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Operators of minimal norm via modified Green's functions

Published online by Cambridge University Press:  14 November 2011

R. E. Kleinman  and
G. F. Roach
R. E. Kleinman
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19711, U.S.A.
G. F. Roach
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow G1 1XH
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Synopsis

The exterior Dirichlet and Neumann problems can be treated very satisfactorily by using a fundamental solution which is modified by adding radiating spherical wave functions. It has been shown [3], that the coefficients of these added terms can be chosen to ensure that the associated boundary integral equation formulation of the problem was uniquely solvable and, in addition, that the modified Green's function was a least squares best approximation to the exact Green's function for the problem. Here we show that the coefficients can be chosen to ensure not only unique solvability but also minimization of the norm of the modified integral operator. This leads to a constructive method of solution. The theory is illustrated when the boundary is a sphere and when it is a perturbation of a sphere.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 94 , Issue 1-2 , 1983 , pp. 163 - 178
Copyright
Copyright © Royal Society of Edinburgh 1983

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References

1Jones, D. S.. Integral equations for the exterior acoustic problems. Quart. J. Mech. Appl. Math. 22 (1974), 129–142.Google Scholar
2Ursell, F.. On the exterior problems of acoustics: II. Math. Proc. Camb. Philos. Soc. 84 (1978), 545–548.CrossRefGoogle Scholar
3Kleinmann, R. E. and Roach, G. F.. On modified Green's functions in exterior problems for the Helmholtz equation. Proc. Roy. Soc. London Ser. A 383 (1982), 313–332.Google Scholar
4Marcus, M. and Mine, H.. A Survey of Matrix Theory and Matrix Inequalities (Allyn and Bacon, Boston: 1964).Google Scholar