Integrals Involving Bessel Functions Research Articles

Gaussian rule for integrals involving Bessel functions

Gaussian rule for integrals involving Bessel functions

Indefinite integrals from Wronskians for Whittaker and Gauss hypergeometric functions

A method related to Wronskians has recently been applied to derive many indefinie integrals involving Bessel functions and associated Legendre functions. Here the same method is applied to derive indefinite integrals of Whittaker functions and Gauss hypergeometric functions. All the integrals presented here appear to be new and are derived either directly from Wronskians or from a differential equation related to Wronskians. All the results presented have been checked numerically using Mathematica.

Strichartz estimates in Wiener amalgam spaces and applications to nonlinear wave equations

In this paper we obtain some new Strichartz estimates for the wave propagator eit−Δ in the context of Wiener amalgam spaces. While it is well understood for the Schrödinger case, nothing is known about the wave propagator. This is because there is no such thing as an explicit formula for the integral kernel of the propagator unlike the Schrödinger case. To overcome this lack, we instead approach the kernel by rephrasing it as an oscillatory integral involving Bessel functions and then by carefully making use of cancellation in such integrals based on the asymptotic expansion of Bessel functions. Our approach can be applied to the Schrödinger case as well. We also obtain some corresponding retarded estimates to give applications to nonlinear wave equations where Wiener amalgam spaces as solution spaces can lead to a finer analysis of the local and global behavior of the solution.

Frequency-explicit convergence analysis of collocation methods for highly oscillatory Volterra integral equations with weak singularities

Filon collocation methods are well known for their good performances in solving the second-kind highly oscillatory Volterra integral equations with weak singularities. As the frequency increases, these methods can provide much more accurate solutions for oscillatory integral equations. This paper is devoted to investigating frequency-related properties of collocation solutions of highly oscillatory Volterra equations with weakly singular kernels. Optimal convergence rates are obtained by analyzing the error equations and studying asymptotic properties of oscillatory integrals involving Bessel functions. Moreover, numerical experiments are conducted to verify the given theoretical results.

Integral Involving Bessel Functions Arising in Propagation Phenomena

An new integral identity involving the product of two modified Bessel functions K 0 and K 1 , previously unreported, is presented and proved.

Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels

In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The other one is piecewise Hermite collocation method, which used a two-points Hermite interpolation in each subinterval. These two methods can calculate the approximate value of function value and derivative value simultaneously. Both methods are constructed easily and implemented well by the fast computation of highly oscillatory integrals involving Bessel functions. Under some conditions, the asymptotic convergence order with respect to oscillatory factor of these two methods are established, which are higher than the existing results. Some numerical experiments are included to show efficiency of these two methods.

Numerical transforms from position to momentum space via Gaussian quadrature in the complex plane

A method for the numerical evaluation of integrals involving Bessel functions of the first kind, via non-standard Gaussian quadrature, is appraised. This method uses specialized Gaussian quadratures based on weight functions which are modified Bessel functions of the third kind. We apply the method to calculate momentum space radial wave functions of the two-dimensional hydrogen atom and harmonic oscillator, via two-dimensional Fourier or Hankel transforms of the position space functions. Results show that the method performs best for the asymptotic or large p regions of the hydrogen atom. We also illustrate that accurate results can be obtained for the smaller argument regions with the use of higher-order quadratures. On the other hand, the method fails in its application to the harmonic oscillator. A discussion of the strengths and weaknesses of the method is presented.

A Hankel Transform Approach to Inverse Quasi-Static Steady-State Thermal Stresses in a Thick Circular Plate

In this paper, we solve the realistic problem of inverse quasi-static steady-state thermal stresses in a thick circular plate, which is subjected to arbitrary interior temperature and determine the unknown temperature and thermal stresses on the upper surface of the thick circular plate, where the fixed circular edge and the lower surface of the circular plate are thermally insulated using Hankel transform. To achieve our objective, first we construct a new stable algorithm for numerical evaluation of Hankel transform of order \(\nu >-1\). The integrand \(rf(r)J_\nu (pr)\) consists of a slowly varying component rf(r) and a rapidly oscillating component \(J_\nu (pr)\). Most of the algorithms proposed in last few decades approximate the slowly varying component rf(r). In the present paper, we take a different approach and replace the rapidly oscillating component \(J_\nu (pr)\) in the integrand by its hat functions approximation. This approach avoids the complexity of evaluating integrals involving Bessel functions. This leads to a very simple, efficient and stable algorithm for numerical evaluation of Hankel transforms. We further give error and stability analysis and corroborate our theoretical findings by various numerical experiments.

A penny-shaped crack in an infinite linear transversely isotropic medium subjected to uniform anti-symmetric heat flux: Closed-form solution

The problem of a penny-shaped crack subjected to anti-symmetric uniform heat flux in an infinite transversely isotropic magneto-electro-thermo-elastic medium is investigated. Although we start our derivations in magneto-electro-thermo-elastic medium, the solution presented in this paper is also applicable for linear transversely isotropic thermopiezoelectric, thermopiezomagnetic, thermoelastic materials. Previous researchers obtained the closed-form solution to the similar problem in crack plane. In regards to full-space solution, they expressed it in multiple integration or line integrals involving Bessel functions. In this article, the authors find that the full-space solution could be in terms of the explicit elementary functions via potential theory method developed by Fabrikant. What's more, the closed form solutions in all kinds of linear transversely isotropic medium can serve as benchmarks for future numerical studies of multiphase composites.

On the evaluation of infinite integrals involving Bessel functions

In this paper we consider the numerical method for computing the infinite highly oscillatory Bessel integrals of the form ∫a∞f(x)Cv(ωx)dx, where Cv(ωx) denotes Bessel function Jv(ωx) of the first kind, Yv(ωx) of the second kind, Hv(1)(ωx) and Hv(2)(ωx) of the third kind, f is a smooth function on [a,∞),limx→∞f(k)(x)=0(k=0,1,2,…),ω is large and a⩾1ωk with k≤1. We construct the method based on approximating f by a combination of the shifted Chebyshev polynomial so that the generalized moments can be evaluated efficiently by the truncated formula of Whittaker W function. The method is very efficient in obtaining very high precision approximations if ω is sufficiently large. Furthermore, we give the error which depends on the endpoint “a”. Numerical examples are provided to confirm our results.

A kernel formula for the action of the Weyl element in the Kirillov model of [formula omitted

We give an explicit description of the action of the Weyl element on smooth functions with compact support in the Kirillov model of non-principal series irreducible representations of SL(2,C) generalizing a result of Motohashi. An important ingredient in the proof of the kernel formula is a new “classical” formula for an integral involving Bessel functions. We also give an analogous formula for GL(2,C).

Infinite integrals involving Bessel functions by an improved approach of contour integration and the residue theorem

Two lemmas are established to evaluate integrals involving Bessel functions or products of two Bessel functions on a large semicircle. These lemmas are as useful as Jordan’s lemma. With the help of them, the approach of contour integration and residue theorem to infinite integrals becomes much more powerful than ever before, yet is still simple and easy. A general formula is derived for infinite integrals involving rational functions, powers and Bessel functions, the index of the power and the order of the Bessel function being independent, and the result is expressed in terms of residues and coefficients of the Laurent series of the rational function. Similar formulae are presented for infinite integrals involving Neumann or Struve functions. Another general formula is derived for infinite integrals involving rational functions, powers and two Bessel functions, the index of the power and the orders of the Bessel functions being independent. The formulation presented may be extended to infinite integrals of similar types. Typical examples are worked out for each type of integral, and many of the results obtained seem not available in the literature.

On Certain Dual Integral Equations

Dual integral equations involving H-Functions have been solved by using the theory of Mellin transforms. The proof is analogous to that of Busbridge on solutions of dual integral equations involving Bessel functions.

On a new class of integrals involving Bessel functions of the first kind

In recent years, several integral formulas involving a variety of special functions have been developed by many authors. Also many integral formulas containing the Bessel function $J_\nu(z)$ have been presented. Very recently, Rakha {\it et al.} presented some generalized integral formulas involving the hypergeometric functions. In this sequel, here, we aim at establishing two generalized integral formulas involving a Bessel functions of the first kind, which are expressed in terms of the generalized Wright hypergeometric function. Some interesting special cases of our main results are also considered.

Infinite integrals involving Bessel functions by contour integration

The approach of contour integration and residue theorem is systematically developed to evaluate infinite integrals involving Bessel functions or closely related ones. Three independent formulations applicable to different types of integrals are presented. The conventional formulation using Hankel functions as auxiliary functions, whose previous development leave considerable room for improvement, is fully developed. The second and third formulations, which appear unnoticed before, uses as auxiliary functions linear combinations of Bessel and Struve functions, and linear combinations of Anger and Weber functions, respectively. These functions play a crucial role as the Hankel functions do in the conventional formulation. Several general formulae are derived, each expressing an infinite integral in terms of residues. The integrands are products of rational functions, powers and Bessel (Neumann, Struve, Anger or Weber) functions, with some relation satisfied between the index of the power and the order of the Bessel or related function. Typical examples are worked out for each type of integral.

Complete solution to elastic field around an elliptic punch

AbstractThe problem of a rigid flat indentor resting on a transversely isotropic medium under external forces is reexamined. The contacting surface is an elliptic region. Both normal and tangential indentation are considered. Unlike the usual approach, we do not use ellipsoidal co‐ordinate system. The integral equations for prescribed normal and tangential indentation are first solved in cartesian co‐ordinate system for the unknown stress on the contact plane. A series of transformation and known results of integrals involving Bessel functions are then used to derive exact and complete solutions to the displacement and stress inside the half space. The result is exact in the sense that it involves a finite integration which must be evaluated numerically. Such a complete solution to the elliptic punch problem in cartesian co‐ordinate has not been obtained before. We also obtain the asymptotic form of the field near the border of the contact region. The corresponding result for an isotropic medium is obtained by a limiting process.

Exponential Approximations of the Bessel Functions I0,1(x), J0,1(x), Y0(x), and H01,2(x), with Applications to Electromagnetic Scattering, Radiation, and Diffraction [EM Programmer's Notebook

Several ordinary and modified Bessel functions are approximated using exponential series. These representations are useful in many scattering and diffraction problems where integrals involving Bessel functions must be computed. Examples are given for diffraction by a circular aperture, radiation from an aperture antenna, transient reflection from Debye and Lorentz media, and scattering by a conducting strip.

On a Class of Integrals Involving a Bessel Function Times Gegenbauer Polynomials

We provide information and explicit formulae for a class of integrals involving Bessel functions and Gegenbauer polynomials. We present a simple proof of an old formula of Gegenbauer. Some interesting special cases and applications of this result are obtained. In particular, we give a short proof of a recent result of A. A. R. Neves et al. regarding the analytical evaluation of an integral of a Bessel function times associated Legendre functions. These integrals arise in problems of vector diffraction in electromagnetic theory.

Eddy currents induced in a conducting rod of finite length by a coaxial encircling coil

This paper describes the calculation of eddy currents in a cylindrical conductive rod of finite length due to a coaxial circular coil carrying an alternating current. The coil impedance variation with frequency is determined from the field for an arbitrary coaxial location of the coil. Expressions for electromagnetic field and impedance of a coil encircling an infinite cylindrical rod are well known, the results being expressed as infinite integrals involving Bessel functions. For a finite length rod, additional boundary conditions must be satisfied at the ends. The extra boundary conditions are accommodated here by recasting the problem in a domain of finite extent in the axial direction. The axial length of the truncated domain is arbitrary and can be large compared with the coil length or the length of the rod. Therefore, the truncated domain solution can yield results that are numerically as close to the infinite domain solution as desired. The results provide a simple means of calculating the impedance of an inductor with a lossy core and can be used to investigate the linear properties of the core material.

Green’s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces

By virtue of two systems of vector functions and the propagator matrix method, Green’s functions for transversely isotropic, piezoelectric functionally graded (exponentially graded in the vertical direction), and multilayered half spaces are derived. It is observed that the homogeneous solution and propagator matrices for each functionally graded layer in the transformed domain are independent of the choice of the two systems of vector functions. For a point force and point charge density applied at any location of the functionally graded half space, Green’s functions are expressed in terms of one-dimensional infinite integrals. To carry out the numerical integral involving Bessel functions, an adaptive Gauss quadrature approach is introduced and modified. The piezoelectric functionally graded Green’s functions include those in the corresponding elastic functionally graded media as special results with the latter being also unavailable in the literature. Two piezoelectric functionally graded half-space models are analyzed numerically: one is a functionally graded PZT-4 half space, and the other a coated functionally graded PZT-4 layer over a homogeneous BaTiO 3 half space. The effects of different exponential factors on Green’s function components are clearly demonstrated, which could be useful in the design and manufacturing of piezoelectric functionally graded structures.