Hankel Transform Research Articles

Gabor Transform Associated with the Dunkl–Bessel Transform and Spectrograms

Time–frequency (or space–phase) analysis plays a key role in signal analysis. In particular, signals that have a very concentrated time–frequency content are of great importance. However, the uncertainty principle sets a limitation to the possible simultaneous concentration of a function and its Dunkl–Bessel transform. For this purpose, we introduce and study a new transformation called Dunkl–Bessel Gabor transform. For this transformation, we define the Toeplitz-type (or time–frequency localization) operators, in order to localize signals on the time–frequency plane. We study these operators; in particular, we give criteria for their boundedness and Schatten class properties. Then, using the special class of concentration operators, which are compact and self-adjoint, we show that their eigenfunctions are maximally time–frequency-concentrated in the region of interest.

A comprehensive approach with DTW-driven IMF selection, multi-domain fusion, and TSA-based feature selection for compound fault diagnosis

Fault diagnosis in industrial machinery ensures operational efficiency and prevents downtimes. However, compound faults present a significant challenge due to their complex nature. This manuscript proposes a novel methodology for identifying compound faults, integrating advanced techniques to address the challenges present in existing methods. Firstly, the ICEEMDAN algorithm is employed for signal decomposition, enabling the extraction of noise-free IMFs crucial for identifying fault signatures. A DTW-based selection criteria is introduced to enhance the accuracy of IMF selection. The Bessel Transform is utilized for time–frequency conversion. HU invariant moments are incorporated as image features, and multi-domain feature fusion is implemented to understand compound faults comprehensively. A novel TSA-based feature selection method is proposed to select optimal features from the fused feature set. The methodology’s effectiveness is evaluated through a case study of compound gear-bearing faults. The results demonstrate the superiority of the approach, with a testing accuracy of 99 %.

Dynamic behavior of the orthotropic hollow cylinder under a magnetic field: a thermoelastic analysis

The present article focuses on the complete analytical solution of an orthotropic hollow cylinder using a proposed magneto-thermoelastic model and finite Hankel transformation. The study considers the Lord–Shulman and Green–Lindsay models of hyperbolic generalized thermoelasticity within the context of thermoelasticity theory. The analytical expressions of the field variables of the cylinder in a magnetic field were obtained under thermo-mechanical loadings. The study also offered closed-form field functions and measured the thermoelastic behavior concerning the magnetic field intensity of the cylinder based on thermoelastic theories involving a second sound. The inclusion of hyperbolic-type heat conduction models enhanced the visualization of the reflection of thermal waves into the cylinder. The study also investigated the impacts of orthotropic material properties on the considered medium and compared the results with the existing literature.

Copper ratio obtained by generalizing the Fibonacci sequence

In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

Abelian Theorems for the Wavelet Transform in Terms of the Fractional Hankel Transform

Abelian Theorems for the Wavelet Transform in Terms of the Fractional Hankel Transform

Asymptotic Expansions of Finite Hankel Transforms and the Surjectivity of Convolution Operators

A compactly supported distribution is called invertible in the sense of Ehrenpreis-Hörmander if the convolution with it induces a surjection from $\mathcal{C}^{\infty}(\mathbb{R}^{n})$ to itself. We give sufficient conditions for radial functions to be invertible. Our analysis is based on the asymptotic expansions of finite Hankel transforms. The dominant term may be the contribution from the origin or from the boundary of the support of the function. For the proof, we propose a new method to calculate the asymptotic expansions of finite Hankel transforms of functions with singularities at a point other than the origin.

A Smart CEEMDAN, Bessel Transform and CNN-Based Scheme for Compound Gear-Bearing Fault Diagnosis

A Smart CEEMDAN, Bessel Transform and CNN-Based Scheme for Compound Gear-Bearing Fault Diagnosis

Photothermal effects induced by laser radiation in a 2D axi-symmetric generalized thermoelastic semiconducting half space using MGL model

In this paper, a novel model based on the strain-temperature rate-dependent theory is investigated to study the coupling influences of thermal, elastic, and plasma waves during the photothermal process in a 2D axi-symmetric semiconducting half space. The surface absorption technique is used to illuminate the surface of the medium by a laser beam whose temporal and spatial profiles are Gaussian, moreover, the surface is also considered traction free. Integral transform method is applied using Laplace and Hankel transformations to obtain a general solution to the considered problem. Inverse Laplace transform is obtained computationally using Rimann-Sum approximation. Silicon element is chosen as an application to show the compatibility of the results of the MGL model with GL and CTE models, moreover the relaxation times effects of the new model were studied.

Fast computation of highly oscillatory Bessel transforms

This study focuses on the efficient and precise computation of Bessel transforms, defined as ∫abf(x)Jν(ωx)dx. Exploiting the integral representation of Jν(ωx), these Bessel transformations are reformulated into the oscillatory integrals of Fourier-type. When a>0, these Fourier-type integrals are transformed through distinct complex integration paths for cases with b<+∞ and b=+∞. Subsequently, we approximate these integrals using the generalized Gauss–Laguerre rule and provide error estimates. This approach is further extended to situations where a=0 by partitioning the integral’s interval into two separate subintervals. Several numerical experiments are provided to demonstrate the efficiency and accuracy of the proposed algorithms.

Asymptotic Estimates for the Growth of Deformed Hankel Transform by Modulus of Continuity

Asymptotic Estimates for the Growth of Deformed Hankel Transform by Modulus of Continuity

Finite Hankel Transform Method–Based Analysis of Axisymmetric Free-Strain Consolidation Theory for Unsaturated Soils

Finite Hankel Transform Method–Based Analysis of Axisymmetric Free-Strain Consolidation Theory for Unsaturated Soils

A convenient switch design for high time resolution and energy resolution in ion velocity imaging

Time-sliced velocity map imaging (VMI) has extensively been applied in photodissociation dynamics studies, thanks to its unique advantages, such as high energy resolution and no requirement of inverse Abel or Hankel transformations. However, its time resolution is generally insufficient for distinguishing adjacent m/z ions with a certain kinetic energy due to the overlapping of time-of-flight distributions. Herein, we have made a novel and convenient switch design for the common ion optics in three-dimensional (3D) VMI. By simply introducing two additional resistors out of the vacuum chamber, the strength ratio of the extraction and acceleration fields is easily changed from 3D VMI to two-dimensional (2D) VMI under optimized conditions, as well as a significant extension of free drift length, leading to a higher time resolution while maintaining the high energy resolution. As a result, 2D and 3D VMI can be quickly switched without breaking the vacuum and replacing the electrostatic plates.

A High Frequency Vibration Compensation Approach for Ultra-High Resolution SAR Imaging Based on Sinusoidal Frequency Modulation Fourier-Bessel Transform

A High Frequency Vibration Compensation Approach for Ultra-High Resolution SAR Imaging Based on Sinusoidal Frequency Modulation Fourier-Bessel Transform

A Unified Approach for Compound Gear-Bearing Fault Diagnosis Using Bessel Transform, Artificial Bee Colony-Based Feature Selection and LSTM Networks

A Unified Approach for Compound Gear-Bearing Fault Diagnosis Using Bessel Transform, Artificial Bee Colony-Based Feature Selection and LSTM Networks

Composition of linear canonical Hankel pseudo-differential operators

In this paper, the study of linear canonical Hankel pseudo-differential operators has been extended to Sobolev-type spaces. Two versions of pseudo-differential operators involving linear canonical Hankel transformation are defined. We have shown that the pseudo-differential operators and composition of pseudo-differential operators are bounded in certain Sobolev-type spaces associated with linear canonical Hankel transformations.

Boas Type and Titchmarsh Type Theorems for Generalized Fourier-Bessel Transform

Let $$n\in \mathbb Z_{+}=\{0,1,\dots \}$$ , $$\nu >-1/2$$ . For a function f in the space $$L^{1}_{n,\nu }(\mathbb R_{+})$$ of special kind, we consider the generalized Fourier-Bessel transform $$\mathcal F_{n,\nu }(f)$$ . We prove a Boas type result about necessary and sufficient conditions for f to belong to the generalized uniform Lipschitz classes in terms of $$\mathcal F_{n,\nu }(f)$$ . Also an analogue of classical Titchmarsh theorem describing Lipschits classes in $$L^{2}$$ space with power weight is established in a more general setting.

Some Relations between Stieltjes Transform and Hankel Transform with Applications

In the present paper four theorems connecting Stieltjes transform and Hankel transform are established. The theorems are general in nature. Four integral formulae involving special functions are obtained with the help of these theorems. Otherwise it is very difficult to evaluate such type of integrals. Other several integrals may be evaluated with the help of these theorems.

Study of fractional-order model on Casson blood flow in stenosed artery with magnetic field effect

In this research work, the impacts of magnetic fields on Casson blood flow with stenosed artery are studied. The flow under investigation is affected by a magnetic field and an oscillating pressure gradient. The Caputo–Fabrizio fractional-order derivative was used to express the governing system of partial differential equations with imposed boundary and initial conditions. The governing problem is solved using Laplace and Finite Hankel transformations. The exact solution for blood velocity, magnetic particle velocity, and temperature profiles are defined. For better understanding of the various physical parameter, numerical results are obtained for blood velocity, magnetic particle velocity and temperature profiles and presented through the graphs. It is seen that the magnetic fields tend to reduce the blood viscosity which can be used for controlling the blood flow rate. It is also seen that Casson fluid parameter tends to improve the blood flow rate. From the results, it is illustrated that the heat transfer process improves with thermal radiation and heat generation parameter.

On Recovery of the Singular Differential Laplace—Bessel Operator from the Fourier–Bessel Transform

This paper is devoted to the problem of the best recovery of a fractional power of the B-elliptic operator of a function on R+N by its Fourier–Bessel transform known approximately on a convex set with the estimate of the difference between Fourier–Bessel transform of the function and its approximation in the metric L∞. The optimal recovery method has been found. This method does not use the Fourier–Bessel transform values beyond a ball centered at the origin.

Diffraction of the Field of a Grounded Cable on an Elongated Dielectric Spheroid in a Conducting Layer

Based on a rigorous solution to the problem, analytical expressions are obtained for calculating the diffraction of the electromagnetic field of a grounded cable on an elongated dielectric spheroid in a conductive layer. The field of a grounded AC cable in a conductive layer is determined by solving the Helmholtz equation for the vector potential by using the method of integral Fourier–Bessel transformations, taking into account the boundary conditions at the bottom and surface of the conductive layer. The process of finding the secondary field of an elongated dielectric spheroid on an alternating current in a conducting layer is divided into two stages. First, we find an exact solution to the problem of an elongated dielectric spheroid at a constant current in a homogeneous field, in free space, decomposing this solution into a Taylor series and retaining the first term, which is a dipole approximation. In the second stage, the resulting field as the sum of the fields of the horizontal and vertical dipoles is analytically continued into the frequency domain. The field of the horizontal and vertical dipoles in the conducting layer is obtained by using the method of integral Fourier–Bessel transformations, taking into account the boundary conditions at the bottom and surface of the conducting layer. The resulting solution is presented in a closed form in elementary functions and has an accuracy level acceptable for the practice. Graphs showing the flow characteristics of an elongated dielectric spheroid modeling a swimmer in a light diving suit are given. The influence of the water–air boundary on the increase in the secondary field of the dielectric spheroid, which leads to an increase in the reliability of object detection, is revealed. The practical implementation of the described device protected by a patent and the experimental data of testing the device layout on the Gulf of Finland are given. A good agreement between the theoretical and experimental flow characteristics of a dielectric object both in shape, amplitude, and phase, is revealed.