Bessel Functions Research Articles

Sturm–Liouville problem for a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems

The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier’s separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions.

Higher Harmonic Generation Arising from the Bessel Function Expansions of the Carrier-Envelope-Phase Driven by the Femtosecond Optical Kerr Effect

Higher Harmonic Generation Arising from the Bessel Function Expansions of the Carrier-Envelope-Phase Driven by the Femtosecond Optical Kerr Effect

Analysis of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic function and related fractional integral inequalities with applications

In the present work we establish for the first time a class of (P,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(P,\\mathrm{m})$\\end{document}-superquadratic functions and look into its features. Using them, we come up with the Jensen and Hermite–Hadamard inequalities, as well as the fractional versions of Hermite–Hadamard inequalities with respect to Riemann–Liouville fractional integral operators. The findings are confirmed by certain numerical calculations and graphical depictions that take a few appropriate examples into account. The study is enhanced by the addition of applications of special means, moments of random variables, and modified Bessel functions of the first kind. This is achieved by considering new functions pertaining to the uniform probability density function and taking the modified Bessel functions of the first kind into consideration. The new results clearly provide extensions and improvements of the work given in the literature.

A Novel Approach to the Fractional Laplacian via Generalized Spherical Means

Although at least ten equivalent definitions of the fractional Laplacian exist in unbounded domains, we introduce an additional equivalent definition based on the generalized spherical mean-value operator—a Fourier multiplier operator involving the normalized Bessel function. Specifically, we demonstrate that this new definition allows us to reduce any n-dimensional fractional Laplacian to a one-dimensional operator, which simplifies computation and enhances efficiency. We propose two methods for computing the generalized spherical means of a given function: one by solving standard wave equations and the other by solving Darboux’s equations.

ANALYSIS ON MULTIPLICATIVELY (P,m)-SUPERQUADRATIC FUNCTIONS AND RELATED FRACTIONAL INEQUALITIES WITH APPLICATIONS

In this work, we, for the first time, establish a class of multiplicatively [Formula: see text]-superquadratic function and look into its various features. In the light of these features, we come up with the several integer order integral inequalities in the frame of multiplicative calculus. Moreover, we develop the fractional version of Hermite–Hadamard’s type inequalities involving midpoints and end points for multiplicatively [Formula: see text]-superquadratic function with respect to multiplicatively [Formula: see text]-Riemann–Liouville fractional integrals. By choosing different values for the parameters of such integral operators, we acquire a simple version of integral inequalities of Hermite–Hadamard’s type as well as its fractional form via multiplicatively Riemann–Liouville fractional integrals for multiplicatively [Formula: see text]-superquadratic function. The findings are confirmed by graphical illustration by taking appropriate examples into account. The study is further enhanced by the addition of applications of special means and first-type modified Bessel functions. The new results clearly provide extensions and improvements of the work available in the literature.

Imaging of upper breakpoints of buried active faults through microtremor survey technology

Detecting buried active faults presents the challenge of precisely locating the upper breakpoint, the shallowest point in the Quaternary system where faults occur. Microtremor survey technology, unaffected by urban electromagnetic interference, offers an eco-friendly and efficient method for investigating buried faults and stratigraphic structures in urban areas. This research uses microtremor survey technology to identify the upper breakpoint of the buried Nankou-Sunhe Fault in Changping, Beijing. For data collection, 17 microtremor survey points were deployed across the northern section of the Nankou-Sunhe fault, employing a three-point nested circular array with a point spacing of approximately 200 m to form a profile spanning approximately 320 m. For data analysis, the spatial autocorrelation method was utilized. Each measurement point was divided into 9 sets of radii, ranging from a minimum of approximately 4 m to a maximum of 28 m. The correlation coefficients for each set were calculated, and the dispersion curve for each measurement point was generated by fitting the average coefficients with the Bessel function of the first kind of order zero. The apparent S-wave velocity was determined directly from the dispersion curve using empirical formulas and interpolated to generate the contour cross-section map. Integrating the section and inverted S-wave velocity data can significantly enhance interpretation accuracy, and based on these data, the spatial development characteristics and upper breakpoint locations of the Nankou-Sunhe fault zone were analyzed, and the strata shallower than 100 m were deduced. The results align well with known geological data, such as luminescence dating and 14C dating from boreholes at nearby locations.Graphical

Transient and Steady-State Analysis of an M/PH2/1 Queue with Catastrophes

In the paper, we consider the PH2-distribution, which is a particular case of the PH-distribution. In other words, The first service phase is exponentially distributed, and the service rate is μ. After the first service phase, the customer can to go away with probability p or continue the service with probability (1−p) and service rate μ′. We study an analysis of an M/PH2/1 queue model with catastrophes, which is regarded as a generalization of an M/M/1 queue model with catastrophes. Whenever a catastrophe happens, all customers will be cleaned up immediately, and the queuing system is empty. The customers arrive at the queuing system based on a Poisson process, and the total service duration has two phases. Transient probabilities and steady-state probabilities of this queuing system are considered using practical applications of the modified Bessel function of the first kind, the Laplace transform, and probability-generating function techniques. Moreover, some important performance measures are obtained in the system. Finally, numerical illustrations are used to discuss the system’s behavior, and conclusions and future directions of the model are given.

Variational Iteration Method for Solving First Kind Bessel Differential Equation of Order Zero

The aim of this paper is to study the first kind Bessel differential equation of order zero. The variational iteration method (VIM) is used to obtain the approximate solutions of first kind Bessel differential equation (BDE) of order zero. The approximate solution of this equation is calculated in the form of a series which its components are computed, can be calculated rapidlyon computer systems and the convergence of this method is studied. Also two numerical examples are given . The findings, which illustrate the effectiveness and precision of the suggested strategy, are presented in tables and figures.

An Exact Solution of the Fractional Transient Electromagnetic Field Inside an Atmospheric Duct

The objective of this paper is to find the exact solution of the fractional electromagnetic (EM) equation within an atmospheric duct (dielectric medium) of the transient EM field caused by a vertical magnetic dipole and discuss the results in both fractional D-dimensional space and integer space. The novelty of this work is the exact solution of the fractional EM equation in terms of Bessel and Mittage-Leffler functions based on Caputo fractional derivative order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} and the Laplacian operator in D -dimensional fractional space. Using the separation of variables method, the resulting magnetic field can be controlled by D=α1+α2+α3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=\\alpha _1+\\alpha _2+\\alpha _3$$\\end{document} and alpha as spatial and time fractional parameters, respectively. The transient magnetic field behaviour inside the duct is plotted through some of figures depending on fractional parameters D and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}. The classical integer results in usual space are recovered from fractional solution at D=α=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=\\alpha =1$$\\end{document}. The main results highlighted are that spatial frequencies grow at a faster rate during EM wave propagation; thus, the amplitude of the propagated wave in integer space is greater than that in fractional space. Increasing the fractional parameters D and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} simultaneously increases the amplitude of the EM wave and can be used to control the power and energy of the EM wave.

Fractional Wiener chaos: Part 1

AbstractIn this paper, we introduce a fractional analogue of the Wiener polynomial chaos expansion. It is important to highlight that the fractional order relates to the order of chaos decomposition elements, and not to the process itself, which remains the standard Wiener process. The central instrument in our fractional analogue of the Wiener chaos expansion is the function denoted as $${\mathcal {H}}_\alpha (x,y)$$ H α ( x , y ) , referred to herein as a power-normalised parabolic cylinder function. Through careful analysis of several fundamental deterministic and stochastic properties, we affirm that this function essentially serves as a fractional extension of the Hermite polynomial. In particular, the power-normalised parabolic cylinder function with the Wiener process and time as its arguments, $${\mathcal {H}}_\alpha (W_t,t)$$ H α ( W t , t ) , demonstrates martingale properties and can be interpreted as a fractional Itô integral with 1 as the integrand, thereby drawing parallels with its non-fractional counterpart. To build a fractional analogue of polynomial Wiener chaos on the real line, we introduce a new function, which we call the extended Hermite function, by smoothly joining two power-normalized parabolic cylinder functions at zero. We form an orthogonal set of extended Hermite functions as a one-parameter family and use tensor products of the extended Hermite functions as building blocks in the fractional Wiener chaos expansion, in the same way that tensor products of Hermite polynomials are used as building blocks in the Wiener chaos polynomial expansion.

Analytical solution for radial consolidation of combined electroosmosis-vacuum-surcharge preloading considering free strain and cyclic loading

In this study, electro-osmotic consolidation considering smear effect and free strain under cyclic loading was investigated. The analytical solution of radial consolidation of electroosmosis-vacuum-surcharge combined preloading is derived by using the Bessel function and eigenfunction methods. Subsequently, the effectiveness of the proposed method is validated through comparison with existing numerical solutions. Based on the derived solutions, the influence of the smear effect, applied voltage, vacuum pressure, and cyclic loading on soil consolidation characteristics was analyzed. The results showed that the smearing effect slows the rate of consolidation, but the final average consolidation and negative excess pore water pressure are enhanced. Compared with only cyclic loading, the combined effect of electroosmosis, vacuum, and surcharge preloading enables the soil to achieve higher strength and consolidation. When the effect of electroosmosis alone on reinforcing low-permeability soils is not significant, the combination of electroosmosis with vacuum preloading helps enhance the soil reinforcement effect.

An Efficient Numerical Algorithm for Abnormal Integrals with Arbitrary Order Double Bessel Functions of the First Type

Abstract The integrals of the first type of double Bessel functions have a wide range of applications in geological exploration, mechanical and electromagnetic responses, signal processing, scattering, and wetting. In this paper, we develop a linear transformation accelerated convergence algorithm (LTACA) that combines the large argument approximate expression of the Bessel function (LAAEBF) and the integral accumulation to provide an efficient numerical algorithm for abnormal integrals with arbitrary order double Bessel functions of the first type. The effectiveness and high efficiency of the algorithm are verified by numerical examples, and its high accuracy is demonstrated by comparison with the Gaver-Stehfest inverse Laplace transform method (GSILTM). This offers a reliable and efficient computational method for the study of signal processing and mechanical problems.

A single layer representation of the scattered field for multiple scattering problems

The scattering of scalar waves by a set of scatterers is considered. It is proven that the scattered field can be represented as an integral supported by any smooth surface enclosing the scatterers. This is a generalization of the series expansion over spherical harmonics and spherical Bessel functions for spherical geometries. More precisely, given a set of scatterers, the field scattered by any subset can be expressed as an integral over any smooth surface enclosing the given subset alone. It is then possible to solve the multiple scattering problem by using this integral representation instead of an expansion over spherical harmonics. This result is used to develop an extension of the Fast Multipole Method in order to deal with subsets that are not enclosed within non-intersecting balls.

Entanglement propagation in integrable Heisenberg chains from a new lens

Abstract The exact single-magnon entanglement evolution in Heisenberg chains is obtained using the Quantum Correlation Transfer Function (QCTF) formulation. A dual, i.e., frequency and time-domain, analysis shows that the transient dynamics of individual spins' entanglement is described via a Bessel function of the first kind. Through QCTF, we bypass the evaluation of the full system's state for the purpose of obtaining entanglement. Although it is known that the observable entanglement edge is formed by the arrival of a stream of quasi-particles that travel with the maximum group velocity, we show how the early quasi-particles travel faster than the maximum group velocity of the chain and contribute to entanglement production. Our results can be extended to the multi-magnon regime, therefore opening up the means to better interpret equilibration dynamics and thermodynamics in Heisenberg chains.

Analytical solutions for the magneto-fluid–structure interaction dynamics with two degrees of freedom

The study focuses on the interaction between fluid and structure under an external magnetic field. The system consists of two rigid cylinders, with the inner cylinder having either translating vibrating or rotating vibrating freedoms. The solutions describing the coupling characteristics among the multi-physics fields are derived, using the zero- and first-order complex modified Bessel functions. The study analyzes the effects of the magnetic field on flow fields, electrical potential distributions, and cylinder vibration characteristics. It also reveals the physical mechanisms behind the coupled effects among fluid flow, cylinder vibration, and the electrical potential field. The study finds that for initial tangential rotating vibration, the magnetic field can stabilize both the flow field and cylinder vibration. Additionally, an increasing magnetic field can transform the cylinder vibration from underdamping to overdamping. On the other hand, for initial translating vibration, the magnetic field can induce instability in the system, with larger magnetic fields leading to divergent, negatively damped vibration. The critical magnetic fields for both rotating and translating vibrations are determined using phase maps. The study also shows that the combined effects of rotating and translating vibrations lead to asymmetry in the flow field. While translating vibration plays a dominant role in the combined vibration, it nonlinearly and non-monotonously influences the system dynamics.

T-matrix of piezoelectric shunt inclusions on a thin plate

Developing piezoelectric-based plate-like metamaterials necessitates an effective modeling method to elucidate the omnidirectional wave properties of piezoelectric coupled inclusions on a thin plate. The commonly used methods, such as the transfer-matrix method and finite element method, are inadequate for analyzing the transmission and reflection of omnidirectional waves in a two-dimensional elastic medium. Unlike the existing methods, the multiple scattering method employs Bessel functions as the displacement-basis methods which can accurately describe the propagation and scattering characteristics of omnidirectional waves. This paper develops a novel T-matrix formulation to support the multiple scattering method, representing the input-output relationship between incident and reflected waves from a piezoelectric shunt inclusion on a host thin plate. The piezoelectric shunt inclusion comprises a varying-thickness substrate bonded with piezoelectric shunting patches. The T-matrix of the piezoelectric shunt inclusion is formulated by integrating the wave-based method with Rayleigh-Ritz method. The derived T-matrix is then used to semi-analytically analyze the far-field scattering and reflection properties of a single inclusion. Numerical results capture the scattering properties resulting from trapped mode resonances of the piezoelectric shunt inclusion. Additionally, the capability of the piezoelectric shunt damping to design and tune multiple critical coupling conditions for axisymmetric modes of thin plates is parametrically investigated by varying the values of inductors and resistors.

Series over Bessel Functions as Series in Terms of Riemann’s Zeta Function

Relying on the Hurwitz formula, we find closed-form formulas for the series over sine and cosine functions through the Hurwitz zeta functions, and using them and another summation formula for trigonometric series, we obtain a finite sum for some series over the Riemann zeta functions. We apply these results to the series over Bessel functions, expressing them first as series over the Riemann zeta functions.

Monotonicity of zeros of derivatives of Bessel functions

Recently Baricz et al., 2018 and Baricz and Singh 2018 gave two different proofs of the fact that the zeros of the nth derivative of the Bessel function of the first kind Jν(x) are all real when ν>n−1. We provide a third alternative proof. The authors of Baricz et al., 2018 conjectured that, for every n∈N, the positive zeros of Jν(n)(x) are increasing functions of the parameter ν, for ν∈(n−1,∞). We provide two apparently distinct proofs of the conjecture.

Uniform Asymptotic Expansions for the Zeros of Bessel Functions

Uniform Asymptotic Expansions for the Zeros of Bessel Functions

Advantages of Direct Laser Writing for Enhancing the Resolution of Diffractive Optical Element Fabrication Processes

A technology for direct laser writing of code sequences on modulation disks has been developed and implemented, ensuring high accuracy and reliability in the process of forming structural elements. The main advantages of applying direct laser writing compared to contact lithography for forming submicron-sized elements have been demonstrated. The proposed technology is characterized by high resolution and flexibility in configuring the parameters of the optical recording system, making it suitable for a wide range of applications in micro-optics. Direct optical writing is presented as a promising approach for enhancing the resolution of optical systems used in recording submicron-sized diffractive optical elements, as this technology enables the creation of complex optical structures. Additionally, a detailed classification of current approaches for further increasing the resolution of the optical recording system was conducted, including the application of a saturated absorber layer on the photosensitive surface, the use of a laser beam with intensity modeled by a Bessel function, and the synthesis of photosensitive materials with optimized exposure characteristics, ensuring high efficiency and accuracy in the process of direct laser writing.