A computational method for a Cauchy problem of Laplace’s equation
Introduction
The Cauchy problem of Laplace’s equation arises from many physics and engineering problems such as nondestructive testing techniques, geophysics, and cardiology. The Cauchy problem is ill-posed, i.e., the solution does not depend continuously on the boundary data, and small errors in the boundary data can amplify the numerical solution infinitely. It is well known that the Cauchy problem is severely ill-posed, hence it is impossible to solve Cauchy problem of Laplace’s equation by using classical numerical methods and it requires special techniques, e.g., regularization methods [1], [8], [9], [15]. Although theoretical concepts and computational implementation related to the Cauchy problem of Laplace equation have been discussed by many authors [4], [5], [10], [11], [14], [3], there are many open problems deserved to be solved. For example, many authors have considered the following Cauchy problem of the Laplace’s equation [16], [6], [12], [13], [17]:However, the boundary condition is very strict. If the boundary condition is replaced by where is a function of y, then their method cannot be applied easily. Therefore, in this paper, we consider the following Cauchy problem for the Laplace equation:It is a classical example of ill-posed problems given by Hadamard in his famous paper [7]. We want to seek the solution in the interval (0, 1) from the Cauchy data pairs located at . Of course, since g and h are assumed to be measured, there must be measurement errors, and we would actually have noisy data function , for which the measurement errors and are small. Here and in the following sections, denotes the norm. Thus (1.2) is a non-characteristic Cauchy problem with appropriate Cauchy data given on the line .
However, the ill-posedness is caused by high frequency. By introducing a “cut-off” frequency we can obtain a well-posed problem. This is discussed in [2]. An error estimate for the proposed method can be found in Section 2. The implementation of the numerical method is explained and a numerical example is reported in Section 3.
Section snippets
Regularization and error estimate
First, let be the Fourier transform of u,Taking Fourier transformation for (1.2), we have a family of problems parameterized by :The solution can easily be verified to beIf , we take the limits of the expression as . Following the idea of Fourier method [2], we consider (2.1) only for by cutting off high frequency, and define a
Numerical implementation and a numerical example
In this section, we use the method of lines [2]. Let,then we can obtain:whereThe matrix D approximate the second order derivative by spectral cut-off method (see [2]). This ordinary differential equation system can easily be solved by various numerical methods.
Example. It is easy to verify that the function
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