Bessel Solutions Research Articles

On the Bessel Solution of Kepler’s Equation

Since its introduction in 1650, Kepler’s equation has never ceased to fascinate mathematicians, scientists, and engineers. Over the course of five centuries, a large number of different solution strategies have been devised and implemented. Among them, the one originally proposed by J. L. Lagrange and later by F. W. Bessel still continue to be a source of mathematical treasures. Here, the Bessel solution of the elliptic Kepler equation is explored from a new perspective offered by the theory of the Stieltjes series. In particular, it has been proven that a complex Kapteyn series obtained directly by the Bessel expansion is a Stieltjes series. This mathematical result, to the best of our knowledge, is a new integral representation of the KE solution. Some considerations on possible extensions of our results to more general classes of the Kapteyn series are also presented.

Radial vibration analysis of single and composite spherical shells by wave approach

This paper presents the wave approach for the analysis of free vibration and transmission response of single and composite spherical shells. From wave standpoint, propagation and refection matrices are combined to obtain the characteristic equation of radial vibration for a single spherical shell with free–free and free–clamped boundary constraints. The corresponding natural frequencies solved by the wave approach are compared with the results by the fundamental analytical solution and classical Bessel solution. Additionally, based on the wave solution, transfer matrix, localization factor, and transmission response of a phononic crystal composite spherical shell are investigated. The predicted attenuation bands by the finite element method (FEM) are exactly the same as the theoretical analytical results. The effect of radial span ratio and inner radius on the attenuation band is also discussed.

A Lie group treatment on a generalized evolution Fisher type equation with variable coefficients

In this study, a general second-order evolution equation of the Fisher-type, with time-dependent variable coefficients, is considered. This equation contains many well-known equations, and obtained results may be applicable in investigating other evolution equations. Lie symmetries and corresponding invariant solutions of the considered problem are studied by a non-traditional Lie symmetry method (LSM). Prolongation of the model is an essential part of the Lie symmetry method, which in the current work, we analyze by the heir-equations. Finally, different types of solutions depending on the variable coefficients, such as the exponential solutions, trigonometric solutions, Bessel solutions, and Mathieu solutions, are extracted.

A modal-based Partition of Unity Finite Element Method for elastic wave propagation problems in layered media

The time-harmonic propagation of elastic waves in layered media is simulated numerically by means of a modal-based Partition of Unity Finite Element Method (PUFEM). Instead of using the standard plane waves or the Bessel solutions of the Helmholtz equation to design the discretization basis, the proposed modal-based PUFEM explicitly uses the tensor-product expressions of the eigenmodes (the so-called Love and interior modes) of a spectral elastic transverse problem, which fulfil the coupling conditions among layers. This modal-based PUFEM approach does not introduce quadrature errors since the coefficients of the discrete matrices are computed in closed-form. A preliminary analysis of the high condition number suffered by the proposed method is also analyzed in terms of the mesh size and the number of eigenmodes involved in the discretization. The numerical methodology is validated through a number of test scenarios, where the reliability of the proposed PUFEM method is discussed by considering different modal basis and source terms. Finally, some indicators are introduced to select a convenient discrete PUFEM basis taking into account the observability of cracks located on a coupling boundary between two adjacent layers.

A Numerical Scheme for Solving Time-Fractional Bessel Differential Equations

The object of this paper devotes on offering an indirect schemebased on time-fractional Bernoulli functions in the sense ofRieman-Liouville fractional derivative which ends up to the highcredit of the obtained approximate fractional Bessel solutions. Inthis paper, the operational matrices of fractional Rieman-Liouvilleintegration for Bernoulli polynomials are introduced. Utilizingthese operational matrices along with the properties of Bernoullipolynomials and the least squares method, the fractional Besseldifferential equation converts into a nonlinear system of algebraic.To solve these nonlinear algebraic equations which are a prominentproblem, there is a need to employ Newton's iterative method. Inorder to elaborate the study, the synergy of the proposed method isinvestigated and then the accuracy and the efficiency of the methodare clearly evaluated by presenting numerical results

Diffusive limit of a two-dimensional well-balanced approximation to a kinetic model of chemotaxis

A $$(2+2)$$ -dimensional kinetic equation, directly inspired by the run-and-tumble modeling of chemotaxis dynamics is studied so as to derive a both “2D well-balanced” and “asymptotic-preserving” numerical approximation. To this end, exact stationary regimes are expressed by means of Laplace transforms of Fourier–Bessel solutions of associated elliptic equations. This yields a scattering S-matrix which permits to formulate a time-marching scheme in the form of a convex combination in kinetic scaling. Then, in the diffusive scaling, an IMEX-type discretization follows, for which the “2D well-balanced property” still holds, while the consistency with the asymptotic drift-diffusion equation is checked. Numerical benchmarks, involving “nonlocal gradients” (or finite sampling radius), carried out in both scalings, assess theoretical findings. Nonlocal gradients appear to inhibit blowup phenomena.

Approximate solution of nonlinear triad interactions of acoustic–gravity waves in cylindrical coordinates

The three-dimensional radial propagation of wave disturbances over a slightly compressible fluid of constant depth is discussed. We focus on resonant triads comprising two gravity modes and one acoustic mode. The derivation of the evolution equations in a non-integral form is made possible by approximating the radial solution by cosine functions in two regions, inner and outer, that are matched at a location where all relevant derivatives are in agreement with the exact Bessel solution. When the interaction takes place in the inner region of all modes, the amplitude evolution equations are found to be similar to the two-dimensional case. However, focusing of one gravity mode and de-focusing of the other is observed when the interaction involves an inner region of the acoustic mode, and outer regions of the gravity modes.

Generation of a Bessel beam in FDTD using a cylindrical antenna.

Bessel beams are becoming a very useful tool in many areas of optics and photonics, because of the invariance of their intensity profile over an extended propagation range. Finite-Difference-Time-Domain (FDTD) approach is widely used for the modeling of the beam interaction with nanostructures. However, the generation of the Bessel beam in this approach is a computationally challenging problem. In this work, we report an approach for the generation of the infinite Bessel beams in three-dimensional FDTD. It is based on the injection of the Bessel solutions of Maxwell's equations from a cylindrical hollow annulus. This configuration is compatible with Particle In Cell simulations of laser plasma interactions. This configuration allows using a smaller computation box and is therefore computationally more efficient than the creation of a Bessel-Gauss beam from a wall and models more precisely the analytical infinite Bessel beam. Zeroth and higher-order Bessel beams with different cone angles are successfully produced. We investigate the effects of the injector parameters on the error with respect to the analytical solution. In all cases, the relative deviation is in the range of 0.01-7.0 percent.

Five ways to solve the Yoshida jet problem of wind-driven equatorial flow

The Yoshida jet is a prototype for wind-driven flow in the tropical ocean. We review the Hermite solution discovered sixty years ago, and show how series acceleration through the Hutton-Moore or Euler schemes is necessary to obtain useful accuracy. An explicit solution in terms of Bessel functions is given here for the first time. It is also shown that much simpler analytic solutions, approximate but accurate, are given by low order rational Chebyshev series and Two-Point Padé approximants. Numerically, the Fourier sine domain truncation method combined with the change of coordinate y ∈ sinh(L t) gives quasi-geometric convergence for most problems. When applied to the Yoshida jet, however, the sine/sinh rate of convergence is awful compared to that of the rational Chebyshev spectral method because wind-driven tropical flows decay very slowly as O(1/y). The log-weakened-geometric rate of convergence of the sine-sinh method, realized for functions that decay exponentially fast for large |y|, is replaced by root-exponential convergence with total error proportional to exp(−qN) for some positive constant q. Except for the explicit Bessel solutions, similar considerations should apply to tropical wind-driven flows in general.

Shukla-Spatschek diffusion effects on the spatial damping of space-charge waves in a turbulent plasma waveguide

The physical characteristics of the spatial damping of the space-charge plasma wave are investigated in a turbulent plasma waveguide. The dispersion relation for the space-charge plasma wave is derived including the propagation wave number as well as the spatial damping rate including the Shukla-Spatschek diffusion effect. The results show that the propagation wave number and the spatial damping rate increase with increasing the wave frequency. It is found that the influence of plasma turbulence enhances the spatial damping rate of the space-charge wave. It is also found that the propagation wave number and spatial damping rate with a higher-order root of the Bessel solution are greater than those with a lower-order root. In addition, it is found that the spatial damping rate decreases with an increase of the thermal energy. Moreover, it is found that the propagation wave number and the spatial damping rate decrease with increasing the radius of the waveguide. The variation of the spatial damping due to the turbulent diffusion, thermal, and geometric effects is also discussed.

Dynamics of accelerating Bessel solutions of Maxwell's equations.

We investigate the propagation dynamics of accelerating beams that are shape-preserving solutions of the Maxwell equations, and explore the contribution of their evanescent field components in detail. Both apodized and nonapodized Bessel beam configurations are considered. We show that, in spite of the fact that their evanescent tails do not propagate, these nonparaxial beams can still accelerate along circular trajectories and can exhibit large deflections. Subsequently, our formulation is extended in other two-dimensional vectorial arrangements. The reported results can be useful in plasmonic and other subwavelength and near-field settings.

Research on Radial Vibration of a Circular Plate

Radial vibration of the circular plate is presented using wave propagation approach and classical method containing Bessel solution and Hankel solution for calculating the natural frequency theoretically. In cylindrical coordinate system, in order to obtain natural frequency, propagation and reflection matrices are deduced at the boundaries of free-free, fixed-fixed, and fixed-free using wave propagation approach. Furthermore, radial phononic crystal is constructed by connecting two materials periodically for the analysis of band phenomenon. Also, Finite Element Simulation (FEM) is adopted to verify the theoretical results. Finally, the radial and piezoelectric effects on the band are also discussed.

Filamentation with nonlinear Bessel vortices.

We present a new type of ring-shaped filaments featured by stationary nonlinear high-order Bessel solutions to the laser beam propagation equation. Two different regimes are identified by direct numerical simulations of the nonlinear propagation of axicon focused Gaussian beams carrying helicity in a Kerr medium with multiphoton absorption: the stable nonlinear propagation regime corresponds to a slow beam reshaping into one of the stationary nonlinear high-order Bessel solutions, called nonlinear Bessel vortices. The region of existence of nonlinear Bessel vortices is found semi-analytically. The influence of the Kerr nonlinearity and nonlinear losses on the beam shape is presented. Direct numerical simulations highlight the role of attractors played by nonlinear Bessel vortices in the stable propagation regime. Large input powers or small cone angles lead to the unstable propagation regime where nonlinear Bessel vortices break up into an helical multiple filament pattern or a more irregular structure. Nonlinear Bessel vortices are shown to be sufficiently intense to generate a ring-shaped filamentary ionized channel in the medium which is foreseen as opening the way to novel applications in laser material processing of transparent dielectrics.

모듈형 수관식 보일러를 위한 열교환 모듈의 열전달 특성

A finned tube type heat exchange module has been proposed for a multi-burner water tube boiler. Fin density and length increase in streamwise direction to equalize the evaporation for each module, which makes it difficult to apply conventional bulk design procedure. The design program has been improved by updating data for every row of tubes along the flow. A numerical simulation has been also conducted to evaluate the effect of inlet conditions and validated with experiment. The heat transfer of the first row has been underpredicted by the conventional Zhukauskas correlation, since the acceleration of the flow due to the blockage is not fully inflected. The fin tip temperature is also underpredicted by Bessel solution, because of the interaction with neighboring fins.

Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem

In this paper, six-node hybrid-Trefftz triangular finite element models which can readily be incorporated into the standard finite element program framework in the form of additional element subroutines are devised via a hybrid variational principle for Helmholtz problem. In these elements, domain and boundary variables are independently assumed. The former is truncated from the Trefftz solution sets and the latter is obtained by the standard polynomial-based nodal interpolation. The equality of the two variables are enforced along the element boundary. Both the plane-wave solutions and Bessel solutions are employed to construct the domain variable. For full rankness of the element matrix, a minimal of six domain modes are required. By using local coordinates and directions, rank sufficient and invariant elements with six plane-wave modes, six Bessel solution modes and seven Bessel solution modes are devised. Numerical studies indicate that the hybrid-Trefftz elements are typically 50% less erroneous than their continuous Galerkin element counterpart.

Rigorous Coupled Wave Analysis of Bipolar Cylindrical Systems: Scattering from Inhomogeneous Dielectric Material, Eccentric, Composite Circular Cylinders

Rigorous Coupled Wave Analysis (RCWA) in bipolar coordinates for the first time is used to study electromagnetic (EM) scattering from eccentric, circular, multi-cylinder systems for which spatially, non uniform material (dielectric permittivity) occupies the regions between the interfaces of the cylinders. The bipolar RCWA algorithm presented herein consists of three basic steps which are; (1) solving Maxwell's equations in bipolar coordinates using a state variable (SV) formulation; (2) solving Maxwell's equations in the spatially uniform regions exterior to the inhomogeneous scattering object in terms of circular, cylindrical Bessel-Hankel functions; and (3) enforcement of EM boundary matching equations which leads to a final matrix equation solution of the system. In the paper extensive use of the residue theorem of complex variable theory was made in order to find fast and exact evaluations of the EM boundary interaction integrals that arose between the bipolar, SV solutions and the Hankel- Bessel solutions. In this paper very extensive reliance on the work of A. A. Kishk, R. P. Parrikar and A. Z. Elsherbeni (22) who studied EM scattering from uniform material multi-eccentric circular cylinders (called herein the KPE algorithm) was made in order to validate the numerical results of the bipolar RCWA algorithm. In this paper, two important system transfer matrices, called the Bessel transfer matrix (based on the KPE algorithm) and called the bipolar SV transfer matrix, were developed in order to validate the numerical accuracy of the RCWA algorithm. The Bessel and SV transfer matrices were very useful for validation purposes because, from the way they were both formulated, they could be meaningfully compared to one another, matrix element to matrix element.

Equatorial Meridional Flows: Rotationally Induced Circulations

—This study examines two-dimensional large-scale atmospheric circulations that are centered over the equator. The influence of terms that contain the Coriolis parameter ƒ′=2ΩcosΘ is highlighted in a simple linear inviscid equatorial beta model. Two general types of oscillatory circulations are identified within the y–z plane. In a neutral or stably stratified atmosphere one circulation is expressed in terms of an analytic solution that contains Hermite polynomials, while a second solution is described in terms of a Bessel function. In the more traditional Hermite polynomial solution the influence of f′ is small as suggested by scale analysis. Neutral stability provides the only exception. In contrast to these findings, the Bessel solution contains frequencies with semiannual periods that depend entirely on ƒ′≠0. This solution describes cross-equatorial flow with a maximum meridional velocity at the equator. Consequently, this indicates that to model the atmosphere it is necessary to include in the model equations all terms containing f′, since they influence oscillatory circulations that describe internal waves with periods that vary from a few days to semiannual.

Electromagnetic tunneling Bessel beams

We obtain the Bessel solutions of Maxwell's equations with axial symmetry and we discuss how these electromagnetic Bessel beams could propagate.

Small acoustic tubes: New approximations including isothermal and viscous effects

Algebraic expressions are given (in fortran format for convenient use) that accurately approximate the value of the per unit-length acoustic impedance of small tubes (radii less than (1)/(16) acoustic wavelength) including isothermal and viscous effects. Their accuracy of typically 3% or better was obtained as a result of adding a shunt conductance term to the traditional approximations and including frequency-dependent multiplicative factors. The multiplicative factors are simple expressions for the real or imaginary terms, providing a factor of 40 increase in calculational speed over the full Bessel-function solution. Experimental verification is presented for various combinations of small tubes and cavities relevant to hearing aid applications, and the differences between the present approximations and previous approximations are illustrated using a 1-mm-diam tube as an example. Normalized complex series impedance and shunt admittance, characteristic impedance, time delay, and attenuation constant plots are provided as a reference utilizing the more nearly exact Bessel solution.

Two-dimensional linearly anisotropic scattering in a finite-thick cylindrical medium exposed to a laser beam

Integro-differential equations are developed for the source function, flux and intensity at the boundaries of a two-dimensional finite-thick medium which scatters in a linear fashion. The incident radiation is collimated, normal to the upper surface of the medium and dependent only on the radial coordinate. Two radial distributions are investigated: (1) a Bessel function and (2) a Gaussian laser beam. The solution for the Gaussian beam is constructed from the Bessel solution. Numerical results are presented in graphical and tabular forms for both boundary conditions. Comparisons are made between forward and isotropic scattering and between the finite and semi-infinite cases.