Quadrature formulas for oscillatory integral transforms

Summary Quadrature formulas are obtained for the Fourier and Bessel transforms which correspond to the well-known Gauss-Laguerre formula for the Laplace transform. These formulas provide effective asymptotic approximations, complete with error bounds. Comparison is also made between the quadrature formulas and the asymptotic expansions of these transforms.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under Contract A7359     

Numerical solution for exterior problems

Summary We solve the Helmholtz equation in an exterior domain in the plane. A perfect absorption condition is introduced on a circle which contains the obstacle. This boundary condition is given explicitly by Bessel functions. We use a finite element method in the bounded domain. An explicit formula is used to compute the solution out of the circle. We give an error estimate and we present relevant numerical results.     

Accurate computation of the zeros of the generalized Bessel polynomials

Summary. A general method for approximating polynomial solutions of second-order linear homogeneous differential equations with polynomial coefficients is applied to the case of the families of differential equations defining the generalized Bessel polynomials, and an algorithm is derived for simultaneously finding their zeros. Then a comparison with several alternative algorithms is carried out. It shows that the computational problem of approximating the zeros of the generalized Bessel polynomials is not an easy matter at all and that the only algorithm able to give an accurate solution seems to be the one presented in this paper. Received July 25, 1997 / Revised version received May 19, 1999 / Published online June 8, 2000     

Space and time inversions of stochastic processes and Kelvin transform

Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion.     

Fast computation of the field of a vertical magnetic dipole above a layered medium

We consider the direct problem of aeroelectrosounding for a layered medium. The specific feature of this problem is that its solution is representable as a Bessel transform of a function ƒ of a certain type. We have developed and implemented a method for fast computation of the Bessel transform of such functions. The method relies on the approximation of ƒ by a linear combination of exponentials with prespecified exponents. The Bessel transform of the original function ƒ is computed analytically as the Bessel transform of the approximating linear combination. Numerical computations have been carried out. __________ Translated from Prikladnaya Matematika i Informatika, No. 26, pp. 39–48, 2007.     

In honor of Professor Dr. Wolfgang Haack on the occasion of his seventieth birthday on April 24, 1972

A discrete transform with a Bessel function kernel is defined, as a finite sum, over the zeros of the Bessel function. The approximate inverse of this transform is derived as another finite sum. This development is in parallel to that of the discrete Fourier transform (DFT) which lead to the fast Fourier transform (FFT) algorithm. The discrete Hankel transform with kernel Jo, the Bessel function of the first kind of order zero, will be used as an illustration for deriving the discrete Hankel transform, its inverse and a number of its basic properties. This includes the convolution product which is necessary for solving boundary problems. Other applications include evaluating Hankel transforms, Bessel series and replacing higher dimension Fourier transforms, with circular symmetry, by a single Hankel transform     

Wavelet-type Transform and Bessel Potentials Associated with the Generalized Translation

Wavelet–type transform associated with singular Laplace–Bessel differential operator is introduced and the relevant Calderón–type reproducing formula is established. Representations of the generalized Bessel potentials 0)$ " align="middle" border="0"> and their inverses via the wavelet–type transform are obtained.     

On the convergence of the multigrid method for a hypersingular integral equation of the first kind

Summary We present a multigrid method to solve linear systems arising from Galerkin schemes for a hypersingular boundary integral equation governing three dimensional Neumann problems for the Laplacian. Our algorithm uses damped Jacobi iteration, Gauss-Seidel iteration or SOR as preand post-smoothers. If the integral equation holds on a closed, Lipschitz surface we prove convergence ofV- andW-cycles with any number of smoothing steps. If the integral equation holds on an open, Lipschitz surface (covering crack problems) we show convergence of theW-cycle. Numerical experiments are given which underline the theoretical results.     

Full Fourier-Bessel transform and the algebra of singular pseudodifferential operators

The one-dimensional full Fourier-Bessel transform was introduced by I.A. Kipriyanov and V.V. Katrakhov on the basis of even and odd small (normalized) Bessel functions. We introduce a mixed full Fourier-Bessel transform and prove an inversion formula for it. Singular pseudodifferential operators are introduced on the basis of the mixed full Fourier-Bessel transform. This class of operators includes linear differential operators in which the singular Bessel operator and its (integer) powers or the derivative (only of the first order) of powers of the Bessel operator act in one of the directions. We suggest a method for constructing the asymptotic expansion of a product of such operators. We present the form of the adjoint singular pseudodifferential operator and show that the constructed algebra is, in a sense, a *-algebra.     

Uniform asymptotic expansion of $J_\nu(\nu a)$ via a difference equation

Summary. There are two ways of deriving the asymptotic expansion of , as , which holds uniformly for . One way starts with the Bessel equation and makes use of the turning point theory for second-order differential equations, and the other is based on a contour integral representation and applies the theory of two coalescing saddle points. In this paper, we shall derive the same result by using the three term recurrence relation . Our approach will lead to a satisfactory development of a turning point theory for second-order linear difference equations. Received December 15, 2000 / Published online September 19, 2001     

Numerical solution of a secular equation

Summary. A method is proposed for the solution of a secular equation, arising in modified symmetric eigenvalue problems and in several other areas. This equation has singularities which make the application of standard root-finding methods difficult. In order to solve the equation, a class of transformations of variables is considered, which transform the equation into one for which Newton's method converges from any point in a certain given interval. In addition, the form of the transformed equation suggests a convergence accelerating modification of Newton's method. The same ideas are applied to the secant method and numerical results are presented. Received July 1, 1994     

The direct updating of damping and gyroscopic matrices

In this paper we develop an efficient numerical method for the finite element model updating of damped gyroscopic systems. This model updating of damped gyroscopic systems is proposed to incorporate the measured modal data into the finite element model to produce an adjusted finite element model on the damping and gyroscopic matrices that closely match the experimental modal data.     

Numerische Behandlung von Verzweigungsproblemen bei gewöhnlichen Differentialgleichungen

Summary We present a new method for the numerical solution of bifurcation problems for ordinary differential equations. It is based on a modification of the classical Ljapunov-Schmidt-theory. We transform the problem of determining the nontrivial branch bifurcating from the trivial solution into the problem of solving regular nonlinear boundary value problems, which can be treated numerically by standard methods (multiple shooting, difference methods).

     

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

Summary Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.     

The collocation method for mixed boundary value problems on domains with curved polygonal boundaries

Summary. We consider an indirect boundary integral equation formulation for the mixed Dirichlet-Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and a cosine approximating space. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. A complete stability and solvability analysis of the transformed integral equations is given by use of a Mellin transform technique, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space. Received April 15, 1995 / Revised version received April 10, 1996     

Towards a computer-assisted proof for chaos in a forced damped pendulum equation

We report on the first steps made towards the computational proof of the chaotic behaviour of the forced damped pendulum. Although, chaos for this pendulum was being conjectured for long, and it has been plausible on the basis of numerical simulations, there is no rigorous proof for it. In the present paper we provide computational details on a fitting model and on a verified method of solution. We also give guaranteed reliability solutions showing some trajectory properties necessary for complicate chaotic behaviour.     

A spatial covariance model with a single wave effect and a finite range

We propose a new spatial covariance model with a single wave effect and a finite range. The construction of this model is based on a radial kernel composed of two concentric spheres and the finite range enables us to obtain an exact simulation from the circulant embedding method. This model responds to the lack of efficient covariance functions to model patterns characterized by concentration areas surrounded by gaps. We compare the single wave model to other existing hole effect models such as the Bessel model, the damped cosine model and the cardinal sine model.     

Some generalizations of Bessel and Flett potentials associated to the Laplace–Bessel differential operator

We introduce the notion of bi-parametric potential-type operators, which generalize the Bessel and Flett potentials associated to singular Laplace–Bessel differential operator. Explicit inversion formulas for them are obtained by using a special wavelet-like transform, generated by the so-called ‘modified beta-semigroup’.     

Numerical algorithms for uniform Airy-type asymptotic expansions

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.     

Continuous and discrete Bessel wavelet transforms

Using the theory of Hankel convolution, continuous and discrete Bessel wavelet transforms are defined. Certain boundedness results and inversion formula for the continuous Bessel wavelet transform are obtained. Important properties of the discrete Bessel wavelet transform are given.