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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The Approximate Solution of Dual Integral Equations by Variational Methods

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500010804 ◽

1958 ◽

Vol 11 (2) ◽

pp. 115-126 ◽

Cited By ~ 13

Author(s):

B. Noble

Keyword(s):

Approximate Solution ◽

Variational Methods ◽

Integral Equations ◽

Separation Of Variables ◽

Circular Disc ◽

Cylindrical Coordinates ◽

Dual Integral Equations ◽

Particular Solution ◽

Image Position

The classic application of dual integral equations occurs in connexion with the potential of a circular disc (e.g. Titchmarsh (9), p. 334). Suppose that the disc lies in z = 0, 0≤ρ≤1, where we use cylindrical coordinates (p, z). Then it is required to find a solution ofsuch that on z = 0Separation of variables in conjunction with the conditions that ø is finite on the axis and ø tends to zero as z tends to plus infinity yields the particular solution.

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FRACTIONAL INTEGRATION AND DUAL INTEGRAL EQUATIONS

10.21236/ad0400963 ◽

1962 ◽

Cited By ~ 11

Author(s):

Ian N. Sneddon

Keyword(s):

Integral Equations ◽

Fractional Integration ◽

Dual Integral Equations

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Dual Integral Equations with Trigonometric Kernels

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500013389 ◽

1966 ◽

Vol 15 (1) ◽

pp. 73-74

Author(s):

J. S. Lowndes

Keyword(s):

Integral Equations ◽

Dual Integral Equations ◽

Image Position

Consider the dual equationswhere

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

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500012682 ◽

1969 ◽

Vol 16 (3) ◽

pp. 185-194 ◽

Cited By ~ 1

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)⌉.

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On dual integral equations with Hankel kernel and an arbitrary weight function

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171286000364 ◽

1986 ◽

Vol 9 (2) ◽

pp. 293-300 ◽

Cited By ~ 2

Author(s):

C. Nasim

Keyword(s):

Weight Function ◽

Integral Equations ◽

Single Equation ◽

Dual Integral Equations ◽

Fredholm Type ◽

Operational Procedure ◽

Mellin Transforms ◽

Special Cases ◽

General Order ◽

Arbitrary Weight

In this paper we deal with dual integral equations with an arbitrary weight function and Hankel kernels of distinct and general order. We propose an operational procedure, which depends on exploiting the properties of the Mellin transforms, and readily reduces the dual equations to a single equation. This then can be inverted by the Hankel inversion to give us an equation of Fredholm type, involving the unknown function. Most of the known results are then derived as special cases of our general result.

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