Hankel Transform Research Articles

Radial transport in a porous medium with Dirichlet, Neumann and Robin-type inhomogeneous boundary values and general initial data: analytical solution and evaluation

The analytical solution is presented to the convection–diffusion equation describing the concentration of solutes in a radial velocity field due to extracting groundwater from or injecting water into an aquifer with arbitrary initial concentration data F(r), with r the radial distance, and an inhomogeneous mixed boundary condition G(t), with t the time, at the well radius r = r 0. The analytical solution is obtained with a generalized Hankel transformation or with a Laplace transformation. The Hankel transformation turns out to be easier for G = 0, F ? 0, while the Laplace transformation is easier for F = 0, G ? 0. Both techniques can, however, deal with the full problem. The representation found by the generalized Hankel transform can also be found by the Laplace transform, through modification of the contour through the complex plane in the Bromwich integral for the inverse Laplace transform to the real axis. In practice, the numerical evaluation of the integral representation is difficult, due to the oscillating behavior of the integrands. A more appropriate numerical inversion procedure is also suggested, which circumvents the integration of the oscillating integrands, by an alternative modification of the contour in the Bromwich integral such that the new contour follows the steepest descent path starting from a saddle point at the real axis.

On the theory of heat and mass exchange between contacting spherical particles

The steady-state mass transfer of a chemically active impurity to the surface of a stationary dumbbell-shaped particle consisting of two solid contacting reacting spheres of different sizes is analyzed theoretically. The surrounding medium is at rest, and the numerical concentration of the reagent at a large distance from the aggregate of the spheres is maintained constant. The first-order chemical heterogeneous reaction occurs at a high finite rate and is assumed to be isothermal. The solution to the boundary-value diffusion problem is described by a Laplace axisymmetric equation in the system of tangential-spherical coordinates of revolution. A system of two second-order linear perturbed ordinary differential equations with variable coefficients are obtained using the zero-order integral Hankel transformation and its properties from the boundary conditions for transformed functions. Partial integrals and the mean auxiliary Sherwood numbers are obtained approximately. The solution to the formulated problem is used in various technological applications associated with combustion or chemical reactions at the interface between the continuous and discrete phases in the dispersed system, in meteorology, in analysis of problems associated with environmental pollution, etc.

Asymptotic expansions for the Bessel transformations with high oscillations by using the homotopy technique

In this paper, we establish the asymptotic expansions for the integration containing the Bessel functions with high oscillations by using the homotopy technique. The proposed method is constructed by approximating the non-oscillatory or low-oscillatory factor in practice. Two cases are considered, that is, the case 0 [a, b] and the case 0∈[a, b], where [a, b] denotes the integration interval. The efficiency of the technique and the correctness of the theory are shown by several examples.

Transference of Bilinear Operators between Jacobi Series and Hankel Transforms

Fan-Sato[8] proved a tranceference theorem with respect to the multilinear operators on ${\bf R}^n$. Also Blasco-Villarroya[3] proved the similar result with repect to the biilinear operators on ${\bf Z}^2$. In this paper, we prove a tranceference theorem of the bilinear operators between Jacobi series and Hankel transforms.

Mathematical modeling of a tubular spectrochemical cell using the finite Hankel transformation

The analytical expression of the concentration of the species, in the electrochemical cell, with tubular electrode and optical fibres inserted into it, from its two ends is derived using the finite Hankel transformation. A series form of an analytical expression for current and absorbance responses of the spectroelec-trochemical cell has been derived. The analytical expressions of current and absorbance in terms of parameters Kf, Kb, eA and eB for chronoamperometry are given. Available numerical results are compared with our analytical results and are found to be in good agreement.

Recurrence Relations for Strongly q-Log-Convex Polynomials

AbstractWe consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.

Analytical Solution of Fick's Law of the TRISO-Coated Fuel Particles and Fuel Elements in Pebble-Bed High Temperature Gas-Cooled Reactors

Two kinds of approaches are built to solve the fission products diffusion models (Fick's equation) based on sphere fuel particles and sphere fuel elements exactly. Two models for homogenous TRISO-coated fuel particles and fuel elements used in pebble-bed high temperature gas-cooled reactors are presented, respectively. The analytical solution of Fick's equation for fission products diffusion in fuel particles is derived by variables separation. In the fuel element system, a modification of the diffusion coefficient from D to D/r is made to characterize the difference of diffusion rates in distinct areas and it is shown that the Laplace and Hankel transformations are effective as the diffusion coefficient in Fick's equation is dependant on the radius of the fuel element. Both the solutions are useful for the prediction of the fission product behaviors and could be programmed in the corresponding engineering calculations.

Nonaxisymmetric problem of the stress-strain state of an elastic half-space with a near-surface circular crack under the action of loads along it

The nonaxisymmetric problem of the influence of a free surface of an elastic semifinite body on the distribution of stresses in the vicinity of a near-surface disk-shaped crack is considered within the framework of linearized mechanics of deformable solids. A joint analysis of two nonclassic mechanisms of fracture, namely, fracture of a material with initial stresses that act parallel to the plane of the crack and fracture of a body in compression along a crack, is performed. Using representations of the general solutions of linearized equilibrium equations in terms of harmonic potential functions and the apparatus of integral Fourier–Hankel transformations, we reduce the problem separately for each harmonic in the angular coordinate to resolving systems of Fredholm integral equations of the second kind. Expressions for stress intensity factors in the vicinity of the crack contour are obtained, and their dependence on the initial stresses and geometrical parameters of the problem is analyzed. For some highly elastic materials, critical compression parameters that correspond to nonaxisymmetric forms of local loss of stability of a material in compression along a near-surface crack are determined.

N-Dimensional Sobolev type spaces involving Hankel transformation

n -Dimensional Sobolev type spaces G μ , p s ( I n ) and G μ , p s ( I n ) , 1 ⩽ p < ∞ , μ ⩾ - 1 / 2 , s ∈ R are defined using Hankel transformation. Properties including completeness, boundedness of product and convolution are investigated in G μ , p s ( I n ) as well as some properties are investigated in G μ , p s ( I n ) .

The deformation of a free surface due to the impact of a water droplet

An attempt was made to compute the free surface deformation due to the impact of a water droplet. The Cauchy Poisson, i.e. the initial value problem, was solved with the kinematic and dynamic free surface boundary conditions linearized. The zero order Hankel transformation and Laplace transform were applied to the related equations. The initial condition for the free surface profile was derived from a captured video image. The effect of the surface tension was not significant with the water mass used in this investigation. The computed and observed free surface deformations were compared.

Modeling of cylindrical resonators by wave concept iterative process in cylindrical coordinates

AbstractThe purpose of this study is the formulation of a wave concept iterative process for the analysis of the microwave planar circuits printed between two dielectric mediums in a cylindrical metallic box. This method is based on the transverse wave formulation. It also uses the Hankel Transform to express the integral relation in a spectral domain. An example of circular patch has been studied and the obtained results validate the new approach. The good agreement between the simulation results and the experimental published data justifies the design procedure and validates the present analysis approach. Copyright © 2010 John Wiley &amp; Sons, Ltd.

Thermal stresses of flexible pavement with consideration of temperature-dependent material characteristics using stiffness matrix method

The asphalt pavement is regarded as a multilayered elastic half space axisymmetrical body. By introducing the relationship between material characteristics and temperature into the fundamental equations of thermoelasticity and using mathematic methods of Laplace and Hankel integral transformation, the stiffness matrix for a layer is derived firstly. Then the global stiffness matrix is established for multilayered elastic half space using the finite element concepts in which layers are completely contacted. Therefore, explicit solution for thermal stresses of the asphalt pavement is obtained from the solution of the algebra equation formed by global stiffness matrix and the inverse Hankel and Laplace integral transformation. Because the elements of matrix do not include positive exponential function, the calculation is not overflowed and the shortages of transfer matrix method are overcome. This approach serves as a better model for real pavement structure as it takes into account the relationships between the material characteristics and temperature in the pavement system.

Laminar pipe flow with time-dependent viscosity

A general solution is obtained for laminar flow in axisymmetric pipes, allowing for prescribed timedependent viscosity and time-dependent pressure gradients. In both cases, the only restriction on the prescribed time dependence is that it must vary continuously; it is not necessary for rates of change to be continuous. The general solution is obtained using the Finite Hankel Transform method. This makes it possible to allow explicitly for time-dependent viscosity, but it does not permit the spatial dependence of viscosity. This contrasts with Laplace transforms, which allow spatial, but not general, temporal variations. The general solution is used to study a selection of particular flows chosen to illustrate distinct forms of physical behaviour and to demonstrate the ease with which solutions are obtained. The methodology is also applied to the simple case of constant (Newtonian) viscosity. In this case, it yields the same solutions as previously published methods, but it does so in a much simpler manner.

Asymptotic expansions of Bessel, Anger and Weber transformations

In this paper we explore higher order numerical quadrature for the integrations of systems containing Bessel, Anger and Weber functions. The method is constructed by finding the approximate solution of the differential equation and truncating the asymptotic series. Numerical examples based on theoretical results are presented to demonstrate the efficiency of the method.

Thermal stresses in a coating layer. I. General theoretical scheme

The evolution of thermal stresses generated by a thermal perturbation of cylindrical symmetry in a flat infinite isotropic thermoelastic layer of constant (before perturbation) thickness on a half-space rigid substrate is theoretically investigated. The problem is considered in the framework of uncoupled linear thermoelasticity, in the quasi-stationary displacement field approximation. Thermal insulation of the layer is considered in two versions: (a) both its surfaces are adiabatically insulated, and (b) the free surface is adiabatically insulated, whereas the contact surface is kept at constant temperature. The free surface of the layer is assumed to be stress free, and the contact surface is kept on a rigid substrate. The proper equations are solved using suitable Hankel and Fourier transformations. Part I of the paper presents a general theoretical scheme of the problem in a general case (without specification of the perturbation), and it is illustrated by a simple, detailed example (time evolution of stresses at the contact surface of the layer, generated by a surface point instantaneous heat pulse in case a). The analysis of stresses in more realistic cases, modeling realistic situations will be presented in Part II.

Reconstruction of optical fields with the Quasi-discrete Hankel transform

Unlike the FFT, the Quasi Discrete Hankel Transform (QDHT) is not sampled on a uniform grid; in particular the field may no longer be sampled on axis. We demonstrate how the generalised sampling theorem may be applied to optical problems, evaluated with the QDHT, to efficiently and accurately reconstruct the optical field at any point. Without sacrificing numerical accuracy this is demonstrated to be typically 50x faster than using an equivalent 2D FFT.

The Cauchy problem for evolution equations with the Bessel operator of infinite order. I

We establish necessary and sufficient conditions under which the Bessel operator of infinite order is bounded in certain spaces. We study properties of the Bessel transformations of distributions from these spaces, those of convolutions, convolutors, and multiplicators.

Analytical Solution for Two-Dimensional Solute Transport in Finite Aquifer with Time-Dependent Source Concentration

Using the Hankel Transform Technique, an analytical solution is derived for two-dimensional solute transport in a homogeneous isotropic aquifer. The aquifer is subjected to time-dependent point source contamination. The solution is derived under two conditions: (1) the flow velocity in the aquifer is a sinusoidally varying function and (2) the flow velocity is an exponentially decreasing function. Initially the aquifer is assumed solute free. The analytical solution is illustrated using an example.

Plasma and thermal waves in a semiconducting cantilever photogenerated by a focused laser beam

The plasma wave (carrier-density wave) and thermal wave in an optically opaque semiconducting cantilever (3D geometry), photogenerated by a tightly focused and intensity-modulated laser beam, were analyzed. The theoretical model for the carrier-density and temperature distribution by using the Green function method and Hankel transformation was given. The amplitude and phase of the carrier-density and temperature space and frequency distribution in the cantilever are calculated, including the thermalization and surface and volume recombination heat processes. These investigations are important for many practical experimental situation (atomic force microscopy, thermal microscopy, thermoelastic microscopy, etc) and sensors and actuators based on cantilevers.

Heuristic geometric “eigenvalue universality” in a one-dimensional neutron transport problem with anisotropic scattering

Abstract In the present work we propose a heuristic construction of a transport equation for neutrons with anisotropic scattering considering only the radial cylinder dimension. The eigenvalues of the solutions of the equation correspond to the positive values for the one dimensional case. The central idea of the procedure is the application of the SN method for the discretisation of the angular variable followed by the application of the zero order Hankel transformation. The basis the construction of the scattering terms in form of an integro-differential equation for stationary transport resides in the hypothesis that the eigenvalues that compose the elementary solutions are independent of geometry for a homogeneous medium. We compare the solutions for the cartesian one dimensional problem for an infinite cylinder with azimuthal symmetry and linear anisotropic scattering for two cases.