首页/论文/2004.01700

arXiv 2004.017002020-04-04math.CAmath-phmath.MPphysics.acc-ph

Fractional Powers of the Differentiation and Integration Operators and its Application in Accelerator Physics and Technology

Mostafa Behtouei, Luigi Faillace, Luigi Palumbo, Bruno Spataro, Alessandro Variola, Mauro Migliorati

备注：: Portions of this manuscript has been published in our previous paper with arXiv:2002.09444. In this paper we describe and analyze the mathematical process used in our previous paper with arXiv:2002.09444

英文摘要：

In this paper we present a solution to a fractional integral of the order 3/2 with the use of a novel method. The integral arises during solving the Biot-Savart equation to find the exact analytical solution for the magnetic field components of a solenoid. We solved the integral by cutting the branch line in order to have an analytic function inside the integral instead of multi-valued operation.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation

A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral.

2001-10-22math-phmath.CAmath.MP

A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration

We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers.

2014-05-12math.CA

On the Laguerre fractional integro-differentiation

A fractional power interpretation of the Laguerre derivative $(DxD)^α,\ D\equiv {d\over dx} $ is discussed. The corresponding fractional integrals are introduced. Mapping and semigroup properties, integral representations and Mellin transform analysis are presented. A relationship with the Riemann-Liouville fractional integrals is demonstrated. Finally, a second kind integral equation of the Volterra-type, involving the Laguerre fractional integral is solved in terms of the double hypergeometric type series as the resolvent kernel.

2020-07-10math.CA

$L^p$ properties of non-Archimedean fractional differentiation operators

Let $D^α, α>0$, be the Vladimirov-Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity $D^αD^{-α}f=f$ was known only for the case where $f$ has a compact support. Following a result by Samko about the fractional Laplacian of real analysis, we extend the above identity in terms of $L^p$-convergence of truncated integrals. Differences between real and non-Archimedean cases are discussed.

2021-07-02math.CAmath.FA

时间尺度上nabla分数阶微分的链式法则公式与广义中值定理

Chain rule formula and generalized mean value theorem for nabla fractional differentiation on time scale

本研究修正了Gogoi等人提出的nabla分数阶导数定义，提出时间尺度上该类微分的链式法则公式与广义中值定理，成果可应用于有限级数求和。

2025-01-15math.CA

Some remarks on stable densities and operators of fractional differentiation

Let $D(s)$ be a fractional derivation of order $s$. For a real $p\ne 0$, we construct an integral operator $A(p)$ in an appropriate functional space such that $A(p) D(s) A(p)^{-1}=D(p s)$ for all $s$. The kernel of the operator $A(p)$ is expressed in terms of a function similar to the stable densities.

2004-04-30math.CAmath.FA

更多相关论文收起更多论文

Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications

In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type differentiation operators with general fractional-order of $n-1 <α\leq n$, $n \in \mathbb{N}$ . A rule of fractional differentiation under integral sign with general order is necessary and applicable tool for verification by substitution for candidate solutions of inhomogeneous multi-term fractional differential equations. We derive explicit analytical solutions o…

2020-12-18math.CA

Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order

The paper investigates the initial problem for a linear system of ordinary differential equations with the fractional differentiation operator Dzhrbashyan -- Nersesyan with constant coefficients. The existence and uniqueness theorems of the solution of the boundary value problem under study are proved. The solution is constructed explicitly in terms of the Mittag-Leffler function of the matrix argument. The Dzhrbashyan -- Nersesyan operator is a generalization of the Riemann -- Liouville, Caputo and Miller-Ross fractional differentiation operators. The obtained results as special cases contain…

2020-07-30math.APmath.CA

Marichev-Saigo-Maeda fractional operator representations of generalized Struve function

The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appells function due to Marichev-Saigo-Maeda, to the generalized Struve function. The results are expressed in terms of generalized Wright function. The results obtained here are general in nature and can easily obtain various known results.

2016-10-01math.CA

Generalized fractional operator representations of Jacobi type orthogonal polynomials

The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appell's function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.

2017-09-25math.CA

A remark on local fractional calculus and ordinary derivatives

In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

2016-12-01math.CA

An Identity Involving Integration with Respect to Variable Order of Fractional Derivative

In 1993, Samko and Ross introduced the study of fractional integration and differentiation when the order is not a constant but a function. This suggestion gave rise to a number of further ideas and results. In particular, this implies a possibility of integration with respect to derivative's order. Here an identity is presented, in which an expression involving Riemann--Liouville fractional derivative is integrated with respect to the derivative's order.

2016-04-14math.CA

Beyond local maximal operators

We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces are derived as consequences of our main result. We also apply our boundedness results by studying both generalized neighbourhood capacities and the Lebesgue differentiation of fractional weighted Sobolev functions.

2015-05-15math.CAmath.FA

A complex analysis approach to Atangana-Baleanu fractional calculus

The standard definition for the Atangana-Baleanu fractional derivative involves an integral transform with a Mittag-Leffler function in the kernel. We show that this integral can be rewritten as a complex contour integral which can be used to provide an analytic continuation of the definition to complex orders of differentiation. We discuss the implications and consequences of this extension, including a more natural formula for the Atangana-Baleanu fractional integral and for iterated Atangana-Baleanu fractional differintegrals.

2019-05-16math.CAmath.CV

Matrix Order Differintegration

The Riemann-Liouville formula for fractional derivatives and integrals (differintegration) is used to derive formulae for matrix order derivatives and integrals. That is, the parameter for integration and differentiation is allowed to assume matrix values. It is found that the computation of derivatives and integrals to matrix order is well defined for any square matrix over the complex numbers. Some properties are worked out for special classes of matrices. It is hoped that this new formalism will be of use in the study of systems of fractional differential equations and sequential fractional…

2003-12-19math-phmath.CAmath.MP

A Conformable Fractional Calculus on Arbitrary Time Scales

A conformable time-scale fractional calculus of order $α\in ]0,1]$ is introduced. The basic tools for fractional differentiation and fractional integration are then developed. The Hilger time-scale calculus is obtained as a particular case, by choosing $α= 1$.

2015-05-12math.CA

The Generalized $K$-Wright Function and Marichev-Saigo-Maeda Fractional Operators

In this paper, the generalized fractional operators involving Appell's function $F_3$ in the kernel due to Marichev-Saigo-Maeda are applied to the generalized $K$-Wright function. These fractional operators when applied to power multipliers of the generalized $K$-Wright function ${}_{p}Ψ^k_q$ yields a higher ordered generalized $K$-Wright function, namely, ${}_{p+3}Ψ^k_{q+3}$. The Caputo-type modification of Marichev-Saigo-Maeda fractional differentiation is introduced and the corresponding assertions for Saigo and Erdélyi-Kober fractional operators are also presented. The results derived in t…

2014-08-17math.CA

About Convergence and Order of Convergence of some Fractional Derivatives

In this paper we establish some convergence results for Riemann-Liouville, Caputo, and Caputo-Fabrizio fractional operators when the order of differentiation approaches one. We consider some errors given by $\left|\left| D^{1-\al}f -f'\right|\right|_p$ for p=1 and $p=\infty$ and we prove that for both Caputo and Caputo Fabrizio operators the order of convergence is a positive real r, 0<r<1. Finally, we compare the speed of convergence between Caputo and Caputo-Fabrizio operators obtaining that they a related by the Digamma function.

2020-01-29cs.NAmath.APmath.CA

Cauchy's formula on nonempty closed sets and a new notion of Riemann--Liouville fractional integral on time scales

We prove Cauchy's formula for repeated integration on time scales. The obtained relation gives rise to new notions of fractional integration and differentiation on arbitrary nonempty closed sets.

2021-05-18math.CA

Fractional Calculus and certain integrals of Generalized multiindex Bessel function

We aim to introduce the generalized multiindex Bessel function $J_{\left( β_{j}\right) _{m},κ,b}^{\left( α_{j}\right)_{m},γ,c}\left[ z\right] $ and to present some formulas of the Riemann-Liouville fractional integration and differentiation operators. Further, we also derive certain integral formulas involving the newly defined generalized multiindex Bessel function $J_{\left( β_{j}\right) _{m},κ,b}^{\left( α_{j}\right)_{m},γ,c}\left[ z\right] $. We prove that such integrals are expressed in terms of the Fox-Wright function $_{p}Ψ_{q}(z)$. The results presented here are of general in nature an…

2019-12-16math.CA