Hankel Transform Research Articles

Fast Hankel transform and its application for studying the propagation of cylindrical electromagnetic fields

We present a fast Hankel transform (FHTn) method for direct numerical evaluation of electromagnetic (EM) field propagation through an axially symmetric system. Comparing with the vector-based plane-wave spectrum (VPWS) method, we present an alternative approach to implement the fast Hankel transform which does not require an additional coordinate transformation for Fourier transform. The proposed FHTn method is an efficient approach for numerical evaluation of an arbitrary integer order of the Hankel transform (HT). As an example to demonstrate the effectiveness of the proposed method, we apply the FHTn technique to the analysis of cylindrical EM field propagation through a diffractive microlens.

An uiniform boundedness for Bochner–Riesz operators related to the Hankel transform

Let Hα be the modified Hankel transform Hα(f, x) = Z ∞ 0 Jα(xt) (xt)α f(t)t dt, defined for suitable functions and extended to some Lp((0,∞), x2α+1) spaces. Given δ > 0, let Mδ α be the Bochner-Riesz operator for the Hankel transform. Also, we take the following generalization

Application of Laplace and Hankel Transforms to Solution of Mixed Nonstationary Boundary Value Problems

The paper is devoted to the study of a heat conduction equations with mixed and unmixed boundary conditions on a surface of a researched half-space. The heat input to an isotropic solid through a circle of a known radius. The method based on the Laplace and Hankel transforms is suggested for the first time to reduce such differential problems to dual integral equations and to obtain the solution in the close form.

Use of Spectra of Normal and Tangential Displacement Components in Lamb Waves for Separating Single-Mode Components

The paper considers one of simple engineering models--an isotropic elastic layer lying on a rigid base where a wave is driven by a vertical periodic load applied to the surface. Using the Hankel transformation and the method of loop integration, formulas have been obtained for the elastic field due to propagating modes generated by an axially symmetrical surface source in a zone where the effect of inhomogeneous waves is insignificant. Calculations of displacement amplitudes and their ratios versus frequency for propagating modes are given.

Axisymmetric contact problems for a two-layer elastic half-space with an annular or circular crack at the interface of the layers

Exact analytical solutions of three related axisymmetric mixed problems of the theory of elasticity are given. These problems are concerned with the pressure of an annular or circular punch on a two-layer half-space taking into account cohesion and the existence of an annular or circular crack at the interface of the layers. The formulation and solution of these problems are based on a special regularization of the solution of the first fundamental boundary-value problem of the theory of elasticity for a single layer when there are arbitrary normal and shear loads on its boundary planes. The solution is constructed by the Hankel transformation method which ensures that the integrals converge for all stresses and displacements.

Exact solutions of some mixed problems of uncoupled thermoelasticity theory for a finite hollow circular cylinder with a groove along the generatrix

Uncoupled problems of thermoelasticity for a finite hollow circular cylinder with a groove along the generatrix for different types of boundary conditions on all the surfaces are considered. These are the conditions for specifying displacements equal to zero or sliding clamping on the cylindrical surfaces and on the edges of the groove and the conditions for specifying stresses on the ends of the cylinder (shear stresses are assumed to be zero). It is assumed that the problem of thermoelasticity has been solved and the temperature is known. Certain auxiliary functions, related to the displacements, are initially introduced, and equations for these functions are derived using Lamé's equations. A finite integral Fourier transformation with respect to the polar angle and a finite Hankel transformation with respect to the radius are then used. As a result a one-dimensional boundary-value vector problem is obtained for a first-order differential equation, which is solved using Green's matrix constructed. Finally, exact solutions of the problems are constructed in series.

Hankel- K\\{M_{p}\\} Spaces

In this paper we introduce the so-called Hankel- K\\{M_{p}\\} spaces, a family which includes as particular instances many of the spaces of test functions arising in connection with the generalized Hankel transformation. Topological properties of the new spaces are discussed.

Hankel Transformation and Hankel Convolution of Tempered Beurling Distributions

In this paper we complete the distributional theory of Hankel transformation developed in [5] and [18]. New Frechet function spaces H μ (w) are introduced. The functions in H μ (w) have a growth in infinity restricted by the Beurling type function w. We study on H μ (w) and its dual the Hankel transformation and the Hankel convolution.

Colombeau Type Generalized Functions of Compact Spectrum with respect to the Hankel Transformation

In this note we introduce an adequate notion of generalized functions with compact spectrum with respect to the Hankel transform on certain spaces of generalized functions of Colombeau type. Also, we present subspaces where the behaviour of the Hankel transform is analogous to the classic one.

A Property Characterizing Montel Hankel-K{Mp} Spaces*

The family of Hankel-K{Mp } spaces includes a number of test function spaces arising in the theory of the generalized Hankel transformation and can be considered as an analogue in this setting of the Gelfand-Shilov K{Mp } spaces. In this work we exhibit a property characterizing those Hankel-K{Mp } spaces which are Montel

Multipliers of hankel-K{m p}spaces *

The family of Hakel-K{M p spaces was introduced by the author as a Hankel transformation related version of the I.M. Gelfand-G.E. Shilov K{M p} spaces. In fact, many spaces of test functions arising in connection with the generalised Hankel tranformation turn out to be Hankel K{M p} spaces. In this paper, under opportune assumptions on the defining sequence {Mp}∞ p=0, the space of multipliers of a Hankel-k{m p} space is characterized in various ways and its topological structure is investigated, thus covering the analysis of the space of multipliers for a wide range of test spaces dealt with in the literature

On Hankel transformation, convolution operators and multipliers on Hardy type spaces

In this paper we study the Hankel transformation on Hardy type spaces. We also investigate Hankel convolution operators and Hankel multipliers on these Hardy spaces.

Isometrim properties of the hankel transfromation in weighted sobolev spaces*

It is shown that the Hankel transformation H v acts in a class of weighted Sobolev spaces. Especially, the isometric mapping property of H v which holds on L 2 is extended to spaces of arbitrary Sobolev order. The novelty in the approach consists in using techniques developed by B.-W. Schulze and others to treat the half-line as a manifold with a conical singularity at r = 0. This is achieved by pointing out a connection between the Hankel transformation and the Mellin transformation. The procedure proposed leads at the same time to a short proof of the Hankel inversion formula. An application to the existence and higher regularity of solutions, including their asymptotics, to the 1+1 dimensional edge-degenerate wave equation is given.

Green's Function for an Infinite Elastic Plateon Winkler's Foundation

In this technical note, an infinite thick plate on Winkler's foundation is studied. The effect of shear between the plate and the foundation on the deflection and the stresses is analyzed. It is assumed that the foundation has a stiffness k (the force needed to produce a unit displacement per area) and reacts in compression as well as tension. The effect of a concentrated normal unit force is investigated. The solution is based on the Airy stress function formulation. The Hankel transformation is employed to solve the biharmonic equation for the stress function. The transformed solution is subject to the transformed boundary conditions and the appli- cation of the inversion theorem leads to the final solution in the integral form. Judiciously selected dimensionless parameters make this solution relatively simple. Numerical results are obtained for some values of those param- eters. In particular, the following two special cases are studied: (1) deflections of a relatively thin plate are compared to the results obtained by Timoshenko and Woinowsky-Krieger, giving an excellent correlation; and (2) with the thickness becoming infinite, the solution of Boussinesq's problem is readily recovered.

Punch problem for initially stressed neo hookean solids

The paper reports the indentation of a semi-infinite, initially stressed elastic medium under the action of an axisymmetric rigid punch pressing the medium normally. Th~ problem has been considered within the framework of incremental deformation theory for neo-Hookean solids. Using the Hankel's transformation, the distributions of incremental stress and strain have been obtained. Indentations by a flat-ended c and a conical punch have been obtained as special cases and these effects have been studied numerically and presented in the form of curves. This problem has defence application as many launching pads and firing machines have some neo-Hookean solids as buffer which bear enormous impact or punch during the action of the machine.

Spaces of ${\mathcal D}_{L^p}-$type and the Hankel convolution

In this paper we introduce new function spaces that are denoted by 7-(,,p, u > -1/2 and 1 < p < oc, and that are spaces of DLP -type where the Hankel convolution and the Hankel transformation are defined. The spaces 7-(,,p will play the same role in the Hankel setting that the spaces DLP play in the theory of Fourier transformation.

Dynamic stress analysis of the interface in a particle-reinforced composite

The analysis of the dynamic stress on the particle-matrix interface in particle-reinforced composite for the reason that this stress may lead to the microvoids' nucleation due to the interfacial debonding were studied. For simplification, a sphere containing a concentric rigid spherical particle was taken as the representative volume element (RVE). The Laplace transformation was used to derive the basic equations, and the analytical solutions were obtained by means of Hankel transformation. Moreover, the influences of the inertia and viscosity on the debonding damage were also discussed.

A computational model of a heat pump system with a hemispherical surface tank as the ground heat source

A computational model for a space heating system utilising a heat pump with a hemispherical surface tank as the ground heat source is presented. An analytical solution is obtained for the transient temperature field outside a hemispherical surface tank by an application of the Complex Finite Fourier Transform (CFFT) and the Finite Bessel Transform (FBT) techniques. This analytical solution, when utilized with expressions valid for the heat pump work and space heat load, leads to the prediction of the annual variation of water temperature in the surface tank as well as the transient earth temperature field surrounding the surface tank. Monthly and annual heat pump work requirements are predicted and presented.

EFFECT OF FLUID DEPTH ON THE HYDROELASTIC VIBRATION OF FREE-EDGE CIRCULAR PLATE

This paper is concerned with the effect of water depth on the free vibration of free-edge circular plates resting on a free fluid surface. The problem addressed is formulated by using the Hankel transformation method, which leads to dual integral equations. The solution of dual integral equations is solved numerically by using Fourier–Bessel series. The fluid is assumed to be inviscid and incompressible. The Kirchhoff theory of plates is used to model the elastic thin plate. Numerical results are given in non-dimensional form for free-edge circular plates, in order to be ready-to-use in applications. To validate the theoretical results, experiments were carried out. Experimental results are in good agreement with theoretical results. It is found that the effect of the fluid depth can be neglected when the fluid depth is greater than the diameter of the circular plates but becomes significant as the water depth decreases. It is also observed in the experimental results that the fluid damping increases as the water depth decreases.

AXI-SYMMETRIC ANALYSIS OF MULTI-LAYERED ELASTIC SYSTEM IN CONSIDERATION OF LAYER INTERFACE SLIPS

Development of AAMES (Axi-symmetric Analysis of Multi-layered Elastic System) is presented, which can evaluate displacements, stresses and strains in pavement structures caused by surface loading from a closed form solution based on Hankel transformation along with Michell displacement function. A shear spring is utilized to model an interface slip between two adjacent layers. AAMES can handle up to ten layers with varying interface conditions and a maximum of 30 loads. Semi-infinite integration involved in the inverse Hankel transformation, which is found a major cause of error near surface, is carefullyexamined to improve accuracy of results. The accuracy is confirmed through identical response data using BISAR.