type Leibniz II, a condition for a given derivative to be considered fractional, according to the criteria established by Ortigueira and Machado. We highlight their respective particular cases. In order to emphasize the applicability and importance of the -Hilfer fractional derivative, we investigated the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of a nonlinear fractional Volterra integro-differential equation with a given initial condition and we discussed some particular cases.

<abstract><p>In this study, we introduce the extended incomplete versions of the Riemann-Liouville (R-L) fractional integral operators and investigate their analytical properties rigorously. More precisely, we investigate their transformation properties in $ L_{1} $ and $ L_{\infty} $ spaces, and we observe that the extended incomplete fractional calculus operators can be used in the analysis of a wider class of functions than the extended fractional calculus operator. Moreover, by considering the concept of analytical continuation, definitions for extended incomplete R-L fractional derivatives are given and therefore the full fractional calculus model has been completed for each complex order. Then the extended incomplete $ \tau $-Gauss, confluent and Appell's hypergeometric functions are introduced by means of the extended incomplete beta functions and some of their properties such as integral representations and their relations with the extended R-L fractional calculus has been given. As a particular advantage of the new fractional integral operators, some generating relations of linear and bilinear type for extended incomplete $ \tau $-hypergeometric functions have been derived.</p></abstract>

Part I: Transmutations, Integral Equations and Special Functions.- Some Recent Developments in the Transmutation Operator Approach.- Transmutation Operators and Their Applications.- Hankel Generalized Convolutions with the Associated Legendre Functions in the Kernel and Their Applications.- Second Type Neumann Series Related to Nicholson's and to Dixon-Ferrar Formula.- On Some Generalizations of the Properties of the Multidimensional Generalized Erdelyi-Kober Operators and Their Applications.- Alternative Approach to Miller-Paris Transformations and Their Extensions.- Transmutation Operators For Ordinary Dunkl-Darboux Operators.- Theorems on Restriction of Fourier-Bessel and Multidimensional Bessel Transforms to Spherical Surfaces.- Necessary Condition for the Existence of an Intertwining Operator and Classification of Transmutations on Its Basis.- Polynomial Quantization on Line Bundles.- Fourier-Bessel Transforms of Measures and Qualitative Properties of Solutions of Singular Differential Equations.- Inversion of Hyperbolic B-Potentials.- One-Dimensional and Multi-Dimensional Integral Transforms of Buschman-Erdelyi Type with Legendre Functions in Kernels.- Distributions, Non-smooth Manifolds, Transmutations and Boundary Value Problems.- Part II: Transmutations in ODEs, Direct and Inverse Problems.- On a Transformation Operator Approach in the Inverse Spectral Theory of Integral and Integro-Differential Operators.- Expansion in Terms of Appropriate Functions and Transmutations.- Transmutation Operators as a Solvability Concept of Abstract Singular Equations.- On the Bessel-Wright Operator and Transmutation with Applications.- On a Method of Solving Integral Equation of Carleman Type on the Pair of Segments.- Transmutation Operators Boundary Value Problems.- Solution of Inverse Problems for Differential Operators with Delay.- Part III: Transmutations for Partial and Fractional Differential Equations.- Transmutations of the Composed Erdelyi-Kober Fractional Operators and Their Applications.- Distributed Order Equations in Banach Spaces with Sectorial Operators.- Transformation Operators for Fractional Order Ordinary Differential Equations and Their applications.- Strong Solutions of Semilinear Equations with Lower Fractional Derivatives.- Mean Value Theorems and Properties of Solutions of Linear Differential Equations.- Transmutations for Multi-Term Fractional Operators.- Fractional Bessel Integrals and Derivatives on Semi-axes.- The Fractional Derivative Expansion Method in Nonlinear Dynamics of Structures: A Memorial Essay.- Boundary Value Problem with Integral Condition for the Mixed Type Equation with a Singular Coefficient.

Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the α-th power of a differential operator corresponds precisely to the α-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures.

This book intends to deepen the study of the fractional calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional calculus (Chapter 1) and of the fractional calculus of variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional calculus and an introduction to the theory of some special functions in fractional calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivatives and integrals are introduced. In the end of this chapter, we review integration by parts formulas for different operators. Chapter 2 presents a short introduction to the classical calculus of variations and review different variational problems, like the isoperimetric problems or problems with variable endpoints. In the end of this chapter, we introduce the theory of the fractional calculus of variations and some fractional variational problems with variable-order. In the second part, we systematize some new recent results on variable-order fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018). In Chapter 3, considering three types of fractional Caputo derivatives of variable-order, we present new approximation formulas for those fractional derivatives and prove upper bound formulas for the errors. In Chapter 4, we introduce the combined Caputo fractional derivative of variable-order and corresponding higher-order operators. Some properties are also given. Then, we prove fractional Euler-Lagrange equations for several types of fractional problems of the calculus of variations, with or without constraints.

Without any doubt, the recently emerging tools from fractional calculus became successful in a manifold of applications and currently the playground of modern engineering sciences. Fractional order differentiation consists in the generalisation of classical integer differentiation to real or complex orders. From a mathematical point of view, several interpretations of fractional differentiation were proposed, but there is still a deep debate about it. However, all these interpretations demonstrate that fractional order differentiation cannot simply be connected to the slope at one point of the derived function for instance. This lack of interpretation is in fact due to the definition of the fractional order operator. This is a nonlocal operator based on an integral with a singular kernel. The same conclusion can be made for the fractional integrator operator; fractional differentiation operator definition being based on the fractional integrator operator definition. This situation explains why these operators are still not well defined and that several definitions still coexist, which impedes the process of becoming standard tools. Since the first recorded referencework in 1695 up to the present day, many articles have been published on this subject, but much progress is still to be done particularly on the relationship of these different definitions with the physical reality of a system (through taking into account the initial conditions for instance). A fractional order system is a system described by an integro-differential equation involving fractional order derivatives of its input(s) and/or output(s). From a physical point of view, linear fractional order systems are not quite conventional linear systems, and not quite conventional distributed parameter systems. They are in fact halfway between these two classes of systems. Fractional order systems exhibit long memory or hereditary effects. Hence, they are a modelling tool well suited to a wide class of phenomena with nonstandard dynamic behaviour and the applications of fractional order systems are now well accepted in the following disciplines:

A review on epidemic models in sight of fractional calculus

On a more general fractional integration by parts formulae and applications

Fractional diffusion equations describe phenomena exhibiting anomalous diffusion that cannot be modeled accurately by second-order diffusion equations. Fractional differential equations raise mathematical difficulties that have not been encountered in the analysis of second-order differential equations. There are two properties of fractional differential operators that make the analysis of fractional differential equations more complicated than that for second-order differential equations. These are (i) fractional differential operators are nonlocal operators, and (ii) the adjoint of a fractional differential operator is not the negative of itself. The wellposedness of a Galerkin weak formulation to fractional elliptic differential equations with a constant diffusivity coefficient and the error analysis for corresponding finite element methods were proved previously. Many subsequent works were carried out to extend the analysis to other numerical methods. A constant diffusivity coefficient has been assumed in all these works. In this paper we present a counterexample which shows that the Galerkin weak formulation loses coercivity in the context of variable-coefficient conservative fractional elliptic differential equations. Hence, the previous results cannot be extended to variable-coefficient conservative fractional elliptic differential equations. We adopt an alternative approach to prove the existence and uniqueness of the classical solution to the variable-coefficient conservative fractional elliptic differential equation and characterize the solution in terms of the classical solutions to second-order elliptic differential equations. Furthermore, we derive a Petrov--Galerkin weak formulation to the fractional elliptic differential equation. We prove that the bilinear form of the Petrov--Galerkin weak formulation is weakly coercive and so the weak formulation has a unique weak solution and is well posed. Finally, we outline potential application of these results in the development of numerical methods for variable-coefficient conservative fractional elliptic differential equations.

The purpose of this tutorial workshop is to introduce the fractional calculus and its applications in controller designs. Fractional order calculus, or integration and differentiation of an arbitrary order or fractional order, is a new tools that extends the descriptive power of the conventional calculus. The tools of fractional calculus support mathematical models that in many cases more accurately describe the dynamic response of actual systems in electrical, mechanical, and automatic control applications etc. The theoretical and practical interest of these fractional order operators is nowadays well established, and its applicability to science and engineering can be considered as emerging new topics. The need to digitally compute the fractional order derivative and integral arises frequently in many fields especially in automatic control and digital signal processing. Fractional order proportional-integral-derivative (PID) controllers are based on the fractional order calculus where the derivative or integral can be of a non-integer order. Due to the extra tuning knobs, it is expected that better control performance can be achieved if the fractional order PID controller is used. Fractional calculus has much to offer science and engineering by providing not only new mathematical tools, but more importantly, its application suggests new insights into the system dynamics as well as controls.

The multivariate Mittag–Leffler function is introduced and used to establish fractional calculus operators. It is shown that the fractional derivative and integral operators are bounded. Some fundamental characteristics of the new fractional operators, such as the semi-group and inverse characteristics, are studied. As special cases of these novel fractional operators, several fractional operators that are already well known in the literature are acquired. The generalized Laplace transform of these operators is evaluated. By involving the explored fractional operators, a kinetic differintegral equation is introduced, and its solution is obtained by using the Laplace transform. As a real-life problem, a growth model is developed and its graph is sketched.