Abstract
This paper is to provide an analysis of an ill-posed Cauchy problem in a half-plane. This problem is novel since the Cauchy data on the accessible boundary is given, whilst the additional temperature is involved on a line. The Dirichlet boundary condition on part of the boundary is an essential condition in the physical meaning. Then we use a redefined method of fundamental solutions (MFS) to determine the temperature and the normal heat flux on the inaccessible boundary. The present approach will give an ill-conditioned system, and this is a feature of the numerical method employed. In order to overcome the ill-posedness of the corresponding system, we use the Tikhonov regularization method combining Morozov’s discrepancy principle to get a stable solution. At last, four numerical examples, including a smooth boundary, a boundary with a corner, and a boundary with a jump, are given to show the effectiveness of the present approach.
Highlights
1 Introduction Cauchy problems, which arise in diverse science areas such as wave propagation, nondestructive testing, and geophysics have been intensively studied in the past decades [1, 7, 13, 33]
On account of the incomplete boundary conditions, Cauchy problems are classified as inverse problems and ill-posed, i.e., the solutions do not depend continuously on the Cauchy data
The Cauchy problem in a half-plane is novel since the solution of the problem should satisfy a Dirichlet boundary condition on part of the boundary AB and the Cauchy data on the accessible boundary Γ1
Summary
Cauchy problems, which arise in diverse science areas such as wave propagation, nondestructive testing, and geophysics have been intensively studied in the past decades [1, 7, 13, 33]. In [2], Chapko and Johansson consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite domain by a direct integral equation method Later, they give a generalization of the situation to three-dimensions with the Cauchy data only partially given. Consider the following Cauchy problem for the Laplace equation: Given Cauchy data f ∈ H1(Γ1) and g ∈ L2(Γ1) on Γ1 and the homogeneous boundary condition u(x) = 0 on AB, find the solution u satisfying u(x) = 0, x ∈ D,. The Cauchy problem in a half-plane is novel since the solution of the problem should satisfy a Dirichlet boundary condition on part of the boundary AB and the Cauchy data on the accessible boundary Γ1. Four numerical examples, including a smooth boundary, a boundary with a corner, and a boundary with a jump, are presented to show the effectiveness of the presented method
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