Stability of a class of entropies based on fractional calculus

Integral equations, fractional calculus and shift operator

We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the exponential shift operator.

Calculus of variations with fractional derivatives and fractional integrals

We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.

Introduction to the techniques of the fractional calculus to investigate some models of the mathematical physics (in Portuguese)

In this paper, we resort to the Laplace transform method in order to show its efficiency when approaching some types of fractional differential equations. In particular, we present some applications of such methods when applied to possible generalizations of certain physical problems in linear viscoelasticity and harmonic oscillators, proving that fractional calculus is well suited for the modelling and solving of problems usually treated by ordinary integer calculus, with the promissing advantages of being able to provide more accurate theoretical predictions to fit with experimental data. OB…

General fractional calculus and Prabhakar's theory

General fractional calculus offers an elegant and self-consistent path toward the generalization of fractional calculus to an enhanced class of kernels. Prabhakar's theory can be thought of, to some extent, as an explicit realization of this scheme achieved by merging the Prabhakar (or, three-parameter Mittag-Leffler) function with the general wisdom of the standard (Riemann-Liouville and Caputo) formulation of fractional calculus. Here I discuss some implications that emerge when attempting to frame Prabhakar's theory within the program of general fractional calculus.

连续介质力学的分数阶微积分 - 各向异性非局部性

Fractional Calculus for Continuum Mechanics - anisotropic non-locality

本文将分数阶连续介质力学公式推广至各向异性非局部性情形，基于非局部的分数阶变形梯度构建理论，其优势在于结构与经典连续介质力学框架类似，便于赋予物理与几何意义。

Analytical solution of local fractional Klein-Gordon equation for the generalized Hulthen potential

One dimensional Klein-Gordon (KG) equation is investigated in the domain of conformable fractional calculus for one dimensional scalar potential namely generalized Hulthen potential. The conformable fractional calculus is based on conformable fractional derivative which is the most natural definition in non integer order calculus. Fractional order differential equations can be solved analytically by means of this derivative operator. We obtained exact eigenvalue and eigenfunction solutions of local fractional KG equation and investigated the evolution of relativistic effects in correspondence …

A Stochastic Fractional Calculus with Applications to Variational Principles

We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler-Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.

A Fractional Calculus of Variations for Multiple Integrals with Application to Vibrating String

We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie. Main results provide fractional versions of the theorems of Green and Gauss, fractional Euler-Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string.

Fractional Calculus in Wave Propagation Problems

Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fract…

A Formulation of Noether's Theorem for Fractional Problems of the Calculus of Variations

Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

Fractional Calculus in C and linear ODE's with Analytic Coefficients, Part 1

The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It is known that if a function is analytic in a simply connected domain, then its fractional derivatives and integrals are also analytic in this region. In multi-connected domains, in general, this property does not hold. However, for the set of hypergeometric functions $_{p+1}F_{p},p\geq 2$, this property is preserved. We propose a new version of fractional ca…

Local Fractional Calculus: a Review

The purpose of this article is to review the developments related to the notion of local fractional derivative introduced in 1996. We consider its definition, properties, implications and possible applications. This involves the local fractional Taylor expansion, Leibnitz rule, chain rule, etc. Among applications we consider the local fractional diffusion equation for fractal time processes and the relation between stress and strain for fractal media. Finally, we indicate a stochastic version of local fractional differential equation.

利用分数阶微积分推导摩擦力

Deriving friction force using fractional calculus

本文利用分数阶微积分，通过考虑耗散过程中时间的不均匀流动来推导摩擦力，所得拉格朗日量中的分数阶项给出正确的欧拉-拉格朗日及哈密顿方程，并在能量测量中消失。

分数Ito calculus用于随机缩放的分数布朗运动及其在演化方程中的应用

Fractional Ito Calculus for Randomly Scaled Fractional Brownian Motion and its Applications to Evolution Equations

本文提出了一种针对随机缩放分数布朗运动的分数Ito calculus，并将其应用于广义时间分数演化方程的研究。

Necessary Optimality Conditions for Fractional Action-Like Integrals of Variational Calculus with Riemann-Liouville Derivatives of Order $(α,β)$

We derive Euler-Lagrange type equations for fractional action-like integrals of the calculus of variations which depend on the Riemann-Liouville derivatives of order $(α,β)$, $α> 0$, $β> 0$, recently introduced by J. Cresson and S. Darses. Some interesting consequences are obtained and discussed.

A Generalized Fractional Calculus of Variations

We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, transversality conditions for free boundary value problems, and a generalized Noether type theorem.

Dirac factorization and fractional calculus

We show that the Dirac factorization method can be successfully employed to treat problems involving operators raised to a fractional power. The technique we adopt is based on an extension of the Pauli matrices and the properties of the roots of unity. We also comment about the possibility of using the method to linearize evolution equations containing the $n$-th root of differential operators and make a comparison with other techniques involving suitable transforms.

Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional calculus

An analogy is drawn between the diffusion-wave equations derived from the fractional Kelvin-Voigt model and those obtained from Buckingham's grain-shearing (GS) model [J. Acoust. Soc. Am. 108, 2796-2815 (2000)] of wave propagation in saturated, unconsolidated granular materials. The material impulse response function from the GS model is found to be similar to the power-law memory kernel which is inherent in the framework of fractional calculus. The compressional wave equation and shear wave equation derived from the GS model turn out to be the Kelvin-Voigt fractional-derivative wave equation …

Variational Calculus with Conformable Fractional Derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Prin…

Fractional Calculus of Variations of Several Independent Variables

We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians depending on generalized partial integrals and derivatives. A generalized fractional Noether's theorem, a formulation of Dirichlet's principle and an uniqueness result are given.

Fractional Calculus: Integral and Differential Equations of Fractional Order

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are r…

Fractional calculus and generalized Mittag-Leffler type functions

In this paper, the generalized fractional integral operators of two generalized Mittag-Leffler type functions are investigated. The special cases of interest involve the generalized Fox--Wright function and the generalized M-series and K-function. In the next Section 2 we first recall some generalized fractional integral operators among the most widely used in fractional calculus. Section 3 is devoted to the definitions of M-series and K-function and their relations to special functions. In Sections 4 and 5, effective fractional calculus of the generalized M-series and the K-function is carrie…

The DuBois-Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler-Lagrange Equation Involving only Derivatives of Caputo

Derivatives and integrals of non-integer order were introduced more than three centuries ago, but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional calculus of variations is currently in fast development. However, all current formulations for the fractional variational calculus f…

Limiting Approach to Generalized Gamma Bessel Model via Fractional Calculus and its Applications in Various Disciplines

The essentials of fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes are established. In addition to this, from this fractional integral one can list out almost all the extended densities for the pathway parameter $q < 1$ and $q \rightarrow 1$. Here we bring out the idea of thicker or thinner tailed models associated with a gamma type distribution as a limiting case of pathway operator. Applications of this extended gamma model in Statistical Mechanics, input-output models, and solar spectral irrad…

Matrix approach to discrete fractional calculus II: partial fractional differential equations

A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach (Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of numerical solution of fractional diffusion equation with various combinations of time/space fractional derivatives (integer/integer, fractional/integer, i…

有限区间上任意阶Petrov-Galerkin格式与一般分数阶微积分的Jacobi卷积级数

Jacobi convolution series for Petrov-Galerkin scheme and general fractional calculus of arbitrary order over finite interval

本研究将有限区间上的一般分数阶微积分扩展至任意阶，并引入Jacobi卷积级数作为基函数开发高效Petrov-Galerkin格式，利用其导数为移位Jacobi多项式的特性实现对角刚度矩阵，适用于任意核。