A Solution of Tranter's Dual Integral Equations Problem

Cooke, J. C.

doi:10.1093/qjmam/9.1.103

Cited by 59 publications

(76 citation statements)

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“…There are many of works that address the way of solving each of obtained DIE, (e.g., [9,10,19,21,22]). Typically, in these works such equations are reduced to the solving of the operator equation (I + K) X = B i.e., the equation of the Fredholm second kind.…”

Section: Set Of Dual Integral Equationsmentioning

confidence: 99%

Dual Integral Equations Technique in Electromagnetic Wave Scattering by a Thin Disk

Balaban¹,

Sauleau²,

Benson³

et al. 2009

PIER B

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Abstract-The scattering of an arbitrary electromagnetic wave by a thin disk located in free space is formulated rigorously in terms of coupled dual integral equations (CDIEs) for the unknown images of the jumps and average values of the normal to the disk scattered-field components. Considered are three cases of the disk: (1) Zero-thickness perfectly electrically conducting (PEC) disk, (2) thin electrically resistive (ER) disk and (3) dielectric disk. Disk thickness is assumed much smaller than the disk radius and the free space wavelength, in ER and dielectric disk cases, and also much smaller than the skinlayer depth, in the ER disk case. The set of CDIEs are "decoupled" by introduction of the coupling constants. Each set of DIEs are reduced to a Fredholm second kind integral equation by using the semi-inversion of DIE integral operators. The set of "coupling" equations for finding the coupling constants is obtained additionally from the edge behavior condition. Thus, each problem is reduced to a set of coupled Fredholm second kind integral equations. It is shown that each set can be reduced to a block-type three-diagonal matrix equation, which can be effectively solved numerically by iterative inversions of the two diagonal blocks and 2 × 2 matrix.

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Section: Set Of Dual Integral Equationsmentioning

confidence: 99%

Dual Integral Equations Technique in Electromagnetic Wave Scattering by a Thin Disk

Balaban¹,

Sauleau²,

Benson³

et al. 2009

PIER B

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“…For example suppose a is fixed and PJf) -{2n where n is a positive integer. Then it is easy to see from (A5) that there will always be a function h in L 2 …”

Section: /> a (O = And-v(i-/ao) (47)mentioning

confidence: 99%

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

Hutson

1969

Proceedings of the Edinburgh Mathematical Society

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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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“…We shall consider dual integral equations [3], Noble [4] and Cook [5]. More recently, Erdelyi and Sneddon gaw a solution using fractional integral operators [6].…”

Section: Introductionmentioning

confidence: 99%