A computational method for a Cauchy problem of Laplace’s equation

Abstract

In this paper, we consider the Cauchy problem for the Laplace’s equation, where the Cauchy data is given at $x=1$ and the solution is sought in the interval $0<x<1$. A spectral method together with choice of regularization parameter is presented and error estimate is obtained. Combining the method of lines, we implement the numerical solution. A numerical example shows that the method works well.

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]:

$$\begin{array}{cc}\hfill & u_{\mathit{xx}}+u_{\mathit{yy}}=0\text{,}0<x<1\text{,}-\infty <y<+\infty \text{,}\hfill \\ \hfill & u(1\text{,}y)=g(y)\text{,}-\infty <y<+\infty \text{,}\hfill \end{array}$$

$$u_x(1\text{,}y)=0\text{,}-\infty <y<+\infty \text{.}$$

However, the boundary condition $u_x(1\text{,}y)=0$ is very strict. If the boundary condition is replaced by $u_x(1\text{,}y)=h(y)$ where $h(y)$ 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:

$$\begin{array}{cc}\hfill & u_{\mathit{xx}}+u_{\mathit{yy}}=0\text{,}0<x<1\text{,}-\infty <y<+\infty \text{,}\hfill \\ \hfill & u(1\text{,}y)=g(y)\text{,}-\infty <y<+\infty \text{,}\hfill \end{array}$$

$$u_x(1\text{,}y)=h(y)\text{,}-\infty <y<+\infty \text{.}$$

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 $(g\text{,}h)$ located at $x=1$. Of course, since g and h are assumed to be measured, there must be measurement errors, and we would actually have noisy data function $g_{\delta }\text{,}{h}_{\delta }\in L^2(\mathbb{R})$, for which the measurement errors $\Vert h-h_{\delta }\Vert$ and $\Vert g-g_{\delta }\Vert$ are small. Here and in the following sections, $\Vert ·\Vert$ denotes the $L^2(\mathbb{R})$ norm. Thus (1.2) is a non-characteristic Cauchy problem with appropriate Cauchy data $[u\text{,}{u}_x]$ given on the line $x=1$.

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.