Asymptotic Behavior of Fractional Derivatives of Bounded Analytic Functions

Abstract The asymptotic behaviour of solutions in an appropriate space is discussed for a fractional problem involving Hadamard left-sided fractional derivatives of different orders. Reasonable sufficient conditions are determined ensuring that solutions of fractional differential equations with nonlinear right hand sides approach a logarithmic function as time goes to infinity. This generalizes and extends earlier results on integer order differential equations to the fractional case. Our approach is based on appropriate desingularization techniques and generalized versions of Gronwall-Bellman inequality. It relies also on a kind of Hadamard fractional version of l'Hopital’s rule which we prove here.

Download Full-text

The space of totally bounded analytic functions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500028303 ◽

1987 ◽

Vol 30 (2) ◽

pp. 229-246

Author(s):

Alan L. Horwitz ◽

Lee A. Rubel

Keyword(s):

Analytic Function ◽

Analytic Functions ◽

Bounded Analytic Functions ◽

Sharp Bounds ◽

Totally Bounded ◽

Bounded Set ◽

Inverse Interpolation ◽

Derivatives Of ◽

Uniformly Bounded

This paper is a continuation of our project on “inverse interpolation”, begun in [6]. In brief, the task of inverse interpolation is to deduce some property of a function f from some given property of the set L of its Lagrange interpolants. In the present work, the property of L is that it be a uniformly bounded set of functions when restricted to the domain of f. In particular (see Section 3), when the domain is a disc, we deduce sharp bounds on the successive derivatives of f. As a result, f must extend to be an analytic function (of restricted growth) in the concentric disc of thrice the original radius.

Download Full-text

Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in Hardy type spaces

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2017-0059 ◽

2017 ◽

Vol 20 (5) ◽

Cited By ~ 3

Author(s):

Alexey Karapetyants ◽

Stefan Samko

Keyword(s):

Analytic Functions ◽

Fractional Derivatives ◽

Mixed Norm ◽

Mixed Norm Spaces ◽

Hardy Type ◽

Type Spaces ◽

Generalized Fractional Derivatives ◽

Definition Of ◽

Derivatives Of ◽

Spaces Of Analytic Functions

AbstractThe aim of the paper is twofold. First, we present a new general approach to the definition of a class of mixed norm spaces of analytic functions 𝓐

Download Full-text

About the Optimal Recovery of Derivatives of Analytic Functions from Their Values at Points That Form a Regular Polygon

Mathematical Physics and Computer Simulation ◽

10.15688/mpcm.jvolsu.2019.4.2 ◽

2019 ◽

pp. 30-38

Author(s):

Mikhail Ovchintsev

Keyword(s):

Approximation Method ◽

Blaschke Product ◽

Analytic Functions ◽

Best Approximation ◽

Extremal Function ◽

Regular Polygon ◽

Optimal Recovery ◽

Bounded Analytic Functions ◽

The Best Approximation ◽

Derivatives Of

In this paper, the author solves the problem of optimal recovery of derivatives of bounded analytic functions defined at the zero of the unit circle. Recovery is performed based on information about the values of these functions at points z1, ... , zn , that form a regular polygon. The article consists of an introduction and two sections. The introduction talks about the necessary concepts and results from the works of Osipenko K.Yu. and Khavinson S.Ya., that form the basis for the solution of the problem. In the first section, the author proves some properties of the Blaschke product with zeros at the points z1, ... , zn. After this, the error of the best approximation method of the derivatives f(N)(0), 1 ≤ N ≤ n − 1, by the values f(z1), ... , f(zn) is calculated. In the same section he gives the corresponding extremal function. In the second section, the uniqueness of the linear best approximation method is established, and then its coefficients are calculated. At the end of the article, the formulas found for calculating of the coefficients are substantially simplified.

Download Full-text

The properties of Bessel-type fractional derivatives and integrals

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.203 ◽

2016 ◽

pp. 3973-3982

Author(s):

V. R. Lakshmi Gorty

Keyword(s):

Fractional Order ◽

Real Line ◽

Computer Algebra System ◽

Fractional Derivatives ◽

Graphical Representation ◽

Fractional Integrals ◽

The Real ◽

Fractional Derivatives And Integrals ◽

Half Axis ◽

Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

Download Full-text

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-2-7 ◽

2020 ◽

Vol 17 (2) ◽

pp. 256-277

Author(s):

Ol'ga Veselovska ◽

Veronika Dostoina

Keyword(s):

Complex Plane ◽

Complex Variable ◽

Analytic Functions ◽

Combinatorial Identities ◽

Independent Interest ◽

Closed Curves ◽

In Series ◽

Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

Download Full-text

FUNCTIONS THAT HAVE NO FIRST ORDER DERIVATIVE MIGHT HAVE FRACTIONAL DERIVATIVES OF ALL ORDERS LESS THAN ONE

Real Analysis Exchange ◽

10.2307/44152475 ◽

1994 ◽

Vol 20 (1) ◽

pp. 140 ◽

Cited By ~ 47

Author(s):

Ross ◽

Samko ◽

Love

Keyword(s):

Fractional Derivatives ◽

Order Derivative ◽

First Order ◽

Derivatives Of

Download Full-text

Time-Domain Analysis of Fractional Electrical Circuit Containing Two Ladder Elements

Electronics ◽

10.3390/electronics10040475 ◽

2021 ◽

Vol 10 (4) ◽

pp. 475

Author(s):

Ewa Piotrowska ◽

Krzysztof Rogowski

Keyword(s):

Fractional Derivative ◽

Fractional Derivatives ◽

Electric Circuit ◽

Theoretical Models ◽

Time Domain Analysis ◽

Electrical Circuit ◽

State Variable ◽

State Space System ◽

Derivatives Of ◽

Different Order

The paper is devoted to the theoretical and experimental analysis of an electric circuit consisting of two elements that are described by fractional derivatives of different orders. These elements are designed and performed as RC ladders with properly selected values of resistances and capacitances. Different orders of differentiation lead to the state-space system model, in which each state variable has a different order of fractional derivative. Solutions for such models are presented for three cases of derivative operators: Classical (first-order differentiation), Caputo definition, and Conformable Fractional Derivative (CFD). Using theoretical models, the step responses of the fractional electrical circuit were computed and compared with the measurements of a real electrical system.

Download Full-text

Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

International Journal of Statistical Mechanics ◽

10.1155/2014/873529 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 3

Author(s):

Vasily E. Tarasov

Keyword(s):

Fractional Derivatives ◽

Three Dimensional ◽

Lattice Models ◽

Fractional Diffusion ◽

Lattice Diffusion ◽

Diffusion Equations ◽

Difference Operators ◽

Fractional Diffusion Equations ◽

Nonlocal Media ◽

Derivatives Of

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

Download Full-text