On the Sobolev Boundedness Results of the Product of Pseudo-differential Operators Involving a Couple of Fractional Hankel Transforms

This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.

A quasi-discrete Hankel transform for nonlinear beam propagation

We propose and implement a quasi-discrete Hankel transform algorithm based on Dini series expansion （DQDHT） in this paper. By making use of the property that the zero-order Bessel function derivative J0^1（0）=0, the DQDHT can be used to calculate the values on the symmetry axis directly. In addition, except for the truncated treatment of the input function, no other approximation is made, thus the DQDHT satisfies the discrete Parsevat theorem for energy conservation, implying that it has a high numerical accuracy. Further, we have performed several numerical tests. The test results show that the DQDHT has a very high numerical accuracy and keeps energy conservation even after thousands of times of repeating the transform either in a spatial domain or in a frequency domain. Finally, as an example, we have applied the DQDHT to the nonlinear propagation of a Gaussian beam through a Kerr medium system with cylindrical symmetry. The calculated results are found to be in excellent agreement with those based on the conventional 2D-FFT algorithm, while the simulation based on the proposed DQDHT takes much less computing time.

Hankel Transform Domain Analysis of Dual-Frequency Stacked Circular Microstrip Antennas

This paper presents an accurate and efficient method for the computation of resonant frequencies and radiation patterns of a dual frequency stacked circular microstrip antenna.The problem is first formulated using the Hankel transform domain approach and expressions are obtained for the Green's function in the Hankel transform domain,which relates the electric surface currents on the circular disks and tangential electric field components on the surfaces of the substrates. Then Galerkin's method together with Parsebal's relation for Hankel transformation is used to solve for the unknown currents, In the derivation process,the resonant frequencies are numerically determined as a function of the radii of two circular disks and thicknesses and relative permittivies of two substrates.Finally,the far zone radiation patterns are directly obtained from the Green's function and the currents. The numerical results for the resonant frequencies and radiation patterns are in excellent agreement with the available experimental data corroborating the accuracy of the present method.

Unsteady flow of viscoelastic fluid between two cylinders using fractional Maxwell model

The unsteady flow of an incompressible fractional Maxwell fluid between two infinite coaxial cylinders is studied by means of integral transforms. The motion of the fluid is due to the inner cylinder that applies a time dependent tor- sional shear to the fluid. The exact solutions for velocity and shear stress are presented in series form in terms of some generalized functions. They can easily be particularized to give similar solutions for Maxwell and Newtonian fluids. Fi- nally, the influence of pertinent parameters on the fluid motion, as well as a comparison between models, is highlighted by graphical illustrations.

Some exact solutions of the oscillatory motion of a generalized second grade fluid in an annular region of two cylinders

The velocity field and the associated shear stress corresponding to the longitudinal oscillatory flow of a generalized second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and at t = 0＋ both cylinders suddenly begin to oscillate along their common axis with simple harmonic motions having angular frequencies Ω1 and Ω2. The solutions that have been obtained are presented under integral and series forms in terms of the generalized G and R functions and satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary second grade fluid and Newtonian fluid are also obtained as limiting cases of our general solutions. At the end, the effect of different parameters on the flow of ordinary second grade and generalized second grade fluid are investigated graphically by plotting velocity profiles.

Elastodynamics of axi-symmetric deformation in magneto-micropolar generalized thermoelastic medium

The present investigation is concerned with an axi-symmetric problem in the electromagnetic micropolar thermoelastic half-space whose surface is subjected to the mechanical or thermal source. Laplace and Hankel transform techniques are used to solve the problem. Various types of sources are taken to illustrate the utility of the approach. Integral transforms are inverted by using a numerical technique to obtain the components of stresses, temperature distribution, and induced electric and magnetic fields. The expressions of these quantities are illustrated graphically to depict the magnetic effect for two different generalized thermoelasticity theories, i.e., Lord and Shulman （L-S theory） and Green and Lindsay （G-L theory）. Some particular interesting cases are also deduced from the present investigation.

Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders

The velocity field and the adequate shear stress corresponding to the longitudinal flow of a fractional second grade fluid,between two infinite coaxial circular cylinders,are determined by applying the Laplace and finite Hankel transforms.Initially the fluid is at rest,and at time t=0^+, the inner cylinder suddenly begins to translate along the common axis with constant acceleration. The solutions that have been obtained are presented in terms of generalized G functions.Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions.The corresponding solutions for ordinary second grade and Newtonian fluids are obtained as limiting cases of the general solutions.Finally,some characteristics of the motion,as well as the influences of the material and fractional parameters on the fluid motion and a comparison between models,are underlined by graphical illustrations.

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 ＋ , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for Newtonian fluid are also obtained as limiting cases of our general solutions.

A new method for solving Biot's consolidation of a finite soil layer in the cylindrical coordinate system

A new method is developed to solve Biot＇s consolidation of a finite soil layer in the cylindrical coordinate system. Based on the governing equations of Biot＇s consolidation and the technique of Laplace transform, Fourier expansions and Hankel transform with respect to time t, coordinate θ and coordinate r, respectively, a relationship of displacements, stresses, excess pore water pressure and flux is established between the ground surface （z = 0） and an arbitrary depth z in the Laplace and Hankel transform domain. By referring to proper boundary conditions of the finite soil layer, the solutions for displacements, stresses, excess pore water pressure and flux of any point in the transform domain can be obtained. The actual solutions in the physical domain can be acquired by inverting the Laplace and the Hankel transforms.

A THEORETICAL SOLUTION FOR AXIALLY SYMMETRIC PROBLEMS IN ELASTODYNAMICS

This paper presents a theoretical solution for the basic equation of axisymmetric problems in elastodynamics.The solution is composed of a quasi-static solution which satisfies inhomogeneous boundary conditions and a dynamic solution which satisfies homogeneous boundary conditions.After the quasi-static so- lution has been obtained an inhomogeneous equation for dynamic solution is found from the basic equation. By making use of eigenvalue problem of a corresponding homogeneous equation,a finite Hankel transform is defined.A dynamic solution satisfying homogeneous boundary conditions is obtained by means of the finite Hankel transform and Laplace transform.Thus,an exact solution is obtained.Through an example of hollow cylinders under dynamic load,it is seen that the method,and the process of computing are simple,effective and accurate.

AXISYMMETRIC BOUNDARY-VALUE PROBLEMS IN A PIEZOELECTRIC MEDIUM

Based on the general solution of three-dimensional problems inpiezoelectric medium, with the method of Green's functins~[2],axisymmetric boundary-value problems are discussed. The purpose ofthis research is for analyzing the effective on mechanics andelectricity of the piezoelectric ceramics caused by voids andinclusions. The displacement, traction and electric Green's functionscorresponding to circular ring loads acting in the interior of apiezoelectric ceramic are obtained. A cylindrical coordinate systemis employed and Hankel transform are applied with respect to radialcoor- dinates. Explicit solutions for Green's functions are presentedin terms of infinite integrals of Lipshitz- Hankel type. By solving atraction boundary-value problem, the solution scheme is illustrate.

A PENNY-SHAPED CRACK IN A MAGNETOELECTROELASTIC LAYER UNDER RADIAL SHEAR IMPACT LOADING

This paper analyzes the dynamic magnetoelectroelastic behavior induced by a pennyshaped crack in a magnetoelectroelastic layer. The crack surfaces are subjected to only radial shear impact loading. The Laplace and Hankel transform techniques are employed to reduce the problem to solving a Fredholm integral equation. The dynamic stress intensity factor is obtained and numerically calculated for different layer heights. And the corresponding static solution is given by simple analysis. It is seen that the dynamic stress intensity factor for cracks in a magnetoelectroelastic layer has the same expression as that in a purely elastic material. And the influences of layer height on both the dynamic and static stress intensity factors are insignificant as h/a 〉 2.

A new analytical solution to axisymmetric Biot's consolidation of a finite soil layer

A new analytical method is presented to study the axisymmetric Biot＇s consolidation of a finite soil layer. Starting from the governing equations of axisymmetric Blot＇s consolidation, and based on the property of Laplace transform, the relation of basic variables for a point of a finite soil layer is established between the ground surface （z= 0） and the depth z in the Laplace and Hankel transform domains. Combined with the boundary conditions of the finite soil layer, the analytical solution of any point in the transform domain can be obtained. The actual solution in the physical domain can be obtained by inverse Laplace and Hankel transforms. A numerical analysis for the axisymmetric consolidation of a finite soil layer is carried out.

EXACT SOLUTION AND ITS BEHAVIOR CHARACTERISTIC OF NONLINEAR DUAL-POROSITY MODEL

A nonlinear dual-porosity model considering a quadratic gradient term is presented. Assuming the pressure difference between matrix and fractures as a primary unknown, to avoid solving the simultaneous system of equations, decoupling of fluid pressures in the blocks from the fractures was furnished with a quasi-steady-state flow in the blocks. Analytical solutions were obtained in a radial flow domain using generalized Hankel transform. The real value cannot be gotten because the analytical solutions were infinite series. The real pressure value was obtained by numerical solving the eigenvalue problem. The change law of pressure was studied while the nonlinear parameters and dual-porosity parameters changed, and the plots of typical curves are given. All these result can be applied in well test analysis.

初始轴对称扰动在具有可渗透海底特征的冰覆盖海洋表面产生的重力波研究

This paper is concerned with the generation of gravity waves due to prescribed initial axisymmetric disturbances created at the surface of an ice sheet covering the ocean with a porous bottom.The ice cover is modeled as a thin elastic plate,and the bottom porosity is described by a real parameter.Using linear theory,the problem is formulated as an initial value problem for the velocity potential describing the motion.In the mathematical analysis,the Laplace and Hankel transform techniques have been used to obtain the depression of the ice-covered surface in the form of a multiple infnite integral.This integral is evaluated asymptotically by the method of stationary phase twice for a long time and a large distance from the origin.Simple numerical computations are performed to illustrate the efect of the ice-covered surface and bottom porosity on the surface elevation,phase velocity,and group velocity of the surface gravity waves.Mainly the far-feld behavior of the progressive waves is observed in two diferent cases,namely initial depression and an impulse concentrated at the origin.From graphical representations,it is clearly visible that the presence of ice cover and a porous bottom decreases the wave amplitude.Due to the porous bottom,the amplitude of phase velocity decreases,whereas the amplitude of group velocity increases.

ANALYSIS OF THE LARGE DEFLECTION DYNAMIC RESPONSE OF RIGID-PLASTIC CIRCULAR PLATE RESTING ON FLUID

A study is presented for the large deflection dynamic response of rigid- plastic circular plate resting on potential fluid under a rectangular pressure pulse load. By virtue of Hankel integral transform technique,this interaction problem is reduced to a problem of dynamic plastic response of the plate in vacuum.The closed-form solutions are derived for both middle and high pressure loads by solving the equations of motion with the large deflection in the range where both bending moments and membrane forces are important.Some numerical results are given.

Adaptive Segmentation Based on Local Image Attribute

The method of moments is used to analyze the effect of a superstrate on the input impedance of an annular ring microstrip antenna. Surface current and vertical unit current ae used to model the microstrip patch and the coaxical probe, respectively. The integral equations for the unknown current on the patch are solved by using Galerkins method applied in the Hankel transform domain. The expression for the input impedance is derived according to the patch current. The numerical results for the input impedance are presented for different values of the thickness and relative permittivity of a superstrate layer. It is shown from the numerical results that the superstrate has an effect of decreasing the resonant frequency and changing the input impedance level.

Penny-shaped interfacial crack between dissimilar magnetoelectroelastic layers

A penny-shaped interfacial crack between dissimilar magnetoelectroelastic layers subjected to magnetoelectromechanical loads is investigated,where the magnetoelectrically impermeable crack surface condition is adopted. By using Hankel transform technique,the mixed boundary value problem is firstly reduced to a system of singular integral equations,which are further reduced to a system of algebraic equations. The field intensity factors and energy release rate are finally derived. Numerical results elucidate the eects of crack configuration,electric and/or magnetic loads,and material parameters of the magnetoelectroelastic layers on crack propagation and growth. This work should be useful for the design of magnetoelectroelastic composite structures.

Axial Couette flow of an Oldroyd-B fluid in an annulus

This paper establishes the velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder which applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. The general solutions can be easily specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid motion are graphically illustrated.

A New Stable Algorithm to Compute Hankel Transform Using Chebyshev Wavelets

A new stable numerical method,based on Chebyshev wavelets for numerical evaluation of Hankel transform,is proposed in this paper.The Chebyshev wavelets are used as a basis to expand a part of the integrand,r f(r),appearing in the Hankel transform integral.This transforms the Hankel transform integral into a Fourier-Bessel series.By truncating the series,an efficient and stable algorithm is obtained for the numerical evaluations of the Hankel transforms of order ν>−1.The method is quite accurate and stable,as illustrated by given numerical examples with varying degree of random noise terms εθ_(i) added to the data function f(r),where θ_(i) is a uniform random variable with values in[−1,1].Finally,an application of the proposed method is given for solving the heat equation in an infinite cylinder with a radiation condition.