Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation
Igor Podlubny
备注：
18 pages, 7 figures, 1 table
发表信息：
Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002, pp. 367--386.
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.
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.
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.
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.
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.
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.
This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results in greater generality than in the literature. Notably we unify the continuous and discrete case. We make many side remarks, for instance on wavelets, Mantica's Fourier-Bessel functions and Greville's minimum R_alpha formulas in connection with discrete smoothing.
We study asymptotic properties of the p-adic version of a fractional integration operator introduced in the paper by A. N. Kochubei, Radial solutions of non-Archimedean pseudo-differential equations, Pacif. J. Math. 269 (2014), 355-369.
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…
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…
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.
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.
In this paper, we extend the Montogomery identities for the Riemann-Liouville fractional integrals. We also use this Montogomery identities to establish some new integral inequalities for convex functions.
In this manuscript we define the right fractional derivative and its corresponding right fractional integral for the recently introduced nonlocal fractional derivative with Mittag-Leffler kernel. Then, we obtain the related integration by parts formula. We use the $Q-$operator to confirm our results. The corresponding Euler-Lagrange equations are obtained and one illustrative example is discussed.
We introduce a 2-parameter Hedberg's method, and use it to prove a Hardy-Littlewood-Sobolev theorem for a new type of fractional integral operators, whose kernel has singularity on the light-cone.
In this first part of our project, we prove a classical Hardy-Littlewood-Sobolev result for a new family of fractional integral operators whose kernel has singularity appeared on the light cone in R^n+1.
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.
We study a family of convolution operators whose kernels have a singularity on the unit sphere. As a result, we prove the regarding L^p-L^q Sobolev inequalities.
We suggest two versions of the Hardy--Littlewood--Sobolev inequality for discrete time martingales. In one version, the fractional integration operator is a martingale transform, however, it may vanish if the filtration is excessively irregular; the second version lacks the martingale property while being analytically meaningful for an arbitrary filtration.
We prove that the discrete fractional integration operators along the primes \[ T^λ_{\mathbb{P}}f(x) := \sum_{p} \frac{f(x-p)}{p^λ} \cdot \log p \] are bounded $\ell^p\to \ell^{p'}$ whenever $ \frac{1}{p'} < \frac{1}{p} - (1-λ), \ p > 1.$ Here, the sum runs only over prime $p$.
The aim of this note is to provide a fractional integration theorem in the framework of Laguerre expansions. The method of proof consists of establishing an asymptotic estimate for the involved kernel and then applying a method of Hedberg \cite{pro}. We combine this result with sufficient $(p,p)$ multiplier criteria of Stempak and Trebels \cite{ST}. The resulting sufficient $(p,q)$ multiplier criteria are comparable with necessary ones of Gasper and Trebels \cite{laguerre}.
In this present paper our aim is to deal with two integral transforms which involving the Gauss hypergeometric function as its kernels. We prove some compositions formulas for such a generalized fractional integrals with k Bessel function. The results are established in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some corresponding assertions for Riemann Liouville and Erdelyi Kober fractional integral transforms.
Inspired by a conjecture of Vladimir Maz'ya on $Φ$-inequalities in the spirit of Bourgain and Brezis, we establish some $Φ$-inequalities for fractional martingale transforms. These inequalities may be thought of as martingale models of $Φ$-inequalities for differential operators. The proofs rest on new simple Bellman functions.
We obtain integral representations of the $n$-th derivatives of the Bessel functions with respect to the order. The numerical evaluation of these expressions is very efficient using a double exponential integration strategy. Also, from the integral representation corresponding to the Macdonald function, we have calculated a new integral. Finally, we calculate integral expressions for the fractional derivatives of the Bessel functions with respect to the order. Simple proofs for some particular cases given in the literature are provided as well.
Two integral operator involving the Appell's functions, or Horn's function in the kernel are considered. Composition of such functions with generalized Bessel functions of the first kind are expressed in term of generalized Wright function and generalized hypergeometric series. Many special cases, including cosine and sine function are also discussed.
We study the inequalities of the type $|\int_{\mathbb{R}^d} Φ(K*f)| \lesssim \|f\|_{L_1(\mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $α- d$ and possibly vector-valued, the function $Φ$ is positively $p$-homogeneous, and $p = d/(d-α)$. Under mild regularity assumptions on $K$ and $Φ$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.
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.
