Dual Integral Equations

1963 ◽  
Vol 15 ◽  
pp. 631-640 ◽  
Author(s):  
E. R. Love
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Electrostatic Field ◽  
Fredholm Integral Equation ◽  
Integral Transforms ◽  
Dual Integral Equations ◽  
Image Position

Erdélyi and Sneddon (4) have reduced the dual integral equations (4, (1.4))where Ψ is unknown, to a single Fredholm integral equation (4, (4.4)), from the solution of which Ψ is explicitly obtainable. Their work extended and clarified an investigation by Cooke (1), placing it in a context of standard integral transforms. Cooke's reduction was obtained after consideration of the Fredholm integral equation obtained by Love (8) in discussing Nicholson's problem of the electrostatic field of two equal circular coaxial conducting disks (9).

scholarly journals On the Fredholm integral equation associated with pairs of dual integral equations

1969 ◽  
Vol 16 (3) ◽  
pp. 185-194 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Sufficient Conditions ◽  
Neumann Series ◽  
Dual Integral Equations ◽  
Image Position

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

scholarly journals On Certain Dual Integral Equations

1961 ◽  
Vol 5 (1) ◽  
pp. 21-24 ◽  
Author(s):  
E. T. Copson
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Dual Integral Equations ◽  
Mellin Transforms ◽  
Abel Integral Equation ◽  
Image Position ◽  
Fourier Integrals

In his book on Fourier Integrals, Titchmarsh [l] gave the solution of the dual integral equationsfor the case α > 0, by some difficult analysis involving the theory of Mellin transforms. Sneddon [2] has recently shown that, in the cases v = 0, α = ±½, the problem can be reduced to an Abel integral equation by making the substitutionorIt is the purpose of this note to show that the general case can be dealt with just as simply by puttingThe analysis is formal: no attempt is made to supply details of rigour.

scholarly journals On a dual integral equation with a trigonometric kernel

1977 ◽  
Vol 18 (2) ◽  
pp. 175-177 ◽  
Author(s):  
D. C. Stocks
Keyword(s):  
Integral Equation ◽  
Elasticity Theory ◽  
Integral Equations ◽  
Crack Problem ◽  
Dual Integral Equation ◽  
Dual Integral Equations ◽  
Cosine Transform ◽  
Fourier Cosine ◽  
Fourier Cosine Transform ◽  
Image Position

In this note we formally solve the following dual integral equations:where h is a constant and the Fourier cosine transform of u–1 φ(u) is assumed to exist. These dual equations arise in a crack problem in elasticity theory.

scholarly journals A note on dual integral equations involving inverse associated Weber-Orr transforms

1996 ◽  
Vol 19 (1) ◽  
pp. 161-169
Author(s):  
Nanigopal Mandal ◽  
B. N. Mandal
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Dual Integral Equations ◽  
Elementary Methods ◽  
Special Case

We consider dual integral equations involving inverse associated Weber-Orr transforms. Elementary methods have been used to reduce dual integral equations to a Fredholm integral equation of second kind. Some known results are obtained as special case.

scholarly journals A Multiplying Factor Method for the Solution of Wiener-Hopf Equations

1961 ◽  
Vol 12 (3) ◽  
pp. 119-122 ◽  
Author(s):  
B. Noble ◽  
A. S. Peters
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Dual Integral Equations ◽  
Factor Method ◽  
Single Integral ◽  
Image Position ◽  
Related Technique

In (1), § 6.2, a multiplying factor method has been used to solve certain dual integral equations. The results are then used to solve a single integral equation of the Wiener-Hopf type. In this note we indicate how a related technique can be used to solve Wiener-Hopf integral equations directly. ConsiderwhereDefinewhere α = σ+iτ, and F+(α) is regular for τ>q; K(α) is regular and non-zero in −p < τ < p. For simplicity we restrict ourselves to the case where

scholarly journals A note on Bessel function dual integral equation with weight function

1988 ◽  
Vol 11 (3) ◽  
pp. 543-549 ◽  
Author(s):  
B. N. Mandal
Keyword(s):  
Integral Equation ◽  
Weight Function ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Bessel Functions ◽  
Dual Integral Equation ◽  
Dual Integral Equations ◽  
Arbitrary Weight

An elementary procedure based on Sonine's integrals has been used to reduce dual integral equations with Bessel functions of different orders as kernels and an arbitrary weight function to a Fredholm integral equation of the second kind. The result obtained here encompasses many results concerning dual integral equations with Bessel functions as kernels known in the literature.

Note on an electrified circular disk situated inside a coaxial infinite hollow cylinder

1962 ◽  
Vol 58 (4) ◽  
pp. 621-624 ◽  
Author(s):  
I. N. Sneddon
Keyword(s):  
Integral Equation ◽  
Charge Density ◽  
Hollow Cylinder ◽  
Electrostatic Field ◽  
Fredholm Integral Equation ◽  
Surface Charge Density ◽  
Present Note ◽  
Circular Disk ◽  
Dual Integral Equations ◽  
Method Of Solution

In a recent paper Collins (l) has shown that Copson's method of solution of the problem of determining the electrostatic field due to an electrified disk (2) can be extended to determine the surface charge density induced on an electrified disk when it is situated inside an earthed coaxial infinitely long hollow cylinder. In Collins's solution the problem is reduced to that of solving a Fredholm integral equation of the second kind. The object of the present note is to show that Collins's solution can be obtained more simply by making use of a technique in the theory of dual integral equations developed recently by the author (3) and Copson(4).

scholarly journals Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

2010 ◽  
Vol 2 (2) ◽  
pp. 264-272 ◽  
Author(s):  
A. Shirin ◽  
M. S. Islam
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Numerical Solutions ◽  
Bernstein Polynomials ◽  
Fredholm Integral Equations ◽  
Matrix Formulation ◽  
Piecewise Polynomials ◽  
The Galerkin Method

In this paper, Bernstein piecewise polynomials are used to solve the integral equations numerically. A matrix formulation is given for a non-singular linear Fredholm Integral Equation by the technique of Galerkin method. In the Galerkin method, the Bernstein polynomials are used as the approximation of basis functions. Examples are considered to verify the effectiveness of the proposed derivations, and the numerical solutions guarantee the desired accuracy.  Keywords: Fredholm integral equation; Galerkin method; Bernstein polynomials. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v2i2.4483               J. Sci. Res. 2 (2), 264-272 (2010) 

A method for the exact solution of a class of two-dimensional diffraction problems

1951 ◽  
Vol 205 (1081) ◽  
pp. 286-308 ◽  
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Integral Equations ◽  
Scattered Field ◽  
Plane Waves ◽  
Angular Spectrum ◽  
Diffraction Theory ◽  
General Interest ◽  
Two Dimensional ◽  
Dual Integral Equations

In the last few years Copson, Schwinger and others have obtained exact solutions of a number of diffraction problems by expressing these problems in terms of an integral equation which can be solved by the method of Wiener and Hopf. A simpler approach is given, based on a representation of the scattered field as an angular spectrum of plane waves, such a representation leading directly to a pair of ‘dual’ integral equations, which replaces the single integral equation of Schwinger’s method. The unknown function in each of these dual integral equations is that defining the angular spectrum, and when this function is known the scattered field is presented in the form of a definite integral. As far as the ‘radiation’ field is concerned, this integral is of the type which may be approximately evaluated by the method of steepest descents, though it is necessary to generalize the usual procedure in certain circumstances. The method is appropriate to two-dimensional problems in which a plane wave (of arbitrary polarization) is incident on plane, perfectly conducting structures, and for certain configurations the dual integral equations can be solved by the application of Cauchy’s residue theorem. The technique was originally developed in connexion with the theory of radio propagation over a non-homogeneous earth, but this aspect is not discussed. The three problems considered are those for which the diffracting plates, situated in free space, are, respectively, a half-plane, two parallel half-planes and an infinite set of parallel half-planes; the second of these is illustrated by a numerical example. Several points of general interest in diffraction theory are discussed, including the question of the nature of the singularity at a sharp edge, and it is shown that the solution for an arbitrary (three-dimensional) incident field can be derived from the corresponding solution for a two-dimensional incident plane wave.

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