Iterative Solutions of Boundary Value Problems.
DELAWARE UNIV NEWARK DEPT OF MATHEMATICS
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Recent work has shown that Neumanns method in potential theory could be extended to solve boundary value and transition problems for the Helmholtz equation. The procedure consists of formulating the problem as a boundary integral equation which is then rewritten, with the use of a homographic transformation of the associated eigenvalue equation, so tht the spectral radius of the resulting integral operator is less than one for small perturbations of the corresponding potential operator. The present paper describes two extensions of this work. The transformation which maximizes the distance to the spectrum of the resolvent point of interest in potential theory wave number equal to zero is shown to be not optimal for non zero wave numbers. Thus while this transformation optimizes the rate of convergence of the Neumann series for zero wave number, this is not true in general. Some analytic and numerical examples are presented. Author
- Theoretical Mathematics
- Operations Research