Hankel Transform Research Articles

Vertical time-harmonic coupling vibration of an impermeable, rigid, circular plate resting on a finite, poroelastic soil layer

Studies associated with dynamic plate–medium interactions generally assumed the plate structures to be permeable for the sake of convenience. But the effect and applicability of such an assumption are still unclear, and then the pore fluid pressure on the plate and medium interfaces cannot be obtained. In this paper, the mentioned problems are discussed by studying the coupling steady-state vibration of an impermeable, rigid, circular plate resting on a finite, fluid-saturated, poroelastic soil layer underlain by rigid base and subjected to a vertical time-harmonic loading. The semi-analytical solutions for the dynamic compliance, displacements, stresses, especially the contact stress including effective stress and pore fluid pressure of the plate and the layer, are proposed. In developing these solutions, the linearly poroelastic model established by de Boer is used to describe the mechanical behaviour of the porous medium. By means of four scalar displacement potentials and the Fourier–Hankel transformation to solve the equations of motion of the poroelastic layer, and then imposing boundary and interfacial conditions, a pair of coupling Fredholm’s integral equations of the second kind formulating the plate–medium interaction are derived and evaluated with numerical methods. The proposed solutions are then verified by comparing with the existing special solutions and the FEM calculation results. Numerical examples are also performed to examine the effects of the permeability of both the plate and the poroelastic layer and the thickness of the layer on the dynamic response of the coupling system.

Efficient Fourier transforms for transverse momentum dependent distributions

Hadron production at low transverse momenta in semi-inclusive deep inelastic scattering can be described by transverse momentum dependent (TMD) factorization. This formalism has also been widely used to study the Drell–Yan process and back-to-back hadron pair production in e+e− collisions. These processes are the main ones for extractions of TMD parton distribution functions and TMD fragmentation functions, which encode important information about nucleon structure and hadronization. One of the most widely used TMD factorization formalism in phenomenology formulates TMD observables in coordinate b⊥-space, the conjugate space of the transverse momentum. The Fourier transform from b⊥-space back into transverse momentum space is sufficiently complicated due to oscillatory integrands that it requires a careful and computationally intensive numerical treatment in order to avoid potentially large numerical errors. Within the TMD formalism, the azimuthal angular dependence is analytically integrated and the two-dimensional b⊥ integration reduces to a one-dimensional integration over the magnitude b⊥. In this paper we develop a fast numerical Hankel transform algorithm for such a b⊥-integration that improves the numerical accuracy of TMD calculations in all standard processes. Libraries for this algorithm are implemented in Python 2.7 and 3, C++, as well as FORTRAN77. All packages are made available open source. Program summaryProgram Title: Fast Bessel Transform (FBT)CPC Library link to program files:http://dx.doi.org/10.17632/65zkb53z8d.1Developer’s repository link:http://github.com/UCLA-TMD/OgataLicensing provisions: MITProgramming language: Python 2/3, C++, FORTRAN77Nature of problem: In order to perform extractions of transverse momentum distribution functions, numerical Hankel transforms must be performed from b⊥-space to momentum space. However, these numerical Hankel transforms are a huge bottleneck in these extractions, making these extractions extremely computationally intensive.Solution method: We develop a numerical Hankel transform algorithm by optimizing Ogata quadrature formula. This algorithm improves the performance of these numerical Hankel transforms by nearly an order of magnitude.

Structure, boundedness, and convergence in the dual of a Hankel-K{Mp } space

ABSTRACT Hankel- spaces, as introduced by the second-named author, play the same role in the theory of the Hankel transformation as the Gelfand-Shilov spaces in the theory of the Fourier transformation. For , under suitable restrictions on the weights , the topology of a Hankel- space E can be generated by norms of -type involving the Bessel operator . In this paper, adapting a technique due to Kamiński, the elements of the dual space are represented as -distributional derivatives of a single continuous function. Corresponding characterizations of boundedness and convergence in (the weak, weak*, strong topologies of) are obtained.

Seismic Reservoir Delineation via Hankel Transform Based Enhanced Empirical Wavelet Transform

To better describe features of nonstationary seismic signals, mode decomposition-based approaches are widely used for seismic processing and analysis, such as empirical mode decomposition (EMD) and empirical wavelet transform (EWT). EWT builds an adaptive filter bank and then decomposes a nonstationary seismic trace into several intrinsic mode functions (IMFs), which has been applied for analyzing the nonstationary seismic signal. In this letter, we propose an enhanced EWT (EEWT) using Hankel transform (HT), which is an integral transform whose kernels are Bessel functions. Compared with sinusoidal functions of Fourier transform (FT), Bessel functions are more effective for describing features of nonstationary signals. Moreover, HT obtains a more compact spectrum than FT for wideband and nonstationary signal analysis, which contributes to the detection of spectral segmentation. To demonstrate the effectiveness of the proposed algorithm, we apply it to both synthetic and field data. Compared with the results provided by EWT, EEWT provides a time–frequency spectrum with higher resolution and offers potentials in precisely highlighting reservoirs.

Fast Discrete Finite Hankel Transform for Equations in a Thin Annulus

An algorithm is proposed for a fast discrete finite Hankel transform of a function in a thin annulus. The transform arises in a natural way in the Neumann boundary-value problem for the Poisson equation in an annulus when spectral methods are applied for its numerical solution. The proposed algorithm uses the limiting properties of eigenvalues and eigenfunctions of the Laplace operator as the annulus thickness goes to zero.

Fast calculation of the nonlinear redshift-space galaxy power spectrum including selection bias

We present a fast implementation of the next-to-leading order (1-loop) redshift-space galaxy power spectrum by using FFTlog-based methods. [V. Desjacques, D. Jeong, and F. Schmidt, JCAP 1812 (12), 035] have shown that the 1-loop galaxy power spectrum in redshift space can be computed with 28 independent loop integrals with 22 bias parameters. Analytical calculation of the angular part of the loop integrals leaves the radial part in the form of a spherical Bessel transformation that is ready to be integrated numerically by using the FFTLog transformation. We find that the original 28 loop integrals can be solved with a total of 85 unique FFTLog transformations, yet leading to a few orders of magnitude speed up over traditional multi-dimensional integration. The code used in this work is publicly available at https://github.com/JosephTomlinson/GeneralBiasPk

Analysis of SESANS data by numerical Hankel transform implementation in SasView

SESANS data analysis has been implemented in the SasView software package, allowing SESANS experiments to be analyzed using a numerical Hankel transformation of isotropic small-angle scattering (SAS) models. The error of the numerical approximation is three orders of magnitude below typical experimental errors. All advanced data fitting features of SasView (multi-model fitting, batch fitting, and simultaneous/constrained fitting) are now also available for SESANS and this is demonstrated by examples of fitting SAS models to SESANS measurements.

A single Rayleigh mode may exist with multiple values of phase-velocity at one frequency

SUMMARYOther than commonly assumed in seismology, the phase velocity of Rayleigh waves is not necessarily a single-valued function of frequency. In fact, a single Rayleigh mode can exist with three different values of phase velocity at one frequency. We demonstrate this for the first higher mode on a realistic shallow seismic structure of a homogeneous layer of unconsolidated sediments on top of a half-space of solid rock (LOH). In the case of LOH a significant contrast to the half-space is required to produce the phenomenon. In a simpler structure of a homogeneous layer with fixed (rigid) bottom (LFB) the phenomenon exists for values of Poisson’s ratio between 0.19 and 0.5 and is most pronounced for P-wave velocity being three times S-wave velocity (Poisson’s ratio of 0.4375). A pavement-like structure (PAV) of two layers on top of a half-space produces the multivaluedness for the fundamental mode. Programs for the computation of synthetic dispersion curves are prone to trouble in such cases. Many of them use mode-follower algorithms which loose track of the dispersion curve and miss the multivalued section. We show results for well established programs. Their inability to properly handle these cases might be one reason why the phenomenon of multivaluedness went unnoticed in seismological Rayleigh wave research for so long. For the very same reason methods of dispersion analysis must fail if they imply wave number kl(ω) for the lth Rayleigh mode to be a single-valued function of frequency ω. This applies in particular to deconvolution methods like phase-matched filters. We demonstrate that a slant-stack analysis fails in the multivalued section, while a Fourier–Bessel transformation captures the complete Rayleigh-wave signal. Waves of finite bandwidth in the multivalued section propagate with positive group-velocity and negative phase-velocity. Their eigenfunctions appear conventional and contain no conspicuous feature.

Discrete two dimensional Fourier transform in polar coordinates part II: numerical computation and approximation of the continuous transform.

The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In the first part of this two-paper series, we proposed and evaluated the theory of the 2D Discrete Fourier Transform (DFT) in polar coordinates. The theory of the actual manipulated quantities was shown, including the standard set of shift, modulation, multiplication, and convolution rules. In this second part of the series, we address the computational aspects of the 2D DFT in polar coordinates. Specifically, we demonstrate how the decomposition of the 2D DFT as a DFT, Discrete Hankel Transform and inverse DFT sequence can be exploited for coding. We also demonstrate how the proposed 2D DFT can be used to approximate the continuous forward and inverse Fourier transform in polar coordinates in the same manner that the 1D DFT can be used to approximate its continuous counterpart.

A fast inverse Hankel Transform of first Order for computing vector-valued weight Functions appearing in Fundamental Measure Theory in cylindrical Coordinates

Application of classical density functional theory in cylindrical coordinates requires a fast Hankel transform algorithm of order zero and its inverse, when the involved convolution integrals are solved in Fourier space. Vector-valued fundamental measure theory requires a fast Hankel transform of first order and its inverse. Compared to naïve real space convolution, this not only reduces complexity of the required computer code, but also increases computational performance due to the efficiency of the fast Hankel transform. This study proposes a new approach to compute the inverse of the first order fast Hankel transform on equidistant grids as a combination of a modified inverse Abel transform and a fast sine transform. Equidistant grids have a significant advantage over alternative implementations that require logarithmic grid spacing, since most problems, such as pores or droplets, require a certain resolution in the outer region of the radial domain, which in the case of a logarithmic grid necessitates an overly high number of grid points in the inner region of the radial domain. The proposed algorithm for the modified inverse Abel transform is straightforward to implement, while for the fast sine transform off-the-shelf algorithms can be used.

Parametric optimization of pulse-echo laser ultrasonic system for inspection of thick polymer matrix composites

One of the main challenges of using laser ultrasonic techniques for non-destructive testing applications is the typically low signal-to-noise ratio of the laser ultrasonic signals. In the case of thick composite structures, this is even more problematic since composite materials have very strong sound attenuation. This article investigates the effects of laser beam size and profile to the amplitude of pulse-echo laser ultrasonic signals with the constraint that the peak energy density (fluence) must be kept constant under the thermal damage threshold of material like polymer matrix composites. Such constraint is very important for the non-destructive feature of non-destructive testing, yet in a number of the existing parameter studies of laser ultrasonics, it was not fully investigated. In this article, a series of A-scan and C-scan experiments on thick composite specimens shows that the amplitude of the direct waves and the reflected waves increases with the increase in laser beam size with constant peak energy density. This amplitude enhancement significantly improves the propagation depth, thereby optimizing the system for inspection of thick composite structures. The validity of experimental results is verified theoretically by solving the thermoelastic model of epicenter displacement using Laplace–Hankel transformation.

МАТЕМАТИЧНІ МОДЕЛІ ТЕПЛООБМІНУ В ЕЛЕМЕНТАХ ТУРБОГЕНЕРАТОРІВ (ПРОДОВЖЕННЯ)

Previously developed [8] and presented new mathematical models for the analysis of temperature regimes in individual elements of turbo generators, which are geometrically described by isotropic half-space and space with an internal heat source of cylindrical shape. Cases are also considered for half-space, when the fuel-releasing cylinder is thin, and for space, when it is heat-sensitive. For this purpose, using the theory of generalized functions, the initial differential equations of thermal conductivity with boundary conditions are written in a convenient form. To solve the obtained boundary value problems of thermal conductivity, the integral Hankel transformation was used, and as a result, analytical solutions in the images were obtained. The inverse Hankel integral transformation was applied to these solutions, which made it possible to obtain the final analytical solutions of the initial problems. The obtained analytical solutions are presented in the form of improper convergent integrals. Computational programs have been developed to determine the numerical values ​​of temperature in the above structures, as well as to analyze the heat transfer in the elements of turbo generators due to different temperature regimes due to heating by internal heat sources concentrated in the cylinder volume. Using these programs, graphs are presented that show the behavior of curves constructed using numerical values ​​of the temperature distribution depending on the spatial radial and axial coordinates. The obtained numerical values ​​of temperature indicate the correspondence of the given mathematical models for determining the temperature distribution to the real physical process. The software also allows you to analyze media with internal heating, concentrated in the spatial figures of the correct geometric shape, in terms of their heat resistance. As a result, it becomes possible to increase it, to determine the allowable temperatures of normal operation of turbo generators, to protect them from overheating, which can cause the destruction of not only individual elements but also the entire structure.

Correcting Wavefront Distortion of Dual-Wavelength Beams Due to Atmospheric Turbulence with a Correction Coefficient

In order to improve the correction effect of the adaptive optical system in coherent optical communication, we investigate the relative distortion between the wavefronts of different wavelengths of the beams transmitted on the near-ground horizontal atmospheric turbulent links emitted by coherent optical communication system. And the situation is analyzed when the wavelength corresponding to the wavefront detected by the wavefront detector and the wavelength corrected by the deformed mirror are different, the influence of the wavelength factor on the adaptive optical system correction. We use a series of trigonometric functions and the Hankel transformation to derive the corrected residual variance and the Strehl ratio between the wavefront distortions of the wavelengths of the dual-wavelength combined beam in atmospheric turbulence. In relation to the parameters of the turbulent environment, the ensemble average of the wavefront difference corresponding to different wavelengths the derived is proposed as the coefficient to correct the dual-wavelength adaptive optical system. The results show that the statistic of the turbulence internal scale has a major influence on the difference between the wavefronts. By adding the correction coefficient, the signal light’s wavefront of the coherent optical communication system can be corrected more effectively by the dual-wavelength adaptive optical closure.

Magnetic field effect on a fractionalized blood flow model in the presence of magnetic particles and thermal radiations

Abstract The presence of magnetic particles is considered in a magneto-hydrodynamic blood flow through a circular cylinder. The fluid inside the tube is acted by an oscillating pressure gradient and an external constant magnetic field. The blood temperature is assumed to change with the blood and particle velocities, and the whole study is based on a mathematical model that includes Caputo fractional-order derivatives. Solutions for the particle and blood velocities, and blood temperature distribution, are obtained via the combination of the Laplace and Hankel transformation methods. Effects of the fractional-order parameter and magnetic field are addressed using numerical simulations. Results show that the applied magnetic field reduces the velocities of the fluid and particles, which remarkably affects the blood temperature. This is obvious for short and long time intervals. However, under long time intervals, particles seem to be accelerated, but their velocity is suitably controlled by the fractional-order parameter which also monitors the increase in blood temperature.

Wave motion in multi-layered transversely isotropic porous media by the method of potential functions

Wave propagation in a multi-layered transversely isotropic porous medium has been considered in this paper, which consists of n parallel layers overlying on a half-space. Potential functions are used to solve elastodynamic differential equations of the poroelastic medium. Time-harmonic excitation is assumed and the procedure of solution is performed in the frequency domain. Generalized reflection and transmission matrices are generated for compressional and shear waves separately. By means of the Hankel transformation method, coupled differential equations are altered to ordinary ones and Riemann surfaces are used to establish the path of integrations. A closed-form solution is described to reach Green’s functions of displacements and stresses. Some special cases of excitations are discussed and verification of the solution is presented. The numerical results of a three-layered medium on a porous half-space are determined and discussed.

A Pair of Linear Canonical Hankel Transformations of Random Order

Present paper is devoted to study a pair of linear canonical Hankel transformations of random order and its inverse. Some interesting properties of these transformations are given. Finally, these transformations are used to obtain the solution of some partial differential equations involving Bessel type differential operators.

Stationary-Phase Method for Hankel Transform of Order Zero

The stationary phase method is applied to investigate the asymptotic behavior at infinity of the Hankel transform of order zero.

A boundary value problem for the Holmgren equation with singular coefficient and spectral parameter

We study a Neumann type problem in the vertical half-strip for the Holmgren equation with singular coefficient and spectral parameter. Applying the method of separation of variables and using Hankel transformation, the solution of the Neumann type problem is represented in an explicit form.

The impact of magnetohydrodynamics and heat transfer on the unsteady flow of Casson fluid in an oscillating cylinder via integral transform: A Caputo–Fabrizio fractional model

Casson fluid flow has numerous functional applications in food processing, metallurgy, drilling and bio-engineering operations. The significance of Casson fluid in cylindrical coordinates has recently attracted researchers because of the numerical and experimental analyses of the fluid. Due to the lack of fractional analytical approaches, this paper is trying to examine the magnetic effect and thermal effect on Casson fluid with oscillatory boundary conditions in cylindrical coordinates. The constitutive model of Casson fluid is solved by using the Caputo–Fabrizio time fractional derivative approach. The fluid moves in the vertical oscillating cylinder under the influence of a transverse applied magnetic field. Closed-form solutions are obtained via integral transforms (Laplace and Hankel transformations) for velocity and temperature distributions. Graphical results are shown for various physical parameters such as the Casson fluid parameter $$\beta $$ , magnetic parameter M, Grashoff number Gr, Prandtl number Pr and fractional parameter $$\alpha $$ . The corresponding expressions for Nusselt number are also evaluated for various embedded parameters in a tabular form.

Linearized non-equilibrium and non-isothermal two-dimensional model of liquid chromatography for studying thermal effects in cylindrical columns

A non-equilibrium and non-isothermal two-dimensional lumped kinetic model (2 D-LKM) is formulated and analytically solved to study the influence of temperature variations along the axial and radial coordinates of a liquid chromatographic column. The model includes convection-diffusion partial differential equations for mass and energy balances in the mobile phase coupled with differential equations for mass and energy in the stationary phase. The solutions are derived analytically through sequential implementation of finite Hankel and Laplace transformations using the Dirichlet inlet boundary conditions. The coupling between the thermal waves and concentration fronts is demonstrated through numerical simulations and important parameters are recognized that influence the column performance. For a more comprehensive study of the considered model, numerical temporal moments are obtained from the derived solutions. Several case studies are conducted and validity ranges of the derived analytical solutions are identified. The current analytical results will play a major role in the improvements of non-equilibrium and non-isothermal liquid chromatographic processes.