Volume 19, Issue 5
The Numerical Solution of First Kind Integral Equation for the Helmholtz Equation on Smooth Open Arcs

J. Comp. Math., 19 (2001), pp. 489-500

Published online: 2001-10

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• Abstract

Consider solving the Dirichlet problem of Helmholtz equation on unbounded region R~2\G with G a smooth open curve in the plane. We use simple-layer potential to construct a solution. This leads to the solution of a logarithmic integral equation of the first kind for the Helmholtz equation. This equation is reformulated using a special change of variable, leading to a new first kind equation with a smooth solution function. This new equation is split into three parts. Then a quadrature method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. An error analysis in a Sobolev space setting is given. And numerical results show that fast convergence is clearly exhibited.

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