Non-singular Integral Equations Research Articles

Nonsingular modified Lippmann–Schwinger equation for two charged particles

We consider the partial waves of the two-body Lippmann–Schwinger equation for the T matrix in the case of the sum of the Coulomb plus a more rapidly vanishing potential. Using the knowledge of explicit forms of the integrals determining the operation of the Lippmann–Schwinger operator on the singularities that characterize the on-shell behavior of the Coulomb-like T matrix, we extract a manifestly nonsingular integral equation from the Lippmann–Schwinger equation. We show that the half- and on-shell inhomogeneous terms and solutions can be made as smooth as the momentum representation partial projection of the non-Coulomb part of the potential.

The meson and quark-gluon degrees of freedom in NN-interaction at low and intermediate energies

A set of nonsingular integral equations has been formulated to calculate the partial amplitudes of hadron-hadron scattering at low and intermediate energies making allowance for the meson and quark-gluon degrees of freedom. The results are presented of the calculations of the np effective range parameters and of the np scattering 1S 0- and 3S 1-phase shifts at incident nucleon energies ranging from 1 to 1050 MeV.

On optimal integration methods for Volterra integral equations of the first kind

Families of methods depending on free parameters are constructed for the solution of nonsingular Volterra integral equations of the first kind in [5]. These parameters are restricted to certain regions in order that a certain polynomial satisfies both a stability and a consistency condition. In this note an optimal choice of the free parameters is outlined in order that the l 2 {l_2} -norm of the roots of the polynomial is minimized.

The numerical stability of multistep methods for convolution Volterra integral equations: Nonsingular equations

A family of test equations is suggested for first and second kind nonsingular Volterra integral equations with convolution kernels. It is shown that the numerical stability of multistep methods when applied to these test equations can be characterised as the requirement that the roots of polynomials lie inside or on the unit circle. Stability results are presented for a sample of methods including some stability regions in graphical form.

Two-Particle System in a Force Field: Nonsingular Integral Equations for Components of T-Operator

Two-Particle System in a Force Field: Nonsingular Integral Equations for Components of T-Operator

Quadrature Rule Methods for Volterra Integral Equations of the First Kind

A new class of quadrature rule methods for solving nonsingular Volterra integral equations of the first kind are introduced; these methods are based on an appropriate modification of the higher-order Newton-Gregory methods which are known to be divergent. Methods up to order six are constructed explicitly and illustrated with numerical examples.

Asymptotic expansions for multistep methods applied to nonlinear Volterra integral equations of the second kind

This paper deals with linear multistep methods applied to nonlinear, nonsingular Volterra integral equations of the second kind. Analogously to the theory of W.B. Gragg, the existence of asymptotic expansions in the stepsizeh is proved. Under certain conditions only even powers ofh occur. As a special case, the midpoint rule is treated, a short numerical example for the applicability to extrapolation techniques is given.

Quadrature rule methods for Volterra integral equations of the first kind

A new class of quadrature rule methods for solving nonsingular Volterra integral equations of the first kind are introduced; these methods are based on an appropriate modification of the higher-order Newton-Gregory methods which are known to be divergent. Methods up to order six are constructed explicitly and illustrated with numerical examples.

Finite-Element Solution of Integral Equations Arising in Radiative Heat Transfer and Laminar Boundary-Layer Theory

Abstract Finite-element solutions of a number of one-dimensional linear and nonlinear (nonsingular) integral equations arising in radiative heat transfer and boundarylayer theory are presented.

A non-singular equation for the three-body scattering problem

This paper derives a non-singular integral equation for the three-body problem. Starting from the three-body equations obtained by Karlsson and Zeiger we introduce a set of algebraic transformations that remove all the Green function pole singularities. For scattering energies on the real axis we find a singularity-free momentum-space integral equation. This equation requires only a finite range of momentum values for its solution. In the case of well-behaved two-body interactions, such as the superposition of Yukawa interactions, we prove that the kernels of this equation have a finite Hilbert-Schmidt norm. This same norm provides a general criteria for establishing when the impulse approximation is accurate.

Null field approach to scalar diffraction I. General method

Invoking the optical extinction theorem (extended boundary condition) the conventional singular integral equation (for the density of reradiating sources existing in the surface of a totally reflecting body scattering monochromatic waves) is transformed into infinite sets of non-singular integral equations, called the null field equations. There is a set corresponding to each separable coordinate system (we say that we are using the ‘elliptic’, ‘spheroidal’, etc., null field method when we employ 'elliptic cylindrical', ‘spheroidal’, etc., coordinates). Each set can be used to compute the scattering from bodies of arbitrary shape, but each set is most appropriate for particular types of body shape as our computational results confirm. We assert that when the improvements (reported here) are incorporated into it, Waterman’s adaptation of the extinction theorem becomes a globally efficient computational approach. Shafai’s use of conformal transformation for automatically accomodating singularities of the surface source density is incorporated into the cylindrical null field methods. Our approach permits us to use multipole expansions in a computationally convenient manner, for arbitrary numbers of separated, interacting bodies of arbitrary shape. We present examples of computed surface source densities induced on pairs of elliptical and square cylinders.

A third order, semi-explicit method in the numerical solution of first kind Volterra integral equations

We present a new numerical method for the solution of nonsingular Volterra integral equations of the first kind. It belongs to a new class of methods that are semi-explicit, provide self-starting algorithms and possess favourable stability properties. The third order convergence of the particular method exhibited is established, under suitable conditions, and numerical results are illustrated.

A simple reduction procedure for the three- and N-body scattering equations

A generalized form of the two-body Kowalski-Noyes method is shown to provide a both simple and powerful unitary reduction of the three- and N-body scattering equations. Employing generalized half-off-shell functions that satisfy of-sshell but real and non-singular integral equations, the reduction directly leads to on-shell integral equations for the scattering amplitudes. Physically, it is simple example of how the scattering problem can be split into an internal and an external part.

Nonsingular integral equation for multichannel scattering

Nonsingular integral equations are derived for a class of multichannel scattering problems. (AIP)

Steady, one-dimensional multigroup neutron transport with anisotropic scattering

The solution of steady, one-dimensional half-space multigroup transport problems with degenerate anisotropic scattering is obtained for L1 sources and incident distributions. The solution is expressed in terms of contour integrals of the resolvent operator (λI−K)−1, where K is the ’’separated’’ transport operator. The connection between this method and the ’’Case eigenfunction’’ method is briefly discussed, and the half-space albedo problem is treated in detail. This problem reduces to obtaining the Wiener–Hopf factorization of the dispersion matrix, hence to solving two coupled nonlinear, nonsingular matrix integral equations.

Numerical solution for the near-field transmission between two H-plane sectoral electromagnetic horns

The effect of higher order interactions in the coupling between two H -plane sectoral electromagnetic horns on the field distribution at the aperture of the transmitting horn is calculated using the point matching technique. A nonsingular integral equation for the induced current in the receiving horn, due to the initially incident wave from the transmitting horn, is formulated using a conformal transformation which maps the cross sectional contour of the receiving horn onto two circles on which the induced currents are finite. The solution leads to a modified field distribution in the transmitting aperture which is then employed in Hu's near-zone power transmission formula to compute a gain correction factor for H -plane sectoral horns. It is shown that the correction factor oscillates about that reported by Chu and Semplak with an amplitude which increases for smaller separation distances.

Still Another Method for Phase-Shift Computation

A new method for computing phase shifts is presented. It is based on solving a nonsingular integral equation for the momentum-space wave function. In a sample case it is more accurate than other methods.

The extended boundary condition and thick axially symmetric antennas

The extended boundary condition of Waterman is applied to axially symmetric antennas. Infinite sets of non-singular integral equations for the surface current density are obtained. The numerical solution of these equations is investigated. Several sets of basis functions are used for expanding the surface current and their rates of numerical convergence are compared. The numerical methods introduced here are more useful than other known techniques for antennas with length to maximum diameter ratios from about 0.25 to 25, and with maximum diameters of about two wavelengths. Detailed numerical results are presented for half-wave cylindrical dipoles with rounded corners of varying curvature and for end-loaded cylindrical dipoles. Some confirmatory experimental results are reported. Comparison is made with results previously reported in the literature.

On Double Chebyshev Series Approximation

Previous article Next article On Double Chebyshev Series ApproximationN. K. BasuN. K. Basuhttps://doi.org/10.1137/0710045PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractMany of the standard practical techniques for Chebyshev polynomial approximation in one-variable problems have been extended to two-variable problems.[1] Ferdinand Freudenstein, Bi-variate, rectangular, optimum-interval interpolation, Math. Comp., 15 (1961), 288–291 MR0129532 0168.14903 CrossrefGoogle Scholar[2] C. W. Clenshaw, A note on the summation of Chebyshev series, Math. Tables Aids Comput., 9 (1955), 118–120 MR0071856 0065.05403 CrossrefGoogle Scholar[3] David Elliott, Error analysis of an algorithm for summing certain finite series, J. Austral. Math. Soc., 8 (1968), 213–221 MR0230452 0159.45104 CrossrefGoogle Scholar[4] David Elliott, The numerical solution of integral equations using Chebyshev polynomials, J. Austral. Math. Soc., 1 (1959/1960), 344–356 MR0128087 0096.10201 CrossrefGoogle Scholar[5] L. Fox and , E. T. Goodwin, The numerical solution of non-singular linear integral equations, Philos. Trans. Roy. Soc. London. Ser. A., 245 (1953), 501–534 MR0054355 0050.12902 CrossrefISIGoogle Scholar[6] Google Scholar[7] L. Fox and , I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London, 1968ix+205 MR0228149 0153.17502 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Analytical and numerical investigation of Poiseuille flow through semi-elliptic annulusPhysics of Fluids, Vol. 34, No. 8 | 1 Aug 2022 Cross Ref Study of the free surface Green function at zero forward speed in finite water depth using Newman’s methodApplied Ocean Research, Vol. 120 | 1 Mar 2022 Cross Ref Comparison of existing methods for the calculation of the infinite water depth free-surface Green function for the wave–structure interaction problemApplied Ocean Research, Vol. 81 | 1 Dec 2018 Cross Ref Nondimensionalized Univariate Equation Characterizing Optimal State Feedback Control for Collision AvoidanceIEEE Transactions on Intelligent Transportation Systems, Vol. 19, No. 10 | 1 Oct 2018 Cross Ref The automatic solution of partial differential equations using a global spectral methodJournal of Computational Physics, Vol. 299 | 1 Oct 2015 Cross Ref Numerical proof of stability of roll waves in the small-amplitude limit for inclined thin film flowJournal of Differential Equations, Vol. 257, No. 8 | 1 Oct 2014 Cross Ref A new kind of double Chebyshev polynomial approximation on unbounded domainsBoundary Value Problems, Vol. 2013, No. 1 | 22 January 2013 Cross Ref On Approximate Solutions of Second-Order Linear Partial Differential EquationsApplied Mathematics, Vol. 03, No. 09 | 1 Jan 2012 Cross Ref A fitting formula for the falloff curves of unimolecular reactions, II: Tunneling effectsInternational Journal of Chemical Kinetics, Vol. 43, No. 1 | 19 October 2010 Cross Ref Direct numerical simulation of preferential particle concentration in decaying turbulence under the influence of magnetic fieldInternational Journal for Numerical Methods in Fluids, Vol. 419 | 1 Jan 2009 Cross Ref Large and small signal distortion analysis using modified Volterra seriesAnalog Integrated Circuits and Signal Processing, Vol. 54, No. 2 | 29 November 2007 Cross Ref Volterra Analysis Using Chebyshev Series2007 IEEE International Symposium on Circuits and Systems | 1 May 2007 Cross Ref Large and Small Signal Distortion Analysis Using Modified Volterra Series2006 NORCHIP | 1 Nov 2006 Cross Ref An efficient numerical approximation for the linear class of mixed integral equationsApplied Mathematics and Computation, Vol. 167, No. 2 | 1 Aug 2005 Cross Ref Chebyshev polynomial solutions of second-order linear partial differential equationsApplied Mathematics and Computation, Vol. 134, No. 1 | 1 Jan 2003 Cross Ref Chebyshev series solution of linear Fredholm integrodifferential equationsInternational Journal of Mathematical Education in Science and Technology, Vol. 29, No. 4 | 1 Jul 1998 Cross Ref Fast polynomial multiplication and convolutions related to the discrete cosine transformLinear Algebra and its Applications, Vol. 252, No. 1-3 | 1 Feb 1997 Cross Ref Chebyshev series solutions of Fredholm integral equationsInternational Journal of Mathematical Education in Science and Technology, Vol. 27, No. 5 | 9 July 2006 Cross Ref An algorithm for the evaluation of multidimensional (direct and inverse) discrete Chebyshev transformsJournal of Computational and Applied Mathematics, Vol. 8, No. 4 | 1 Dec 1982 Cross Ref Optimum acoustic design of free-running low speed propellersAircraft Systems and Technology Meeting | 16 August 1977 Cross Ref Clenshaw sums in several variablesConstructive Theory of Functions of Several Variables | 2 October 2006 Cross Ref Volume 10, Issue 3| 1973SIAM Journal on Numerical Analysis History Submitted:05 March 1971Published online:14 July 2006 InformationCopyright © 1973 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0710045Article page range:pp. 496-505ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

Continuum eigenmodes of an inhomogeneous plasma

The van Kampen-Case treatment of the linearized Vlasov equation is generalized to cases of non-uniform plasmas in non-uniform equilibrium fields. This is done by transforming the Vlasov operator into the representation of the eigenfunctions of the streaming operator. The Vlasov operator is thus reduced to the form of a multiplication operator perturbed by an integral operator so that the continuum eigenfunctions may be constructed by adapting a method originally devised in neutron transport theory. The calculation leads to solving a non-singular Fredholm-type integral equation. Various kinds of degeneracy of the continuous spectrum are discussed and an example is given.