Bessel Functions Research Articles

The Memory Function of an Impurity in Mass and Hooke Constant in a Diatomic Chain

The memory function of an impurity with different mass and Hooke constant in a classical diatomic chain is studied by means of the recurrence relations method. The Laplace transform of the memory function has two pairs of resonant poles and three branch cuts. The poles contribute a cosine and the cuts contribute acoustic and optic branches, which are expressed as a convolution of a difference of two sines and an expansion of even-order Bessel functions. The expansion coefficients are integrals of Jacobian elliptic function sn(u) along the real axis in a complex u+iv plane for the acoustic branch, and integrals of nd(v) along a contour parallel to the imaginary axis of the plane for the optical branch, respectively. Different special cases are discussed in detail. It shows that a perfect monatomic or diatomic chain and such a chain with a mass impurity share the same memory function except for a constant factor, and that the pole contribution exists only if the impurity has both different mass and Hooke constant.

Certain geometric properties of generalized Bessel-Maitland function

"In the present study, we first introduce Generalized Bessel-Maitland function $\mathbb{J}^{\xi }_{\zeta,a}(z)$ and then derive sufficient conditions under which the Generalized Bessel-Maitland function $\mathbb{J}^{\xi}_{\zeta,a}(z)$ have geometric properties like univalency, starlikeness and convexity in the open unit disk $\mathscr{D}$. Keywords: Univalent, starlike, convex and close-to-convex function, subordination, Bessel functions, Bessel-Maitland functions."

New algorithms for approximating oscillatory Bessel integrals with Cauchy-type singularities

In this paper, we present an efficient numerical algorithm for approximating integrals involving highly oscillatory Bessel functions with Cauchy-type singularities. By employing the technique of complex line integration, the highly oscillatory Bessel integrals are transformed into oscillatory integrals with a Fourier kernel. When the integration interval does not contain zeros, we use Cauchy’s theorem to transform the integration path to the complex plane and then use the Gaussian–Laguerre formula to compute the integral. For cases in which the integration interval contains zeros, we decompose the integral into two parts: the ordinary and the singular integral. We give a stable recursive formula based on Chebyshev polynomials and Bessel functions for ordinary integrals. For singular integrals, we utilize the MeijerG function for efficient computation. Numerical examples verify the effectiveness of the new algorithm and the fast convergence.

A multivariate heavy-tailed integer-valued GARCH process with EM algorithm-based inference

A new multivariate integer-valued Generalized AutoRegressive Conditional Heteroscedastic (GARCH) process based on a multivariate Poisson generalized inverse Gaussian distribution is proposed. The estimation of parameters of the proposed multivariate heavy-tailed count time series model via maximum likelihood method is challenging since the likelihood function involves a Bessel function that depends on the multivariate counts and its dimension. As a consequence, numerical instability is often experienced in optimization procedures. To overcome this computational problem, two feasible variants of the expectation-maximization (EM) algorithm are proposed for estimating the parameters of our model under low and high-dimensional settings. These EM algorithm variants provide computational benefits and help avoid the difficult direct optimization of the likelihood function from the proposed process. Our model and proposed estimation procedures can handle multiple features such as modeling of multivariate counts, heavy-tailedness, overdispersion, accommodation of outliers, allowances for both positive and negative autocorrelations, estimation of cross/contemporaneous-correlation, and the efficient estimation of parameters from both statistical and computational points of view. Extensive Monte Carlo simulation studies are presented to assess the performance of the proposed EM algorithms. Two empirical applications of our approach are provided. The first application concerns modeling bivariate count time series data on cannabis possession-related offenses in Australia, while the second one involves modeling intraday high-frequency financial transactions data from multiple holdings in the U.S. financial market.

Logarithmic singularity subtraction for matrix‐friendly layered medium Green's functions

AbstractA subtraction procedure is proposed to evaluate the logarithmic singularity for matrix‐friendly layered medium Green's functions. In this subtraction procedure, an integral of the Bessel function combined with the exponential function and power function, where the sum of the power of and the order of the Bessel function is −1, is designed to fit the oscillation of the tail integral. After subtracting the designed integral, the integrand of the remainder decreases rapidly. Accordingly, the remainder can be calculated without finding any surface poles or steepest descent path, while the subtracted part can be calculated analytically. Several numerical examples are provided to reveal the feasibility and the accuracy of the proposed method.

EDL Induced Electro-magnetized Modified Hybrid Nano-blood Circulation in an Endoscopic Fatty Charged Arterial Indented Tract.

The electrokinetic process for streaming fluids in magnetic environments is emerging due to its immense applications in medical and biochemical industrial domains. In this context, our proposed model seeks to inquire into the hemodynamic characteristics of electro-magnetized blood blended with trihybrid nanoparticles circulation induced by electro-osmotic forces in an endoscopic charged arterial annular indented tract. This steaming model also invokes the consequences of variable Lorentz attractive force, buoyancy force, heat source, viscous and Joule warming, arterial wall properties, and sliding phenomena for featuring more realistic problems in blood flows. Different shapes of suspended trihybrid nanoparticles, such as spheres, bricks, cylinders, and platelets, are included in the model formation. Electro-magnetized modified hybrid nano-blood is an electro-conductive solution comprising blood as base fluid and magnetized trihybrid nanoparticles (copper, gold, and alumina). Closed-form solution in terms of Bessel's functions is gotten for electro-osmotic potential due to the electric double layer (EDL). The homotopy perturbation methodology is implemented in order to track down the convergent series solutions of non-linear coupled flow equations being elicited. The physical attributes of distinct evolving parameters on the different dimensionless hemodynamic profiles and quantities of interest are elucidated evocatively via a sort of graphs and charts. The ancillary outcomes proved that the Debye-Hückel parameter and Helmholtz-Smoluchowski velocity have a dual impact on the ionized bloodstream. The bloodstream rapidity is alleviated/boosted for the assisting/opposing electroosmosis process. Cooling of ionized blood in the endoscopic arterial conduit is achieved with lower Hartmann numbers. Copper-gold-alumina/blood exhibits a superior heat transmission rate across the arterial wall than copper-gold-blood, copper-blood, and pure blood. Additionally, the contour topology for the bloodstream in the flow domain is briefly elaborated. The contour distribution is significantly amended due to the variant of the Debye-Hückel parameter. The model's new findings may be invaluable in electro-magneto-endoscopic operation, electro-magneto-treatment for cancer, surgical process, etc.

ANALYTICAL ANALYSIS OF COMBINED FOUNDATION PLATES, SUBJECTED TO AN ACTION OF ANTISYMMETRIC LOADS

The combined foundation plates, having circular form and consisting of several parts which have different laws of thickness variation, are under study. The constructions under examination are subjected to an action of antisymmetric load distributed along circumferences according to the laws or . Similar problems occur in the cases of seismic and wind loads. The interaction with elastic foundation is taking into account. The conditions of the construction’s different parts conjugation are fulfilled. In this work for solution of such problems the analytical approach is used for the first time. The method of compensating loads (MCL) is applied. The solutions are obtained in closed form in terms of Bessel functions. As an example the foundation plate, consisting of two parts, is considered in detail. The inner part of the slab under study has the variable thickness, the outer one has the constant thickness.

Analytical Solution for the Steady Seepage Field of an Anchor Circular Pit in Layered Soil

An analytical study was carried out on an anchored circular pit with a submerged free surface in layered soil. The seepage field around the anchor circular pit was divided into three zones. Separate variable method was used to obtain the graded solution forms of head distribution in the column coordinate system for each of the three regions. Combined with the continuity condition between the regions the Bessel function orthogonality was used to obtain the explicit analytical solution of the seepage field in each region, and the infiltration line was determined. Comparison with the calculation results of Plaxis 2D 8.5 software verified the correctness of the analytical solution. Based on the analytical solution, the influence of the radius of the pit and the distance of the retaining wall from the top surface of the impermeable layer on the total head distribution on both sides of the retaining wall was analyzed. And the variation in the infiltration line was determined with the above parameters. The results show that as the pit radius, r, decreased, the total head on both sides of the retaining wall gradually increased, and the height of the submerged surface drop also increased. As the distance, a, between the retaining wall and the impermeable boundary at the bottom increased, the hydraulic head on the outer side of the retaining wall decreased and the head on the inner side increased. The height of the submerged surface drop increased with decreasing depth of insertion of the retaining wall. The depth of insertion of the retaining wall had a greater influence on the degree of diving surface drop than the pit radius.

Neumann series of Bessel functions for inverse coefficient problems

Consider the Sturm–Liouville equation with a real‐valued potential . Let be its solution satisfying certain initial conditions for a number of , where , and are some complex numbers. Denote , where . We propose a method for solving the inverse problem of the approximate recovery of the potential and number from the following data . In general, the problem is ill‐posed; however, it finds numerous practical applications. Such inverse problems as the recovery of the potential from a Weyl function or the inverse two‐spectra Sturm–Liouville problem are its special cases. Moreover, the inverse problem of determining the shape of a human vocal tract also reduces to the considered inverse problem. The proposed method is based on special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid the problem is reduced to the classical inverse Sturm–Liouville problem of recovering from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples.

Geometric Nature of Special Functions on Domain Enclosed by Nephroid and Leminscate Curve

In this work, the geometric nature of solutions to two second-order differential equations, zy′′(z)+a(z)y′(z)+b(z)y(z)=0 and z2y′′(z)+a(z)y′(z)+b(z)y(z)=d(z), is studied. Here, a(z), b(z), and d(z) are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution F(z) (with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve 1+z. We construct several examples by the judicious choice of a(z), b(z), and d(z). The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion.

Analytic solution for pulse wave propagation in flexible tubes with application to a patient-specific arterial tree

In this paper, we present an analytic solution for pulse wave propagation in a flexible arterial model with tapering, physiological boundary conditions and variable wall properties (wall elasticity and thickness). The change of wall properties follows a profile that is proportional to $r^\alpha$ , where $r$ represents the lumen radius and $\alpha$ is a material coefficient. The cross-sectionally averaged velocity and pressure are obtained by solving a hyperbolic system derived from the mass and momentum conservations, and they are expressed in Bessel functions of order $(4-\alpha )/(3-\alpha )$ and $1/(3-\alpha )$ , respectively. The solution is successfully validated by comparing it with numerical results from three-dimensional (3-D) fluid–structure interaction simulations. Subsequently, the solution is employed to study pulse wave propagation in an arterial model, revealing that the wall properties and the physiological outlet boundary conditions, such as the resistor–capacitor–resistor (RCR) model, play a crucial role in characterizing the input impedance and reflection coefficient. At low-frequency range, the input impedance is found to be insensitive to the wall properties and is primarily determined by the RCR parameters. At high-frequency range, the input impedance oscillates around the local characteristic impedance, and the oscillation amplitude varies non-monotonically with $\alpha$ . Expressions for the input impedance at both low-frequency and high-frequency limits are presented. This analytic solution is also successfully applied to model flow inside a patient-specific arterial tree, with the maximum relative errors in pressure and flow rate never exceeding $1.6\,\%$ and $9.0\,\%$ when compared with results from 3-D numerical simulations.

ON A CLASS OF NONSTATIONARY CURVES IN HILBERT SPACE

Stationary random processes have been studied quite well over recent years starting with the works of A. N. Kolmogorov. The possibility of building nonstationary random process correlation theory was considered in the monographs by M. S. Livshits, A. A. Yantsevich, V. A. Zolotarev and others. Some classes of nonstationary curves were investigated by V. E. Katsnelson. In this paper nonstationary random processes are represented as curves in Hilbert space which "slightly deviate" from random processes with the correlation function of special kind. The infinitesimal correlation function has been introduced; in essence, this function characterizes the deviation from the correlation process with the given correlation function. The paper discusses the cases of nonstationary random processes, the operator of which has one‑dimensional imaginary component. Cases of a dissipative operator with descrete spectrum are also considered in this work. It is shown that the nonstationarity of the random process is closely related to the deviation of the operator from its conjugated operator. Using the triangle and universal models of non‑self‑ajoint operators it is possible to obtain the representation for the correlation function in the case of nonstationary process which replaces the Bochner – Khinchin representation for stationary random processes. The expresson for the infinitesimal correlation function was obtained for different cases of operator spectrum: for the descrete spectrum placed in the upper half‑plane and for the contrast‑free spectrum at zero. In the case of dissipative operator with descrete spectrum the infinitesimal function can be found in terms of special lambda function. For Lebesque spaces of compex‑valued squared integratable functions the expresson of infinitesimal function was found in terms of special zero order modified Bessel function. It was shown that a similar approach can be applied for the evolutionarily represented sequences in Hilbert spaces.

Vibration Analysis of Internally Perforated Annular Plate with Discontinuous Boundary Conditions

Because of the introduction of modal coupling, discontinuous boundaries have a significant impact on the vibration characteristics of annular plates. Based on the spectral element method (SEM) and parametric equivalent method (PEM), the vibration calculation and analysis of internally perforated annular plate with discontinuous boundary are explained. Firstly, a uniform annular plate is regarded as a spectral element, and the vibration response of an annular plate with general elastic boundaries is obtained based on the motion equation of the annular plate and the virtual spring method (VSM). The displacement solution is described by the Bessel function, and a fitting method of discontinuous spring stiffness coefficient based on the weighted least square norm least square method is proposed. Based on the force equilibrium condition, the vibration response of an annular plate with a discontinuous boundary is derived, and the reasons for the change of vibration characteristics with a discontinuous boundary are explained from modal superposition. The dynamic stiffness matrix method is used to establish the vibration equation of an annular plate with a discontinuous boundary, and the influence of internal openings on the vibration characteristics of an annular plate is effectively calculated by the parametric equivalent method. Finally, a finite element simulation model is established to test the accuracy and stability of the proposed method. The influence of general discontinuous boundary conditions on structural dynamic behavior is analyzed by numerical examples, which provides useful guidance for structural vibration analysis with discontinuous boundary conditions.

Modes of a Nonlinear Fiber Containing a Parabolic Graded-Index Core with Light Induced Variable Radius

"New features of the light localization in a nonlinear and graded-index medium in the case of radial symmetry are described analytically. The model of nonlinear graded-index fiber assumes that the dielectric permittivity changes abruptly when the electric field amplitude reaches a certain level. The dielectric permittivity depends on the polar radius according to a parabolic law, in the regions where the electric field amplitude exceeds a certain level. Explicit exact solution to the wave equation is found, in terms of the Whittaker function and the modified Bessel function of the second kind, describing a new type of optical localized structure. The influence of the propagation constant and parameters of the nonlinear graded-index dielectric permittivity on the field profile over the fiber radius is analyzed. It is derived that the core radius and the dielectric permittivity of the core depend on the propagation constant. The dependence of the core radius on the propagation constant causes the appearance of local dispersion."

On the Multiple Spectrum of a Problem for the Bessel Equation of an Integer Order with Squared Spectral Parameter in the Boundary Condition

The problem for the Bessel equation of an integer order with complex physical and spectral parameters in the boundary condition is considered. The spectral parameter enters the boundary condition quadratically. The question of the basis property of the system of eigenfunctions in the case of the appearance of a multiple eigenvalue is studied

Multi-user communications for line-of-sight large intelligent surface systems

In this paper, we propose a new interference-suppression beamformer design and a Direction-of-Arrival (DoA) estimation method for Line-of-Sight (LoS) multi-user communication systems with circular Large Intelligent Surfaces (LISs). This symmetrical circular-shaped LIS allows signals defined on it to be formed by combining Fourier and Bessel series. The Fourier harmonics are orthogonal along the rotational direction and the Bessel functions constituting the Bessel series are orthogonal along the radial direction. Synthesizing basis functions from these two series results in the proposed interference-suppression beamformer, with their coefficients designed to cancel inter-user interference. Also by exploiting signal structure of a circular LIS communicating with Mobile Stations (MSs), we devise a DoA estimation method to obtain directional information of incoming signals. Under LoS channel, the DoA corresponds to elevation and Azimuth angles of MSs relative to the LIS. This method is numerically demonstrated to have promising performance with high-frequency carriers.

Multipole solution with nonlinear pressure loss for oblique waves action on a submerged partially perforated semi-circular breakwater

This study is an extension of the research on semi-circular breakwater based on Lyu et al. (2020). This time, we focus on the changes in the flow field when waves obliquely interact with the axis of the partially perforated semi-circular breakwater, as well as the nonlinear effects resulting from variations in wave height to some extent. The analytical solution of the flow field is still obtained using the multipole expansion method based on potential flow theory and the method of separation of variables. The analytical solution for oblique incidence is primarily achieved through the cylindrical form of the Bessel equation. Separately considering viscous effects and potential flow, we estimate the impact of wave height variation on the perforated boundary through the quadratic pressure loss term. By employing an iterative solving approach, we comprehensively discuss the influence of wave height, water depth, perforation form, and oblique incidence angle on the flow field and the forces acting on the breakwater.

Задача о колебаниях в цепи связанных линейных осцилляторов с демпфирующей и антидемпфирующей границами

The problem of oscillations in a finite homogeneous chain of coupled harmonic oscillators is considered under special boundary conditions that ensure a stable flow of energy from one end of the chain to the other. The problem covers, as special cases, the classical problem of oscillations in a chain with free and fixed ends, as well as the problem of oscillations in a chain with absorbing and anti-absorbing boundaries. Absorbing boundary conditions are used in numerical simulation of wave propagation to minimize the influence of non-physical boundaries. An exact analytical solution to the considered problem is obtained. The dynamical system of the problem is studied. In particular, a description of the invariant subspaces of the system is given. The oscillatory properties of the system are investigated. The phenomenon of a significant increase in the amplitude of low-frequency oscillations has been discovered and studied. The problem is solved in Schrodinger variables. The solution is presented both in terms of infinite series of Bessel functions and in terms of natural oscillations (eigenmodes) of the system.

Infinite Series Based on Bessel Zeros

An interesting series based on Bessel function roots (zeros) is discussed and numerically analyzed. The novel-derived simplified general solutions are based on Lommel polynomials. This kind of series can have a large practical use in many scientific areas, such as solid mechanics, fluid mechanics, thermodynamics, electronics, physics, etc. Some practical examples connected with fluid mechanics are provided in this paper. The errors in Afanasiev solutions are corrected. In addition, the main solution for the series analyzed by Baricz and Angel is presented.

Electromagnetic plasma waves in dark energy cosmology

We explore the electromagnetic wave modes that can exist in a cosmological plasma dominated by dark energy due to a cosmological constant. It is found that, in the cold and hot plasma cases, electromagnetic plasma wave modes can be found exactly. The effect of this cosmology appears as a time-dependent potential in the wave equation for the electromagnetic modes, which effectively modifies the frequency response of the plasma. This potential depends on the metric of the spacetime and on the thermodynamical properties of the plasma. For both cases, cold and hot, the solutions are found in terms of Airy and Bessel functions, respectively. When those solutions are required to have vanishing initial conditions, a discretization on the wavelengths of the electromagnetic plasma waves is imposed. Thus, only some specific wave modes can exist in this dark energy cosmology. Relaxing those conditions, we obtain other solutions that approximate to plane waves only in the very hot plasma limit.