| Summary: | 碩士 === 國立臺灣海洋大學 === 機械與機電工程學系 === 94 === We consider a new method that it is developed to solve inverse Cauchy problems and inverse Robin problems for the Laplace equation, which is named the regularized integral equation method (RIEM). The inverse problem for the Laplace equation by recoverning boundary values on the inaccessible boundary of the body from available overspecified data on the accessible boundary. The numerical results can be used to determine the Robin type inverse problem, the Cauchy type inverse problem, the problem of detecting crack position, the unknown shape of Zero potential problem through the measurements at the accessible boundary.
The Fourier series expansion is used to formulate the first kind Fredholm integral equation for the unknown data on the inaccessible boundary. Then we consider a Lavrentiev regularization, by adding an extra term to obtain a second kind Fredholm integral equation. The termwise separable property of kernel function allows us to obtain a closed-form solution of the missing boundary condition. The uniform convergence and error estimate of the regularization solution are proved. Then We apply this method to the Cauchy type inverse problems, the Robin type inverse problems, the unknown shape of Zero-potential problem, as well as the problem of detecting crack position. These numerical examples show the effectiveness of the new method in providing excellent estimates of the unknown data from the given data.
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