An averaging method for the Helmholtz equation

S. L. Ma'u; P. Ramankutty

doi:10.3934/proc.2003.2003.604

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2003, Volume 2003: 604-609. Doi: 10.3934/proc.2003.2003.604 open access

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An averaging method for the Helmholtz equation

S. L. Ma'u¹ and
P. Ramankutty¹

1.
Department of Mathematics, University of Auckland, Private Bag 92019 Auckland

Published: March 2003

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Abstract

The well-known J. Schauder result on the existence of Lip$_\alpha(bar(\Omega))$ solutions of the Dirichlet problem for bounded domains with smooth boundaries is true for the Helmholtz equation $\Delta u + \lambda u = 0$ for $\lambda =< 0$. We suggest a method of constructing the solution based on an averaging procedure and mean-value theorem. We show some conditions under which, for $0 < \alpha < 1$, and $\lambda =< 0$, a sequence of iterated averages of an initial approximation converges geometrically to the solution.

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Mathematics Subject Classification: Primary: 35J05, 35A35; Secondary: 65N06.

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