Showing papers on "Integro-differential equation published in 1975"
Dissertation•
01 Aug 1975
TL;DR: In this paper, the authors define finite-part integrals within the framework of distribution theory, and present a formal definition of finite part integrals and a formal formula for finite part integral integration.
Abstract: it •• ' •••••••••••••••••••••••••••••••••••••••••••••• ' ••••••••• ~ ACKNOWLEDGEMENTS .............................. ',' . iii NOTATION ....... ' '.',' ' ' .. 'Vi Chapter, I THE DEFINITION OF A FINITE-PART INTEGRAL ......•.•. 1.1 Historical introduction •...........•..•••.... 1.2 Physical examples of finite-part integrals 22 1.3 Direct definition of finite-part integrals within the framework of distribution theory 26 1.4 The general case '................... 37 Chapter II PROPERTIES OF FINITE-PART INTEGRALS •.. ~".•.•.••...• 45 2.1 The basic rules of classic integration applied to finite-part integrals .•.......•..••..••.•. 45 2.2 Basic transformations of a finite integration interval " -................. "48 2.3 Transformation of an infinite integration in t e rv a1 ~. . . . . . . . . . . . . . . . . . . . . ~. . . . . . 52 2.4 The continuity of the finite-part integral as a functional ...............................•• 55 Chapter III AN INTERPOLATORY QUADRATURE FORMULA ...•...•...•.•. 66 3.1 Derivation of the formula .........•.......... 66 3.2 Computation of the coefficients w. and c. ..•. 71 ~ ~ 3.3 General properties of the w. and c. ..•.•..... 75 ~ ~ (iv) Stellenbosch University http://scholar.sun.ac.za
98 citations
TL;DR: It is proved that every state that is sufficiently small in a sense made precise in the paper can be reached from the origin in a timeT depending on the coefficients of the equation.
Abstract: We obtain results on local controllability (near an equilibrium point) for a nonlinear wave equation, by application of an infinite-dimensional analogue of the Lee-Markus method of linearization. Controllability of the linearized equation is studied by application of results of Russell, and local controllability of the nonlinear equation follows from the inverse function theorem. We prove that every state that is sufficiently small in a sense made precise in the paper can be reached from the origin in a timeT depending on the coefficients of the equation.
40 citations
32 citations
22 citations
14 citations
01 Mar 1975
TL;DR: A survey of the backward beam equation approach in numerical computation of backwards parabolic equations is given in this paper, which includes problems with variable coefficients as well as an example of non-linear equation, Burgers' equation.
Abstract: : A survey is given of the backward beam equation approach in the numerical computation of backwards parabolic equations. The discussion includes problems with variable coefficients as well as an example of non-linear equation, Burgers' equation. Numerical experiments are presented.
13 citations
TL;DR: In this article, a two-dimensional model for the cochlea is developed, and an integral equation is derived that describes the pressure difference between the scalae of the scala, which is an improvement of the well-known Peterson-Bogert equation.
Abstract: A two-dimensional model for the cochlea is developed. An integral equation is derived that describes the pressure difference between the scalae. For the main quantity, the transmembrane pressure, an ordinary differential equation is obtained, which appears to be an improvement of the well-known Peterson-Bogert equation. The results are valid for all frequencies; an assumption of long or short wavelengths is not necessary at all.
13 citations
TL;DR: In this paper, a solution to the integro-differential equation of batch grinding has been presented which maps the trajectory of particle size spectra of continuous size and time domains.
13 citations
TL;DR: DigiZeitschriften e.V. as discussed by the authors gewährt ein nicht exklusives, nicht übertragbares, persönliches and beschränktes Recht auf Nutzung dieses Dokuments.
Abstract: DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich für den persönlichen, nicht kommerziellen Gebrauch bestimmt. Das Copyright bleibt bei den Herausgebern oder sonstigen Rechteinhabern. Als Nutzer sind Sie sind nicht dazu berechtigt, eine Lizenz zu übertragen, zu transferieren oder an Dritte weiter zu geben. Die Nutzung stellt keine Übertragung des Eigentumsrechts an diesem Dokument dar und gilt vorbehaltlich der folgenden Einschränkungen: Sie müssen auf sämtlichen Kopien dieses Dokuments alle Urheberrechtshinweise und sonstigen Hinweise auf gesetzlichen Schutz beibehalten; und Sie dürfen dieses Dokument nicht in irgend einer Weise abändern, noch dürfen Sie dieses Dokument für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, aufführen, vertreiben oder anderweitig nutzen; es sei denn, es liegt Ihnen eine schriftliche Genehmigung von DigiZeitschriften e.V. und vom Herausgeber oder sonstigen Rechteinhaber vor. Mit dem Gebrauch von DigiZeitschriften e.V. und der Verwendung dieses Dokuments erkennen Sie die Nutzungsbedingungen an.
10 citations
TL;DR: In this paper, a new difference equation to the two dimensional diffusion equation for x-y geometry is derived by using the finite Fourier transformation, which has a form of a coupled equation of the 3 point difference equations for each coordinate and can be easily solved by the iterative method of the alternative direction implicit method.
TL;DR: In this paper, the problem of the excitation, detection, and scattering of electroelastic surface (Bleustein) waves is solved exactly by determining the charge distribution on the fingers of an interdigital transducer.
Abstract: The problem of the excitation, detection, and scattering of electroelastic surface (Bleustein) waves is solved exactly by determining the charge distribution on the fingers of an interdigital transducer The approach is to solve an integral equation, in the Fourier transform domain, that relates the charge density on the fingers to the electric potential of the fingers The solution of the integral equation is accomplished by expanding the charge distribution in a series of pulses and then transforming the problem to a vector matrix,one which is readily handled by a computer In this manner the charge distribution is determined for a variety of conditionsSubject Classification: [43]35,54
TL;DR: In this article, an exact solution of a non-linear integro-differential equation for describing turbulent diffusion is found, and the physical significance of the solution obtained is analyzed in spaces of one, two and three domensions.
Abstract: AN EXACT solution of a non-linear integro-differential equation is found. It is shown that this equation is a model equation for describing turbulent diffusion, and in this connection the physical significance of the solution obtained is analysed in spaces of one, two and three domensions.
TL;DR: Solutions of Laplace's equation for azimuthally symmetrical potentials in cylindrical coordinates are found which can be correlated with two-dimensional multipoles in planes as mentioned in this paper.
TL;DR: In this paper, the role of the continuum approximation in the theory of electrolytes is re-examined, and a possible generalization of this approximation is considered, and the authors show that the relaxation of the approximation leads to a significant enhancement of the potential felt by a counterion in the immediate neighborhood of the central ion.
Abstract: In this paper, the role of the continuum approximation in the theory of electrolytes is re−examined, and a possible generalization of this approximation is considered. For several reasons (cited in the manuscript), the model of Debye and Huckel is used as the starting point in our analysis. We proceed by postulating a functional form for the dependence of the permittivity e (r) on the distance from the central ion. When this expression for the permittivity is used within the context of Poisson−Boltzmann theory, there results a nonlinear differential equation whose analytic properties are investigated in detail; in particular, it is proved that solutions of this equation exist and are unique. By construction of the Green’s function, we obtain the associated nonlinear integral equation and solutions to this equation for a choice of parameters corresponding to an electrolyte system considered previously by Guggenheim. Our main conclusions follow from a comparison of our results with those obtained previously using the continuum approximation. We find that the relaxation of this approximation leads to a significant enhancement of the potential felt by a counterion in the immediate neighborhood of the central ion, with an attendent accelerated damping of the potential as one moves away from the central ion. A second, rather unexpected result which emerges from our study is that the computed potential is surprisingly insensitive to the explicit down−range behavior of the function postulated to describe the change in permittivity as a function of distance. This paper concludes with some remarks on future problems to be studied. In the following paper detailed ion−distribution profiles are reported, and the question of the internal consistency of the augmented Poisson−Boltzmann equation, introduced in this paper, is examined.
Journal Article•
TL;DR: Sommerfeld's formula for approximate evaluation of integrals involving the Fermi-Dirac function is derived by the Laplace transform method, and its limitations are discussed in this article.
Abstract: Sommerfeld's formula for approximate evaluation of integrals involving the Fermi-Dirac function is derived by the Laplace transform method, and its limitations are discussed.
TL;DR: In this paper, an integral equation for the Gel'fand-Levitan kernel is given in terms of the scattering potential, which may be regarded as complementary to the GFL equation.
Abstract: An integral equation for the Gel’fand–Levitan kernel is given in terms of the scattering potential. This integral equation may be regarded as complementary to the Gel’fand–Levitan equation which is an integral equation for the kernel in terms of the Fourier transform of the reflection coefficient.
TL;DR: In this article, an initial value method for the integral equation of the column is presented for determining the buckling load of columns, which is reduced to a set of ordinary differential equations with prescribed initial conditions to find the Fredholm resolvent.
TL;DR: In this article, it was confirmed once more that there exists the general solution of Laplace's equation in ellipsoidal coordinates which satisfies the Stackel theorem and which was derived earlier by M. Jarov-Jarovoi and S. J. Madden.
Abstract: In this paper it is confirmed once more that there exists the general solution of Laplace's equation in ellipsoidal coordinates which satisfies the Stackel theorem and which was derived earlier by M. Jarov-Jarovoi and S. J. Madden. The author interprets physically the general solution in real space as potentials of layers of charge and double layers in which the distribution of densities is defined by Green's formula.
TL;DR: In this paper, the authors considered the photon counting statistics with polarized thermal light having arbitrary rational spectral profile and used the standard techniques of contour integration to obtain an explicit expression for the generating function valid for an arbitrary value of the counting interval.
Abstract: We consider the photon-counting statistics with polarized thermal light having arbitrary rational spectral profile. The method is illustrated for a light beam having a two-peak symmetric Lorentzian spectrum. The associated integral equation is converted into a differential equation and the eigenvalues are determined. Using the standard techniques of contour integration, an explicit expression is obtained for the generating function valid for an arbitrary value of the counting interval.
01 Mar 1975
TL;DR: In this article, the response of a clamped panel to supersonically convected turbulence is considered and a theoretical model in the form of an integro-differential equation is employed that takes into account the coupling between the panel motion and the surrounding acoustic medium.
Abstract: The response of a clamped panel to supersonically convected turbulence is considered. A theoretical model in the form of an integro-differential equation is employed that takes into account the coupling between the panel motion and the surrounding acoustic medium. The kernels of the integrals, which represent induced pressures due to the panel motion, are Green's functions for sound radiations under various moving and stationary sources. An approximate analysis is made by following a finite-element Ritz-Galerkin procedure. Preliminary numerical results, in agreement with experimental findings, indicate that the acoustic damping is the controlling mechanism of the response.
TL;DR: In this article, the generalized transport equation was investigated in a Hilbert space of square integrable functions and the existence of its unique solution was proved, which provided a firm basis for investigating various approximate methods for numerical reactor calculations.
TL;DR: The generalized kinetic equation derived by Balescu is equivalent to the Zwanzig asymptotic equation deduced by means of Bogoliubov's boundary conditions as discussed by the authors.
Abstract: We show that the generalized kinetic equation derived by Balescu is equivalent to the Zwanzig asymptotic equation deduced by means of Bogoliubov’s boundary conditions and the five assumptions of Balescu are reducible to one assumption on an operator whose formal expression is known. We also show that this generalized kinetic equation has the structure of a generalized Boltzmann equation in the sense that the collision term is a sum of collisions operators of two, three, etc. particles. As a confirmation we derive the Boltzmann equation from this generalized kinetic equation.
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