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On a non-linear integral equation occurring in diffraction theory

Published online by Cambridge University Press:  24 October 2008

R. F. Millar
R. F. Millar
Affiliation:
Laboratory of Electromagnetic Theory, The Technical University of Denmark, Lyngby, Denmark
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Abstract

The problem of diffraction of a plane wave by a semi-infinite grating of iso-tropic scatterers leads to the consideration of a non-linear integral equation. This bears a resemblance to Chandrasekhar's integral equation which arises in the study of radiative transfer through a semi-infinite atmosphere. It is shown that methods which have been used with success to solve Chandrasekhar's equation are equally useful here. The solution to the non-linear equation satisfies a more simple functional equation which may be solved by factoring (in the Wiener-Hopf sense) a given function. Subject to certain additional conditions which are dictated by physical considerations, a solution is obtained which is the unique admissible solution of the non-linear integral equation. The factors and solution are found explicitly for the case which corresponds to closely spaced scatterers.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 2 , April 1966 , pp. 249 - 261
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

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