Hankel Transform Research Articles

K-functional related to the Deformed Hankel Transform

Abstract The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness for the Deformed Hankel Transform. Before that, we introduce the K-functional associated to the Deformed Hankel Transform.

Equivalence of K-functional and modulus of smoothness constructed by Fourier-Bessel transform in the space L2,γ(ℝn+)

Using a generalized shift operator, we define generalized modulus of smothness in the space L2,γ(ℝn+). Based on the Laplace-Bessel differential operator we define Sobolev-type space and K-functionals. In this paper paper we prove the equivalence theorem for a K-functional and a modulus of smoothness for the Fourier-Bessel transformation on ℝn+.

An Efficient Quadrature Rule for the Oscillatory Infinite Generalized Bessel Transform with a General Oscillator and Its Error Analysis

An Efficient Quadrature Rule for the Oscillatory Infinite Generalized Bessel Transform with a General Oscillator and Its Error Analysis

Analytically Regularized Evaluation of the Coupling of Planar Concentric Conducting Rings

In this paper, an accurate and efficient method for the analysis of coupled perfectly conducting annular rings is presented. The problem is first formulated as a couple of Integral Equation (IEs) in the Vector Hankel Transform (VHT) domain, considered as unknowns in the cylindrical harmonics of the unknown surface current density. As a second step, Galerkin’s method is applied with suitable expansion functions. The selected functions have two main properties: they reconstruct the expected physical behavior of the nth cylindrical harmonic at the edges of the annular rings, and their VHT transform is analytical and can be expressed in closed-form. Consequently, the method is effective and the problem is regularized, as testified by the truncation error. Comparisons with the commercial software CST Microwave Studio have been carried out and are presented to validate the method.

Correction to: Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix

Correction to: Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix

Visco-Elastic Fluid Model in an Inclined Porous Stenosed Artery with Slip Effect and Body Acceleration

The present paper analyzes an unsteady magnetohydrodynamic blood flow model of an visco-elastic fluid through an inclined porous stenosed artery with body acceleration and slip effect. Navier-Stokes equations have been used to describe the blood flow model. The governing equation of blood flow is solved by an analytic method by considering blood as an incompressible, visco-elastic fluid, and suspension of RBC’s in plasma. Axial velocity, blood acceleration, flow rate, and shear stress are derived numerically by using the finite Laplace and Hankel transformation and their inverse. The effect of parameters such as the visco-elasticity parameter, Womersley number, Hartmann number, inclination angle, parameter of slip, and body acceleration frequency is analyzed. Axial velocity reduces as the Hartmann number and visco-elasticity parameter enhance and it enhances with the enhancement of the slip parameter and inclination angle. The study is beneficial for finding the effect of slip parameter, porosity factor and Hartmann number when a human body is exposed to MRI and CT scan.

Dynamic response of an orthotropic hollow cylinder under thermal shock based on Green–Lindsay theory

This research deals with the thermoelastic response of an orthotropic hollow cylinder under thermal shock, based on generalized thermoelasticity. In order to consider both Lord–Shulman and Green–Lindsay theories of generalized thermoelasticity, a unified formulation is considered for governing equations. The heat conduction equation and the equation of motion for the orthotropic cylinder are derived based on this unified formulation, and an analytical solution using Finite Hankel transformation is presented. A thermal shock in the form of heat flux is applied on the inner surface of the orthotropic cylinder, and then the temperature distribution and the history of stress components are plotted for the classical dynamic, Lord–Shulman and Green–Lindsay theories. The effects of considering different values for the relaxation times in the Green–Lindsay theory and considering orthotropic material properties for the cylinder are shown in the figures. The results are validated by comparing with the known data in the literature, which show good agreement.

Coil optimization of ultra-deep azimuthal electromagnetic resistivity logging while drilling tool based on numerical simulation

This paper presents a coil optimization method of the ultra-deep azimuthal electromagnetic (EM) resistivity Logging While Drilling (LWD) to obtain the best parameters of the tool for formation boundary detection by theory. First, a rapid numerical simulation algorithm, i.e. the Fast Hankel Transform (FHT) method, is developed to simulate the logging responses of the layered formation. A system of double inclined coils is selected based on the response characteristics and the depth of detection (DOD) from four common coils. Simulation results demonstrated that the optimal coils are perfectly suitable for anisotropic formation. Then, the crucial parameters of the frequency and spacing of the selected coils with a DOD of 30 m are determined using the Picasso diagram. In addition, the influences of the electrical parameter of the drilling collar and resistivity anisotropy of the formation on the response are investigated. The results show that the conductivity of the drilling collar, the thickness and the resistivity anisotropy of the formation all affect the DOD. Finally, formation models with different layers are created to verify the performance and signal intensity. The results demonstrate that the optimized coils not only have a large DOD but also possess a high signal intensity, further confirming the advantages of the preferred coils. Compared with the conventional physical experiment tests, the method of coil optimization proposed in this paper is more economical and efficient. The findings of this study can help better understand the detection characteristics of the different coils, which can provide theoretical support for designing and developing a tool with the best boundary capability.

Effect of generalized thermal transport on MHD free convection flows of nanofluids: A generalized Atangana-Baleanu derivative model

The purpose of this article is to analyse the influence of a the generalized thermal transport on the free convection flow of a viscous nanofluid inside a cylindrical tube. The generalized Atangana-Baleanu time-fractional differential operator is used to generalized the heat transport equation. The effect of transverse magnetic field is also considered. The close form solutions are found by use of integral transformations, more precisely Laplace and finite Hankel transformations. The graphs depict the influence of fractional and other physical characteristics. Tables illustrate the effect of the volume fraction parameter. It has found that the heat flux given by the fractional equation influence heat transfer because the values of the temperature gradient are attenuated by the weight function in the heat flux expression. The temperature in the nanofluid has a different evolution concerning the fractional parameter.

Tomographic reconstruction techniques optimized for velocity-map imaging applications.

Examples of extracting meaningful information from image projection data using tomographic reconstruction techniques can be found in many areas of science. Within the photochemical dynamics community, tomography allows for complete three-dimensional (3D) charged particle momentum distributions to be reconstructed following a photodissociation or photoionization event. This permits highly differential velocity- and angle-resolved measurements to be made simultaneously. However, the generalized tomographic reconstruction strategies typically adopted for use with photochemical imaging-based around the Fourier-slice theorem and filtered back-projection algorithms-are not optimized for these specific types of problems. Here, we discuss pre-existing alternative strategies-namely, the simultaneous iterative reconstruction technique and Hankel Transform Reconstruction (HTR)-and introduce them in the context of velocity-map imaging applications. We demonstrate the clear advantages they afford, and how they can perform considerably better than approaches commonly adopted at present. Most notably, with HTR we can set a bound on the minimum number of projections required to reliably reconstruct 3D photoproduct distributions. This bound is significantly lower than what is currently accepted and will help make tomographic imaging far more accessible and efficient for many experimentalists working in the field of photochemical dynamics.

Fractional Thermal Analysis in a Generalized two Dimensional Infinite half Space with Heat Source

A generalized time fractional thermoelastic study under one relaxation time consideration is presented here for the two dimensional model having infinite half space. To analyze the thermal response of the problem, the infinite space is subjected to periodical varying heat source with respect to time. Futher, time derivative involved in heat transfer equation is of Caputo type having order  . Certains thermomechanical boundaries are applied at lower surface. Here the solution is obtained directly by employing Laplace and Hankel transformation but inversion for Laplace is evaluated by numerical method as given by Gaver-Stehfast algorithm. Properties of Copper metal is selected for the numerical analysis and results of temperature and stress distributions are presented graphically by using MATLAB software.

Improvement of Frequency–Bessel Phase-Velocity Spectra of Multicomponent Cross-Correlation Functions from Seismic Ambient Noise

ABSTRACT The frequency–Bessel (F–J) transformation method is effective for the extraction of multimode surface wave dispersion data from ambient noise cross-correlation functions (CCFs). Recently, this method has been improved in terms of increasing resolution and reducing artifacts (or cycle skipping) in Rayleigh wave dispersion measurements. However, these advances are restricted to the ZZ-component F–J method, which is only applicable to Rayleigh waves. In contrast, they have not been extended to Love waves, which are fundamental for determining the horizontally polarized shear-wave velocity and the radial anisotropy associated with it. Furthermore, there is still a lack of a methodology for combining these advances. In this study, we propose a modified multicomponent F–J (MMFJ) method to extract high-quality dispersion data of both the Rayleigh and Love waves. To achieve high resolution, we derive formulas to optimize the MMFJ spectra. With synthetic ambient noise data and USArray data, we demonstrated the effectiveness of the MMFJ method in eliminating “crossed” artifacts and enhancing resolution. In particular, the “crossed” artifacts are greatly reduced using the MMFJ with small seismic arrays when the interstation distances in the seismic array are not dense enough. As such, the new MMFJ method has significant potential for handling seismic arrays with a limited number of receivers and the subsequent tomography of radial anisotropies at high precision.

Using the multi-precision fast Hankel transformation to calculate the constant current field of wide-area layered earth

The constant current field of wide-area layered earth has adverse effects on many industries. But it presents a high degree of numerical singularity, resulting in difficulty in solving. In this paper, the derivation of Green’s function and the fast Hankel transformation are presented based on the basic principle of linear filtering method for Green’s function of wide-area layered earth. A method of Multi-Precision Fast Hankel Transformation (MP-FHT) is proposed, which solves the problem of numerical singularity of integral kernel function in Green’s function by multi-precision algorithm, thus ensuring the accuracy of Hankel integral. The accuracy and reliability of the MP-FHT are proved by comparison of MP-FHT, commercial software CDEGS and improved version of high-order complex image method (HCIM). Compared with the existing algorithms, this method has high precision and low difficulty in algorithm implementation, which is a new method with high practical value.

Static Electro-Mechanical Response of Axisymmetric One-Dimensional Piezoelectric Quasicrystal Circular Actuator.

The piezoelectric effect of piezoelectric quasicrystalline materials is coexcited by phonon and phason fields. Piezoelectric quasicrystalline materials have excellent properties of both piezoelectric materials and quasicrystalline materials, which are expected to be used as actuators in the fields of aerospace, automotive, and intelligent manufacturing. Based on the three-dimensional elastic theory of piezoelectric quasicrystals, the state space equation for axisymmetric piezoelectric quasicrystal circular plate actuators is derived by using the state space method. Afterwards, the finite Hankel transformation is performed on the state equation, and a system of ordinary differential equations and corresponding boundary conditions are obtained. Finally, the exact solution of axisymmetric bending of one-dimensional hexagonal piezoelectric quasicrystal circular actuators under generalized elastic simply supported boundary conditions is obtained by using the propagator matrix method. Numerical results are given to compare the degradation results in this paper with those in the literature, and present the influences of the thickness-to-span ratio and stacking sequence on the phonon, phason, and electric fields when the surface of the laminated circular actuators is subjected to mechanical load. The exact solution obtained does not introduce any deformation assumption; therefore, the exact solution can provide references for numerical calculations of the mechanical behavior of piezoelectric quasicrystals.

Pattern synthesis of linear and ring arrays with minimum number of elements using FFT and Bessel transformation

In antenna engineering, to reduce the final cost of array design, it is often necessary to design an array with the minimum number of elements. So, proper selection of the number of array elements and the location of the elements are two important factors in array design. In this work, a novel analytical approach for designing an array with optimal parameters is described. The Nyquist–Shannon sampling theorem is used to determine the number of array elements and the distance between them. The array's excitation coefficients are then determined using the recursive least square approach and the Bessel transform of the array factor. It is also demonstrated that the introduced procedure can be extended to include the concentric ring arrays. Several practical arrays are evaluated to verify the performance of the suggested approach. The results show that the introduced approach is a good candidate for designing practical arrays with acceptable accuracy.

Discrete fractional Hankel transform based on a nonsymmetric kernel matrix

A discrete fractional Hankel transform (DFRHT) is developed based on the eigen decomposition of a nonsymmetric involutory kernel matrix Y of a discrete Hankel transform (DHT). The two eigen spaces pertaining to the two distinct eigenvalues of Y are not orthogonal because Y is not a normal matrix. Explicit expressions are derived for the oblique projection matrices of Y on its eigen spaces by applying a contributed generalized spectral theorem of linear algebra. Expressions are derived for the dimensions of the two eigen spaces in terms of the trace of matrix Y. An analytic form is derived for the inverse of a modal matrix of Y in order to circumvent the need for numerically evaluating it when computing the sophisticated fractional transform.The fact that the Hankel transform (HT) is self-reciprocating implies that taking the HT once can be viewed as a rotation by an angle π radians in the space – Hankel frequency plane. Being a generalization of the HT, the fractional Hankel transform (FRHT) of fractional order a corresponds to a rotation by an arbitrary angle α where α=πa.Initial orthonormal bases are individually generated for the two eigen spaces by the singular value decomposition of the oblique projection matrices on those spaces. In order for the DFRHT to approximate its continuous counterpart, namely the FRHT, it is desirable to define it in terms of eigenvectors which are as close as possible to samples of the eigen functions of the FRHT. After deriving a three term recursion formula for the eigen functions, a sampling scheme is proposed such that a finite number of samples will be representative of the behavior of those functions. Final optimal orthonormal eigenvectors are individually generated for the two eigen spaces.

Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix

Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix

Effect of Natural Heat Convection on Fractional MHD Second-Grade Fluid in an Infinite Vertically Oscillating Cylinder with Hankel Transform

Effect of Natural Heat Convection on Fractional MHD Second-Grade Fluid in an Infinite Vertically Oscillating Cylinder with Hankel Transform

High accurate method to calculate a singular integral related to Hankel transform

In this paper we are interested in the approximation of the integral \[I_0(f,\omega)=\int_0^\infty f(t)\,e^{-t}\,J_0(\omega t)\,dt\] for fairly large $\omega$ values. This singular integral comes from the Hankel transformation of order $0$, $f(x)$ is a function with which the integral is convergent. For fairly large values of $\omega$, the classical quadrature methods are not appropriate, on the other side, these methods are applicable for relatively small values of $\omega$. Moreover, all quadrature methods are reduced to the evaluation of the function to be integrated into the nodes of the subdivision of the integration interval, hence the obligation to evaluate the exponential function and the Bessel function at rather large nodes of the interval $]0,+\infty[$. The idea is to have the value of $I_0(f,\omega)$ with great precision for large $\omega$ without having to improve the numerical method of calculation of the integrals, just by studying the behavior of the function $I_0(f,\omega)$ and extrapolating it. We will use two approaches to extrapolation of $I_0(f,\omega)$. The first one is the Padé approximant of $I_0(f,\omega)$ and the second one is the rational interpolation.

Spectral Solver for Cauchy Problems in Polar Coordinates Using Discrete Hankel Transforms

Spectral Solver for Cauchy Problems in Polar Coordinates Using Discrete Hankel Transforms