Kind Of Bessel Function Research Articles

An Efficient Quadrature Rule for Highly Oscillatory Integrals with Airy Function

In this work, our primary focus is on the numerical computation of highly oscillatory integrals involving the Airy function. Specifically, we address integrals of the form ∫0bxαf(x)Ai(−ωx)dx over a finite or semi-infinite interval, where the integrand exhibits rapid oscillations when ω≫1. The inherent high oscillation and algebraic singularity of the integrand make traditional quadrature rules impractical. In view of this, we strategically partition the interval into two segments: [0,1] and [1,b]. For integrals over the interval [0,1], we introduce a Filon-type method based on a two-point Taylor expansion. In contrast, for integrals over [1,b], we transform the Airy function into the first kind of Bessel function. By applying Cauchy’s integration theorem, the integral is then reformulated into several non-oscillatory and exponentially decaying integrals over [0,+∞), which can be accurately approximated by the generalized Gaussian quadrature rule. The proposed methods are accompanied by rigorous error analyses to establish their reliability. Finally, we present a series of numerical examples that not only validate the theoretical results but also showcase the accuracy and efficacy of the proposed method.

Analytical model for a circular lined tunnel embedded in the layered soil under seismic waves

AbstractIn this study, based on the wave function expansion method and exact stiffness method (ESM), an analytical model for a circular lined tunnel embedded in the layered soil under seismic waves is established. For convenience, the wave field in the layered‐soil‐tunnel (LST) system is divided into three parts, that is, the free wave field and scattering wave field in the layered soil, as well as the transmitted wave field in the lining. The free wave field in the layered soil is determined by the ESM for the layered soil. The scattering wave field in the layered soil is further decomposed into three parts, namely, the principal and complementary wave fields in the soil layer with the tunnel (the tunnel layer) as well as the modified wave field occurring in the other soil layers. These scattering wave fields are represented by the corresponding principal, complementary, and modified wave functions, respectively. The principal wave functions associated with the direct cylindrical waves emanated from the tunnel are represented by the Hankel functions. The complementary and modified wave functions correspond to the additional responses of the tunnel layer and the other soil layers due to the direct cylindrical waves from the tunnel, and can be obtained by expanding direct cylindrical waves into plane waves together with the application of the ESM. The transmitted wave field in the lining is simply represented by two kinds of Bessel functions. With the aforementioned wave field representations, the boundary value problem for the LST system under seismic waves is formulated. Based on the proposed analytical model, some numerical results and corresponding analyses are presented. Presented numerical results indicate that the dynamic stress concentration factor (DSCF) associated with the stiffer tunnel layer is usually larger than that associated with the softer tunnel layer for the incidence of plane P waves with not very low frequencies, while the circumstance for the incidence of plane SV waves shows an opposite tendency.

Numerical model of seepage flows by reformulating finite element method based on new spherical Hankel shape functions

The water penetration in soil is investigated numerically using the finite element method (FEM) in a novel way. In the suggested method, new spherical Hankel shape functions are used and the finite element method is reformulated based on them. These new functions are obtained from the first and second kind of Bessel functions. The properties of Hankel shape functions lead to having more accuracy and robustness for the proposed method with low number of elements. To validate the suggested approach, at first, a boundary value problem is solved and the results are compared with the available analytical solution. Then, in order to prove the efficiency and applicability of the present model in the seepage problems, five examples including saturated and unsaturated flow in porous media are studied and the hydraulic head is calculated. Afterward, the results obtained from the classical and new method are compared together. The comparisons indicate that the suggested method with the low number of elements is more precise than the classic FEM with the same mesh.

Expansion of some well-known functions and classical polynomials in series of the generalized Laguerre polynomials.1

The generalized Laguerre polynomials form a complete set (orthogonal and normalized) in the space ** with respect to a certain weighting function because they are the Eigen functions of a second-order differential operator. Here we shall show how to expand some well-known classical polynomials such as the Legendre and the Hermite polynomials in series of generalized Laguerre polynomials. Since the generalized Laguerre orthogonal polynomials are set of polynomials that are mutually orthogonal to each other with respect to a measure of weighting function that is just the integrand of the gamma function, thus it grants us the guarantee of the ability to expand the first kind of Bessel functions and the gamma function in terms of the generalized Laguerre polynomials. The series expansion in terms of the generalized Laguerre polynomials can be achieved by following various approaches. For instance, the series expansion of the first kind Bessel functions is gained by the generalized hypergeometric function approach, whereas the series expansion of gamma function was obtained directly by the usual way that is by calling the orthogonality and orthonormality properties of the generalized Laguerre polynomials. Another powerful technique to gain the series expansion of the classical polynomials is the generating function approach as it has been followed here to obtain a series expansion of the Legendre and the Hermite polynomials in terms of the generalized Laguerre polynomials. The series expansion in series of the generalized Laguerre polynomials has a variety of applications in mathematics, physics, and engineering

SIMPLIFIED FORMULAS FOR SOME BESSEL FUNCTIONS AND THEIR APPLICATIONS IN EXTENDED SURFACE HEAT TRANSFER

Bessel functions find many applications in Physics and Engineering fields. Some of these applications are in the analysis of extended surface heat transfer where the cross-sections vary. Tables of various kinds of Bessel functions are available in most handbooks of mathematics. However, the use of tables is not always convenient, particularly for applications where many values must be computed. In the applications of Bessel functions in extended surface heat transfer, graphs are also available to provide quick evaluations of the values needed. However, reading these graphs always needs interpolation; this will be cumbersome and time-consuming if there are many readings to be taken. Mathematical formulas for Bessel functions are available but they are usually complicated. Software to calculate values of Bessel functions is also available. Excel, Maple, and Mathematica can also be used to compute the values of Bessel functions. A user can write a program for an application that involves Bessel functions. However, the use of Bessel functions in Excel is limited while Maple and Mathematica are expensive commercial software. In this paper, formulas for Bessel functions of and are simplified with adequate accuracy that can be used to easily compute values needed in the extended surface heat transfer analysis. It is found that errors for and are relatively small (maximum errors are 0.004% and 0.003%, respectively) in the range of 0.05 to 3.75 while the maximum error for is 3.678% for the same range. However, the maximum error for is reduced to 0.166 if the range is from 0.25 to 3.75.

Experimental and numerical analysis on the equilibrium shape of sessile droplets on elastic thin membranes

The equilibrium shape of sessile droplets on elastic thin membranes is determined by the balance among three types of surface tensions at the interface, namely vapor-liquid, liquid-solid, and solid-vapor. We studied the cross-sectional contour of the membrane equilibrium state and sessile drops by focusing on the global characteristics of the deformation, both numerically and experimentally. To determine the initial tension, which is required for numerical analysis, we carried out a modal test. A laser Doppler vibrometer (LDV) was used to measure the velocity of the membrane vibration on a table isolated from external forces. With the measured the first and second modal frequencies, the initial tension of the membrane was derived using a circular membrane vibration equation by solving a first kind of Bessel function. Furthermore, a high-resolution reflective displacement sensor and micro linear actuator with micrometer accuracy were used to measure the cross-section deformation of the membrane on the sessile drop, and the droplet contact radius in the experimental environment. The membrane sample was assembled on a thin, light aluminum ring with a resonance frequency of about 700 Hz, and three different diameters were employed. Using this technique, we investigated the trend in membrane deformation due to the increase in the weight of the water droplets and the surface tension, which is an intrinsic property of the membrane. In addition, we found that the experimental results related to the contact radius and contour of a water drop agreed with the simulation data well.

Spherical Hankel-Based Free Vibration Analysis of Cross-ply Laminated Plates Using Refined Finite Element Theories

In this paper, spherical Hankel basis functions and Carrera unified formulation (CUF) are utilized for the free vibration analysis of laminated composite plates. The spherical Hankel shape functions are derived from the corresponding radial basis function. These functions have excellent ability to satisfy the first and second kind of Bessel function as well as polynomial function fields which leads to more accurate results. Also, CUF presents an effective formulation to employ any order of Taylor expansion to expand solution field. Higher-order theories supposed by CUF are free from Poisson locking phenomenon, and they do not need any shear correction factor. Therefore, coupling spherical Hankel basis functions and CUF ends in a suitable methodology to analyze laminated plates. To investigate the proposed approach, several numerical examples are provided and the superiority and robustness of the suggested approach are shown.

A time-dependent discontinuous Galerkin finite element approach in two-dimensional elastodynamic problems based on spherical Hankel element framework

In this paper, a time-discontinuous finite element method (TD-FEM) based on a new class of shape functions is suggested for solving two-dimensional elastodynamic problems. The main idea is to use spherical Hankel shape functions derived from corresponding radial basis functions instead of classical Lagrange shape functions. Among the advantages of the proposed interpolation, containing the first and second kind of Bessel functions in complex space as well as the polynomial ones can be pointed out, while the classic Lagrange interpolation is only able to contain polynomial functions. The validity and accuracy of the present method are fully demonstrated using several numerical examples. For this purpose, five numerical examples are considered, and the results obtained from the proposed method (Hankel-based TD-FEM) and the Lagrange-based TD-FEM are compared with the exact analytical solutions. The results show that the use of Hankel functions in TD-FEM leads to more accurate and stable solutions rather than those obtained from the classic TD-FEM.

The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

SummaryIn this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.

Analyzing the effect of the forces exerted on cantilever probe tip of atomic force microscope with tapering-shaped geometry and double piezoelectric extended layers in the air and liquid environments

The aim of the present study is to assess the force vibrational performance of tapering-shaped cantilevers, using Euler–Bernoulli theory. Tapering-shaped cantilevers have plan-view geometry consisting of a rectangular section at the clamped end and a triangular section at the tip. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. In this model, a micro cantilever, which is covered by two piezoelectric layers at the top and the bottom, is modeled at angle α. Both of these layers are subjected to similar AC and DC voltages. This paper attempts to determine the effect of the capillary force exerted on the cantilever probe tip of an atomic force microscope. The capillary force emerges due to the contact between thin water films with a thickness of hc which have accumulated on the sample and the probe. In addition, an attempt is made to develop the capillary force between the tip and the sample surface with respect to the geometry obtained. The smoothness or the roughness of the surfaces as well as the geometry of the cantilever tip have significant effects on the modeling of forces applied to the probe tip. In this article, the Van der Waals and the repulsive forces are considered to be the same in all of the simulations, and only is the capillary force altered in order to evaluate the role of this force in the atomic force microscope based modeling. We also indicate that the tip shape and the radial distance of the meniscus greatly influence the capillary force. The other objective of our study is to draw a comparison between tapering-and rectangular-shaped cantilevers. Furthermore, the equation for converting the tip of a tapering-shaped cantilever into a rectangular cantilever is provided. Moreover, the modal analysis method is employed to solve the motion equation. The mode shape function for the two tapering-shaped sections of the first and the second kind of Bessel functions is utilized. The nonlinear governing equation is solved by employing the Forward Time Simulation (FST). As the Atomic Force Microscopy cantilever switches from the attractive mode to the contact repulsive mode upon proximity to the surface and the reverse occurs during the departure from the sample surface, a hybrid mode is developed which is illustrated in the graphs.

Analytical Solution for Long-Wave Reflection by a Rectangular Obstacle with Two Scour Trenches

In this paper, linear long-wave reflection by a rectangular obstacle with two scour trenches of power function profile is explored. A closed-form analytical solution in terms of the first and second kinds of Bessel functions is obtained, which finds two classic analytical solutions as its special cases, i.e., the wave reflection from a rectangular obstacle and from an infinite step. The phenomenon of zero reflection coefficient for a single rectangular obstacle with the same depths in front of and behind the obstacle still remains for a rectangular obstacle with two scour trenches as long as the bathymetry is symmetrical about the obstacle. The periodicity of the reflection coefficient as a function of the relative length of the middle rectangular obstacle disappears if two scour trenches are attached to the middle rectangular obstacle. Finally, the wave reflection by a rectangular obstacle with two scour trenches generally increases when the trenches become wide and deep. The wave reflection by a degener...

SIMULATION OF BUBBLE FORMATION AND HEAPING IN A VIBRATING GRANULAR BED

The Eulerian-Eulerian approach is used to simulate flow in a dense granular bed subjected to a slight, vertical, sinusoidal vibration. The bottom of the bed was subjected to a vertical vibration, ASin(ω v t)J 0(k m x), where J represents the profile of the bottom wall and is the first kind of Bessel function. The governing equations of the gas phase and particle phase are described and the results of the model with different vibration frequencies and amplitudes are presented. Bubble formation and upward and downward heaping were observed at different amplitudes and frequencies. The results of the model confirm that there is a change from upward to downward heaps according to the strength of the vibration. In addition, the location of the bubble and the shape of the heaping at the interface of the bed depend on the strength of the vibration and the profile of the bottom of the container.

A main inequality for several special functions

By using a recent generalization of the Cauchy–Schwarz inequality we obtain several new inequalities for some basic special functions such as beta and incomplete beta functions, gamma, polygamma and incomplete gamma functions, error functions, various kinds of Bessel functions, exponential integral functions, Gauss hypergeometric and confluent hypergeometric functions, elliptic integrals, moment generating functions and the Riemann zeta function. We also apply the generalized Cauchy–Schwarz inequality to some classical integral transforms. All these new inequalities obey the general form f 2 ( x ) ≤ k ( x ) f ( p x + q ) f ( ( 2 − p ) x − q ) , in which p , q are real parameters and k ( x ) is a specific positive function.

Analytical Study of Linear Long-Wave Reflection by a Two-Dimensional Obstacle of General Trapezoidal Shape

In this paper, an analytical solution for linear long-wave reflection by an obstacle of general trapezoidal shape is explored. A closed-form expression in terms of first and second kinds of Bessel functions is obtained for the wave reflection coefficient, which depends on the relative lengths of the two slopes and top of the obstacle as well as the depth ratios in front of and behind the obstacle versus that above the obstacle. The analytical solution obtained in this study finds a few well-known analytical solutions to be its special cases, which include the wave reflection from a rectangular obstacle, an infinite step, and an infinite step behind a linear slope. The present analytical solution, however, covers a much wider range of problems. It is found that the periodicity of the wave reflection coefficient as the function of the relative length of the obstacle remains when two slopes are present but with a reduced magnitude. The phenomenon of zero wave reflection from the structure is special to a rectangular obstacle only, which disappears with the addition of a slope in front or at the rear. The new solution may be very useful in some engineering applications, for example, the design of a submerged breakwater of trapezoidal shape.

Improved recursive algorithm for light scattering by a multilayered sphere.

An improved recurrence algorithm to calculate the scattering field of a multilayered sphere is developed. The internal and external electromagnetic fields are expressed as a superposition of inward and outward waves. The alternative yet equivalent expansions of fields are proposed by use of the first kind of Bessel function and the first kind of Hankel function instead of the first and the second kinds of Bessel function. The final recursive expressions are similar in form to those of Mie theory for a homogeneous sphere and are proved to be more concise and convenient than earlier forms. The new algorithm avoids the numerical difficulties, which give rise to significant errors encountered in practice by previous methods, especially for large, highly absorbing thin shells. Various calculations and tests show that this algorithm is efficient, numerically stable, and accurate for a large range of size parameters andrefractive indices.

Fluid Flow in a Circular Cylinder Subject to Circulatory Oscillation-Theoretical Analysis

A fluid flow inside a circular cylinder subject to horizontal and circular oscillation is analyzed theoretically. Under the assumption of small-amplitude oscillation, the governing equations take linear forms. The velocity field is obtained in terms of the first kind of Bessel function of order 1. It was found that a particle describes an orbit close to a circle in the central region and an arc near the side wall. We also obtained the Stokes' drift velocity induced by the traveling wave along the circumferential direction. The Eulerian streaming velocities at the edge of the bottom and side boundary layers were also obtained. It was shown that the vertical component of the steady streaming velocity on the side wall is almost proportional to the amplitude of the free surface motion.

Mathematical approximation techniques in biophysical models

The mathematical characterization of biophysical problems often leads to certain transcendental functions which must be evaluated to fully appreciate the significance of the mathematical model. In illustration, many mathematical models concerned with wave propagation and transmission of fluids through elastic tubes require treatment of the various kinds of Bessel functions and their ratios. In my recent work ( The Special Functions and Their Approximations, Volumes 1 and 2, Academic Press, 1969) polynomial, rational and infinite series expansions in series of Chebyshev polynomials are developed for a wide class of transcendental functions which include Bessel functions as special cases. In numerous instances, tables of coefficients are presented. In the present paper, we show how these ideas can be exploited to yield economic methods for the evaluation of Bessel functions arising in the problems cited. As the approximations can be given in closed form, they can be easily applied to further simplify available closed form analyses.