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Two regularization methods for a Cauchy problem for the Laplace equation. (English) Zbl 1132.35493

Summary: A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for \(y=0\), and boundary data are prescribed for \(x=0\) and \(x=\pi \). The solution is sought for \(0<y\leqslant 1\). We propose two different regularization methods for the ill-posed problem based on separation of variables. Both methods are applied to find regularized solutions which are stably convergent to the exact one with explicit error estimates.

MSC:

35R25 Ill-posed problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
35B35 Stability in context of PDEs
35A35 Theoretical approximation in context of PDEs
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