American Institute of Mathematical Sciences

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2003, 2003(Special): 604-609. doi: 10.3934/proc.2003.2003.604

An averaging method for the Helmholtz equation

S. L. Ma'u 1, and  P. Ramankutty 1,

1. 

Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand, New Zealand

Published  April 2003

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.
Keywords: averaging method, spherical means, Helmholtz, boundary value problem..
Mathematics Subject Classification: Primary: 35J05, 35A35; Secondary: 65N0.
Citation: S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604
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