\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 106, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/106\hfil Modified quasi-boundary value method]
{Modified quasi-boundary value method for cauchy problems
 of elliptic equations with variable coefficients}

\author[H. Zhang\hfil EJDE-2011/106\hfilneg]
{Hongwu Zhang} 

\address{Hongwu Zhang \newline
 School of Mathematics and Statistics,
 Lanzhou University, Lanzhou city, Gansu Province, 730000, China.
\newline
 School of Mathematics and Statistics, Hexi University,
 Zhangye city, Gansu Province, 734000, China}
\email{zh-hongwu@163.com}

\thanks{Submitted May 4, 2011. Published August 23, 2011.}
\subjclass[2000]{35J15, 35J57, 65G20, 65T50}
\keywords{Ill-posed problem; Cauchy problem; elliptic equation;
\hfill\break\indent
quasi-boundary value method; convergence estimates}

\begin{abstract}
 In this article, we study a Cauchy problem for an elliptic
 equation with variable coefficients.  It is well-known that such a
 problem is severely ill-posed; i.e., the solution does not depend
 continuously on the Cauchy data. We propose a modified quasi-boundary
 value regularization method to solve it. Convergence estimates are
 established under two a priori assumptions on the exact solution.
 A numerical example is given to illustrate our proposed method.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}\label{sec1}

In this article, we consider the following Cauchy problem 
for an elliptic equation with variable coefficients in a strip,
 as in \cite{H+2007},
\begin{equation} \label{e1.1}
\begin{gathered}
u_{xx}+a(y)u_{yy}+b(y)u_y+c(y)u=0, \quad x\in\mathbb{R},\; y\in(0,1)\\
u(x,0)=\varphi(x),\quad x\in\mathbb{R},\\
u_y(x,0)=0,\quad x\in\mathbb{R},
\end{gathered}
\end{equation}
where $a, b, c$ are given functions such that for some given
positive constants $\lambda\leq \Lambda$,
\begin{gather}\label{e1.2}
\lambda\leq a(y)\leq\Lambda, \quad y\in[0,1], \\
\label{e1.3}
a(y)\in C^2[0,1], \quad b(y)\in C^1[0,1],\quad
c(y)\in C[0,1],\quad c(y)\leq0.
\end{gather}
Without loss of generality, in the following we suppose that
$\lambda \geq 1$.

This problem is well-known to be severely ill-posed; i.e., a small
perturbation in the given Cauchy data may result in a very large
error on the solution \cite{D+1992, I+V+2006, J+1997, L+1986}.
Therefore, it is very difficult to solve it using classic numerical
methods.  In order to overcome this difficulty, the regularization
methods are required \cite{H+T+2000, I+V+2006, K+1996(9),
E+H+H+M+N+1996(8)}.

It should be mentioned that, for the Cauchy problem of the elliptic
equations, many regularization methods have been
proposed: such as Tikhonov regularization method
\cite{H+J+1953(12),T+A+A+V+1977(14)}, the modified method
\cite{E+1987(11),Q+F+L+2008(13)}, the moment method \cite{W+2003},
the center difference method \cite{F+M+1986(15),R+H+H+H+D+1999(16)},
etc.  For the Cauchy problem of elliptic equations with variable
coefficients \eqref{e1.1}, in 2007, H\`ao and his
group \cite{H+2007} applied the mollification
method to solve it, and prove some stability estimates of
H\"older type for the solution and its derivatives.  In 2008,
Qian \cite{Q+2008} used a wavelet regularization
method to treat it. In the present article, following H\`ao
\cite{H+2007} and Qian \cite{Q+2008}, we continue to
consider  problem \eqref{e1.1}.

In $1983$, Showalter presented a method called the quasi-boundary
value (QBV) to regularize the linear homogeneous ill-posed problem
\cite{S+1983}.  The main idea of this method is making an
appropriate modification to the final data.   Recently many authors
have successfully used this method to solve the backward heat
conduction problem (BHCP) \cite{C+1994, M+2005, N+2008, M+1992,
F+2001}.  In \cite{D+N+V+2009}, this method was used to solve a
Cauchy problem for elliptic equation in a cylindrical domain (where
the authors {called it a} non-local boundary value
problem method).  In this paper, we shall apply a modified
quasi-boundary value method to solve problem \eqref{e1.1}.  Here our
idea mainly comes from Showalter's method (see Section 3).

This paper is constructed as follows.  In Section \ref{sec2},
we give some required results for this paper.
In Section \ref{sec3}, we present our regularization method.
Section \ref{sec4} is devoted to the convergence estimates.
Numerical results are shown in Section \ref{sec5}, and 
some conclusions are given.

\section{Some required results} \label{sec2}

For a function $f\in L^2(\mathbb{R})$, its Fourier transform is
defined by
\begin{equation}\label{e2.1}
\widehat{f}(\xi):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}
f(x)e^{-i\xi x}dx, \quad \xi\in \mathbb{R}.
\end{equation}

Let the exact data $\varphi\in L^2(\mathbb{R})$ and the measured
data $\varphi^\delta\in L^2(\mathbb{R})$ satisfy
\begin{equation}\label{e2.2}
\|\varphi^\delta-\varphi\|\leq\delta,
\end{equation}
where $\|\cdot\|$ denotes the $L^2$-norm, the constant $\delta>0$
denotes a noise level, and there exists a constant $E>0$, such that
the following a-priori bounds exist,
\begin{equation}\label{e2.3}
\|u(\cdot,1)\|\leq E.
\end{equation}
or
\begin{equation}\label{e2.4}
\|u(\cdot,1)\|_p\leq E.
\end{equation}
Here $\|u(\cdot,1)\|_p$ denotes the Sobolev space $H^p$-norm defined
by
\begin{equation}\label{e2.5}
\|u(\cdot,1)\|_p=\Big(\int_{-\infty}^{\infty}(1+\xi^2)^p
|\widehat{u}(\cdot,1)|^2d\xi\Big)^{1/2}.
\end{equation}

Now, we firstly consider the following Cauchy problem in the
frequency domain,
\begin{equation} \label{e2.6}
\begin{gathered}
-\xi^2v(\xi,y)+a(y)v_{yy}(\xi,y)+b(y)v_y(\xi,y)+c(y)v(\xi,y)=0,
\quad \xi\in\mathbb{R},\;y\in(0,1)\\
v(\xi,0)=1, \quad \xi\in\mathbb{R},\\
v_y(\xi,0)=0, \quad \xi\in\mathbb{R}.
\end{gathered}
\end{equation}
The following Lemma is very important to our
analysis, and its proof can be found in \cite{H+2007}.

\begin{lemma}\label{lem2.1}
There exists a unique solution of \eqref{e2.6} such that
\begin{itemize}
\item[(i)] $v(\xi,y)\in W^{2,\infty}(0,1)$ for all $\xi\in\mathbb{R}$,
\item[(ii)] $v(\xi,1)\neq 0$  for all $\xi\in\mathbb{R}$,
\item[(iii)] there  exist positive constants $c_1, c_2$, such that for
$\xi\in\mathbb{R}$,
\begin{gather}\label{e2.7}
|v(\xi,y)|\leq c_1e^{|\xi|A(y)},\quad \forall \;y\in[0,1],\\
\label{e2.8}
|v(\xi,1)|\geq c_2e^{|\xi|A(1)},
\end{gather}
where,
\begin{equation}\label{e2.9}
A(y)=\int^y_0\frac{ds}{\sqrt{a(s)}},\;y\in[0,1].
\end{equation}
\end{itemize}
\end{lemma}

\section{A modified quasi-boundary value regularization method}
\label{sec3}

Taking the Fourier transform in problem \eqref{e1.1} with respect to
$x$, we have
\begin{equation} \label{e3.1}
\begin{gathered}
a(y)\widehat{u}_{yy}(\xi,y)+b(y)\widehat{u}_y(\xi,y)
 +c(y)\widehat{u}(\xi,y)-\xi^2\widehat{u}(\xi,y)=0, \quad
\xi\in\mathbb{R},\;y\in(0,1)\\
\widehat{u}(\xi,0)=\widehat{\varphi}(\xi), \quad \xi\in\mathbb{R},\\
\widehat{u}_y(\xi,0)=0, \quad \xi\in\mathbb{R}.
\end{gathered}
\end{equation}
It can be shown that the solution of \eqref{e1.1} in the frequency
domain is
\begin{equation}\label{e3.2}
\widehat{u}(\xi,y)=v(\xi,y)\widehat{u}(\xi,0)
=v(\xi,y)\widehat{\varphi}(\xi).
\end{equation}
Then, the exact solution of  \eqref{e1.1} is
\begin{equation}\label{e3.3}
u(x,y)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}v(\xi,y)
\widehat{\varphi}(\xi)e^{i\xi x}d\xi.
\end{equation}
From Lemma \ref{lem2.1} and $v(\xi,1)\neq 0$, we have
\begin{equation}\label{e3.4}
\widehat{\varphi}(\xi)=\widehat{u}(\xi,0)
=\frac{\widehat{u}(\xi,1)}{v(\xi,1)},
\end{equation}
and from \eqref{e3.4}, we can note that $\widehat{u}(\xi,1)\neq0$.

If $\widehat{\varphi}(\xi), \widehat{u}(\xi,1)>0$, we consider the
following Cauchy problem in the frequency domain
\begin{equation} \label{e3.5}
\begin{gathered}
a(y)\widehat{u}_{yy}(\xi,y)+b(y)\widehat{u}(\xi,y)
+c(y)\widehat{u}(\xi,y)-\xi^2\widehat{u}(\xi,y)=0, \quad
\xi\in\mathbb{R},\;y\in(0,1)\\
\widehat{u}(\xi,0)+\alpha\widehat{u}(\xi,1)
=\widehat{\varphi}_\delta(\xi), \quad \xi\in\mathbb{R},\\
\widehat{u}_y(\xi,0)=0, \quad \xi\in\mathbb{R}.
\end{gathered}
\end{equation}
Denoting $\widehat{u}^\delta_{\alpha1}(\xi,y)$ as the solution of
\eqref{e3.5}, we obtain
\begin{equation}\label{e3.6}
\widehat{u}^\delta_{\alpha1}(\xi,y)=\frac{v(\xi,y)}{1+\alpha
v(\xi,1)}\widehat{\varphi}_\delta(\xi).
\end{equation}

If $\widehat{\varphi}(\xi)>0$, $\widehat{u}(\xi,1)<0$, we consider
the following Cauchy problem in the frequency domain
\begin{equation} \label{e3.7}
\begin{gathered}
a(y)\widehat{u}_{yy}(\xi,y)+b(y)\widehat{u}(\xi,y)+c(y)
 \widehat{u}(\xi,y)-\xi^2\widehat{u}(\xi,y)=0, \quad
 \xi\in\mathbb{R},\;y\in(0,1)\\
\widehat{u}(\xi,0)-\alpha\widehat{u}(\xi,1)
 =\widehat{\varphi}_\delta(\xi), \quad \xi\in\mathbb{R},\\
\widehat{u}_y(\xi,0)=0, \quad \xi\in\mathbb{R}.
\end{gathered}
\end{equation}
Denoting by $\widehat{u}^\delta_{\alpha2}(\xi,y)$
the solution of \eqref{e3.7}, we have
\begin{equation}\label{e3.8}
\widehat{u}^\delta_{\alpha2}(\xi,y)=\frac{v(\xi,y)}{1-\alpha
v(\xi,1)}\widehat{\varphi}_\delta(\xi).
\end{equation}

If $\widehat{\varphi}(\xi)>0$, $\widehat{u}(\xi,1)$ can be positive
or negative, we define the following modified regularization
solution to \eqref{e1.1} in the frequency domain:
\begin{equation}\label{e3.9}
\widehat{u}^\delta_{\alpha}(\xi,y)=\frac{v(\xi,y)}{1+\alpha
|v(\xi,1)|}\widehat{\varphi}_\delta(\xi).
\end{equation}

By the above analysis, for $\widehat{\varphi}(\xi)>0$, we define a
modified regularization solution of form \eqref{e3.9} to problem
\eqref{e1.1} in the frequency domain.

Equivalently, the regularization solution of  \eqref{e1.1}
is given by
\begin{equation}\label{e3.10}
u^\delta_\alpha(x,y)=\frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{\infty}\frac{v(\xi,y)}{1+\alpha
|v(\xi,1)|}\widehat{\varphi}_\delta(\xi)e^{i\xi x}d\xi.
\end{equation}
Adopting similar analysis, when $\widehat{\varphi}(\xi)<0$, we can
also define the modified regularization solution of form
\eqref{e3.10}.

In the following section, we will prove that the regularization
solution $u^\delta_\alpha(x,y)$ given by \eqref{e3.10} is a stable
approximation to the exact solution $u(x,y)$ given by \eqref{e3.3},
and the regularization solution $u^\delta_\alpha(x,y)$ depends
continuously on the measured data $\varphi^\delta$ for a fixed
parameter $\alpha>0$.

\section{Convergence Estimates}\label{sec4}

In this section, we give the convergence estimates for $0<y<1$ and
$y=1$ under two different a-priori assumptions for the exact
solution $u$, respectively.

\begin{theorem}\label{thm4.1}
  Suppose that $u$ is defined by \eqref{e3.3} with the exact
data $\varphi$ and $u^\delta_\alpha$ is defined by \eqref{e3.10}
with the measured data $\varphi^\delta$.
Let the measured data $\varphi^\delta$ satisfy
\eqref{e2.2}, and let the exact solution $u$ at $y=1$ satisfy
\eqref{e2.3}.  If the regularization parameter $\alpha$ is chosen as
\begin{equation}\label{e4.1}
\alpha=\frac{\delta}{E},
\end{equation}
then for fixed $0<y<1$ we have the following convergence estimate
\begin{equation}\label{e4.2}
\|u^\delta_\alpha(\cdot,y)-u(\cdot,y)\|
\leq 2C_yE^\frac{A(y)}{A(1)}\delta^{1-\frac{A(y)}{A(1)}}.
\end{equation}
\end{theorem}

\begin{proof}
 From \eqref{e3.2}, \eqref{e3.9}, \eqref{e2.2}, \eqref{e2.3}, we
have
\begin{equation} \label{e4.3}
\begin{split}
\|u^\delta_\alpha(\cdot,y)-u(\cdot,y)\|
&=\|u^\delta_\alpha(\xi,y)-u(\xi,y)\|\\
&=\|\frac{v(\xi,y)\widehat{\varphi}(\xi)(1+\alpha|v(\xi,1)|)
-v(\xi,y)\widehat{\varphi}_\delta(\xi)}{1+\alpha|v(\xi,1)|}\|\\
&=\|\frac{v(\xi,y)(\widehat{\varphi}_\delta(\xi)-\widehat{\varphi}(\xi))
 +\alpha|v(\xi,1)|v(\xi,y)\widehat{\varphi}(\xi)}{1+\alpha|v(\xi,1)|}\|\\
&\leq\delta\sup_{\xi\in\mathbb{R}}\frac{|v(\xi,y)|}{1+\alpha|v(\xi,1)|}
+\alpha E\frac{|v(\xi,y)|}{1+\alpha|v(\xi,1)|}\\
&:=\delta \sup_{\xi\in\mathbb{R}}I_1
 +\alpha E\sup_{\xi\in\mathbb{R}}I_1.
\end{split}
\end{equation}
From Lemma \ref{lem2.1}, we can derive that
\begin{equation} \label{e4.4}
I_1=\frac{|v(\xi,y)|}{1+\alpha|v(\xi,1)|}\leq\frac{ c_1
e^{|\xi|A(y)}}{1+\alpha
c_2e^{|\xi|A(1)}}\leq\frac{c_1}{\min\{1,c_2\}}\cdot\frac{
e^{|\xi|A(y)}}{1+\alpha e^{|\xi|A(1)}}.\\
\end{equation}
Let $f(s)= e^{sA(y)}/(1+\alpha e^{sA(1)})$, $s\geq0$, then
\begin{equation}\label{e4.5}
f'(s)=f(s)\frac{ A(y)-\alpha(A(1)-A(y)) e^{A(1)s}}{1+\alpha
e^{|s|A(1)}}.
\end{equation}
Setting $f'(s)=0$, we have
\begin{equation}\label{e4.6}
\alpha (A(1)-A(y))e^{A(1)s}=A(y).
\end{equation}
Note that $A(1)\geq0$, $A(1)\geq A(y)\geq0$ for $0\leq y\leq1$, it is
easy to see that $f(s)$ has a unique maximal value point $s^*$ such
that
\begin{equation}\label{e4.7}
\alpha e^{A(1)s^*}=\frac{A(y)}{A(1)-A(y)}.
\end{equation}
Thus,
\begin{equation}\label{e4.8}
f(s)\leq f(s^*)=c_y\alpha ^{-\frac{A(y)}{A(1)}},
\end{equation}
where
\[
c_y=\frac{(A(y))^{\frac{A(y)}{A(1)}}}{A(1)}(A(1)-A(y))
^{\frac{A(y)}{A(1)}-1}.
\]
Then
\begin{equation}\label{e4.9}
I_1\leq\frac{c_1}{\min\{1,c_2\}}\cdot\frac{ e^{|\xi|A(y)}}{1+\alpha
e^{|\xi|A(1)}}\leq\frac{c_1c_y}{\min\{1,c_2\}}\alpha
^{-\frac{A(y)}{A(1)}}:=C_y\alpha ^{-\frac{A(y)}{A(1)}},
\end{equation}
By \eqref{e4.1}, \eqref{e4.3}, \eqref{e4.9}, for fixed $0<y<1$, we
obtain
\[
\|u^\delta_\alpha(\cdot,y)-u(\cdot,y)\|\leq2C_yE^\frac{A(y)}{A(1)}\delta^{1-\frac{A(y)}{A(1)}}.
\]
\end{proof}

From Theorem \ref{thm4.1}, we note that $u^\delta_\alpha$ defined by
\eqref{e3.10} is an effective approximation to the exact solution $u$
for the fixed $0<y<1$.  But the estimate \eqref{e4.2} gives no
information about the error estimate at $y = 1$ as the constraint
\eqref{e2.3} is too weak for this purpose.   To retain the
continuity, as common, we suppose that $u(x, y)$ satisfies a
stronger a-priori assumption \eqref{e2.4} at $y = 1$.

 \begin{theorem}\label{thm4.2}
Let the exact solution $u$ and the regularization solution
$u^\delta_\alpha$ be defined by \eqref{e3.3}, \eqref{e3.10},
respectively. Assume that the measured data $\varphi^\delta$
satisfies $\|\varphi^\delta-\varphi\|\leq \delta$, and let the exact
solution $u$ satisfy \eqref{e2.4}.  If the regularization parameter
$\alpha$ is chosen as
\begin{equation}\label{e4.10}
\alpha=\sqrt{\delta/E},
\end{equation}
then we have the following convergence estimate at $y=1$,
\begin{equation}\label{e4.11}
\|u(\cdot,1)-u^\delta_\alpha(\cdot,1)\|
\leq\sqrt{\delta E}+CE\max\big\{
\big(\frac{\delta}{E}\big)^{1/3},
\big(\frac{1}{6}\ln\frac{E}{\delta}\big)^{-p}\big\}.
\end{equation}
\end{theorem}

\begin{proof}
By \eqref{e3.2}, \eqref{e3.9}, \eqref{e2.2}, \eqref{e2.4}, we have
\begin{equation} \label{e4.12}
\begin{aligned}
\|u^\delta_\alpha(\cdot,1)-u(\cdot,1)\|
&=\|u^\delta_\alpha(\xi,1)-u(\xi,1)\|\\
&=\|\frac{v(\xi,1)\widehat{\varphi}(\xi)(1+\alpha|v(\xi,1)|)
 -v(\xi,1)\widehat{\varphi}_\delta(\xi)}{1+\alpha|v(\xi,1)|}\|\\
&=\|\frac{v(\xi,1)(\widehat{\varphi}_\delta(\xi)-\widehat{\varphi}(\xi))
 +\alpha|v(\xi,1)|v(\xi,1)\widehat{\varphi}(\xi)}
 {1+\alpha|v(\xi,1)|}\|\\
&\leq\delta\sup_{\xi\in\mathbb{R}}\frac{|v(\xi,1)|}{1+\alpha|v(\xi,1)|}
+ E\sup_{\xi\in\mathbb{R}}\frac{\alpha(1+\xi^2)
 ^{-\frac{p}{2}}|v(\xi,1)|}{1+\alpha|v(\xi,1)|}\\
&:=\delta \sup_{\xi\in\mathbb{R}}I_2+ E\sup_{\xi\in\mathbb{R}}I_3.
\end{aligned}
\end{equation}
It is easy to know that
\begin{equation}\label{e4.13}
I_2=\frac{|v(\xi,1)|}{1+\alpha|v(\xi,1)|}\leq\frac{1}{\alpha},
\end{equation}
then by \eqref{e4.10}, we know
\begin{equation}\label{e4.14}
\delta\sup_{\xi\in\mathbb{R}}{I_2}\leq\sqrt{\delta E}.
\end{equation}
In the following, we estimate $I_3$.  From Lemma \ref{lem2.1}, we obtain
\begin{equation}\label{e4.15}
I_3=\frac{\alpha(1+\xi^2)^{-\frac{p}{2}}|v(\xi,1)|}{1+\alpha|v(\xi,1)|}
\leq\frac{c_1}{\min\{1,c_2\}}\cdot\frac{
\alpha(1+\xi^2)^{-\frac{p}{2}} e^{|\xi|A(1)}}{(1+\alpha
e^{|\xi|A(1)})}.
\end{equation}

Case 1: For the large values with
$|\xi|\geq\ln\frac{1}{\sqrt[3]{\alpha}}$, we have
\begin{equation}\label{e4.16}
\frac{c_1}{\min\{1,c_2\}}\cdot\frac{ \alpha(1+\xi^2)^{-\frac{p}{2}}
e^{|\xi|A(1)}}{(1+\alpha
e^{|\xi|A(1)})}\leq\frac{c_1}{\min\{1,c_2\}}
\big(\ln\frac{1}{\sqrt[3]{\alpha}}\big)^{-p}
:=C\big(\ln\frac{1}{\sqrt[3]{\alpha}}\big)^{-p}.
\end{equation}

Case 2: For $|\xi|<\ln\frac{1}{\sqrt[3]{\alpha}}$, since
$1\leq\lambda\leq a(y)\leq\Lambda$,
$A(1)=\int^1_0\frac{1}{\sqrt{a(s)}}ds\leq1$, then
\begin{equation}\label{e4.17}
\frac{c_1}{\min\{1,c_2\}}\cdot\frac{ \alpha(1+\xi^2)^{-\frac{p}{2}}
e^{|\xi|A(1)}}{(1+\alpha
e^{|\xi|A(1)})}\leq\frac{c_1}{\min\{1,c_2\}}\alpha e^{|\xi|A(1)}
\leq C\alpha^{\frac{2}{3}}.
\end{equation}
By \eqref{e4.16}, \eqref{e4.17}, we obtain
\begin{equation}\label{e4.18}
I_3\leq C\max\big\{\alpha^{2/3},
\big(\ln\frac{1}{\sqrt[3]{\alpha}}\big)^{-p}\big\}.
\end{equation}
Then, from \eqref{e4.10}, \eqref{e4.12}, \eqref{e4.14}, \eqref{e4.18},
for $y=1$, we have
\[
\|u(\cdot,1)-u^\delta_\alpha(\cdot,1)\|\leq\sqrt{\delta
E}+CE\max\big\{\big(\frac{\delta}{E}\big)^{1/3},
\big(\frac{1}{6}\ln\frac{E}{\delta}\big)^{-p}\big\}.
\]
\end{proof}

\begin{remark}\label{rmk4.3} \rm
 In the convergence estimate \eqref{e4.11}, we can see that the
logarithmic term with respect to $\delta$ is the dominating term.
Asymptotically this yields a convergence rate of order
$O(\ln\frac{E}{\delta})^{-p}$.  The first
term {is} asymptotically negligible compared to this
term.
\end{remark}

\section{Numerical implementations}\label{sec5}

 In this section, we use a numerical example to verify the
stability of our proposed regularization method.  For
simplicity, we consider the following Cauchy problem for the Laplace
equation,
\begin{equation} \label{e5.1}
\begin{gathered}
u_{xx}+u_{yy}=0, \quad x\in\mathbb{R},\; y\in(0,1)\\
u(x,0)=\varphi(x),\quad x\in\mathbb{R},\\
u_y(x,0)=0,\quad x\in\mathbb{R}.
\end{gathered}
\end{equation}
It is easy to verify that
\begin{equation}\label{e5.2}
u(x,y)=e^{y^2-x^2}\cos(2xy),
\end{equation}
is the exact solution  of problem \eqref{e5.1}, with initial data
\begin{equation}\label{e5.3}
\varphi(x)=e^{-x^2}.
\end{equation}
In this case, the solution of \eqref{e2.6} becomes
\begin{equation}\label{e5.4}
v(\xi,y)=\cosh(|\xi| y).
\end{equation}

We define all functions to be zero for
$x\in(-\infty,-3\pi)\cup(3\pi,\infty)$, so we choose the interval
$[-3\pi,3\pi]$ to complete our numerical experiment by using the
discrete Fourier transform and inverse Fourier transform (FFT and
IFFT).

The measured data $\varphi_\delta $ is given by
$\varphi^\delta(x_i)=\varphi(x_i)+\varepsilon \operatorname{rand}(i)$, where
$\varepsilon$ is the error level,
\begin{gather}\label{e5.5}
\varphi(x)=(\varphi(x_1), \dots, \varphi(x_N)), \\
\label{e5.6}
x_j=-3\pi+\frac{6\pi(j-1)}{N-1}, \quad j=1, 2, \dots, N, \\
\label{e5.7}
\delta=\|\varphi_\delta-\varphi\|_{l_2}
=\Big(\frac{1}{N}\sum_{j=1}^N
 |\varphi_\delta(x_j)-\varphi(x_j)|\Big)^{1/2}.
\end{gather}
the function $\operatorname{rand}(\cdot)$ denotes arrays of random
numbers whose elements are uniformly distributed in the interval
$[0,1]$.  The relative root mean square error between the exact
and approximate solution is given by
\begin{equation}\label{e5.8}
\epsilon(u)=\frac{\sqrt{\frac{1}{N}\sum_{j=1}^N\left(u_{j}
-(u^\delta_\alpha)_{j}\right)^2}}{\sqrt{\frac{1}{N}
\sum_{j=1}^N(u_{j})^2}}.
\end{equation}
Then we obtain the regularization solution $u^\delta_\alpha$
computed by \eqref{e3.10}.

Numerical results are shown in Figures \ref{fig1}-\ref{fig2}. The
numerical result for $u(\cdot,y)$ and $u^\delta_\alpha(\cdot,y)$ at
$x=0.2$, $x=0.5$, and $x=0.8$ with $\varepsilon=1\times10^{-4},
10^{-3}$ are shown in Figure \ref{fig1}.
In Figure \ref{fig1}, we choose
the a-priori bound $E=1$ and the regularization parameter $\alpha$
is chosen by \eqref{e4.1}.  The numerical results for $u(\cdot,1)$
and $u^\delta_\alpha(\cdot,1)$ with $\varepsilon=1\times 10^{-4},
\varepsilon=10^{-3}$ are shown in Fig.\ref{fig2}, where the
regularization parameter $\alpha$ is chosen by \eqref{e4.10} and the
a-priori bound $E=1$.  The relative root mean square errors at
$y=0.6, y=1$ for the computed solution versus the error levels
$\varepsilon$  are shown in Tables $1-2$.

From Figures \ref{fig1}-\ref{fig2}, we find the stability of our 
proposed method.  From Tables 1--2, we note that the
smaller the $\varepsilon$ is, the better the computed solution is,
which means that our proposed regularization method is sensitive to
the noise level $\varepsilon$.  In addition, we can note that
numerical results become worse when $y$ approaches to $1$.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig1a}
\includegraphics[width=0.45\textwidth]{fig1b}\\
 $y=0.2$, $\varepsilon=1\times10^{-4}$ \hfil
 $y=0.2$, $\varepsilon=1\times10^{-3}$\\
\includegraphics[width=0.45\textwidth]{fig1c}
\includegraphics[width=0.45\textwidth]{fig1d} 
 $y=0.5$, $\varepsilon=1\times10^{-4}$ \hfil
 $y=0.8$, $\varepsilon=1\times10^{-3}$.\\
\includegraphics[width=0.45\textwidth]{fig1e}
\includegraphics[width=0.45\textwidth]{fig1f}\\
$y=0.8$, $\varepsilon=1\times10^{-4}$ \hfil
$y=0.8$, $\varepsilon=1\times10^{-3}$
\end{center}
\caption{Graph of $u(\cdot,y)$ and $u^\delta_\alpha(\cdot,y)$} \label{fig1}
\end{figure}


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig2a}
\includegraphics[width=0.45\textwidth]{fig2b} 
$\varepsilon=1\times10^{-4}$\hfil
$\varepsilon=1\times10^{-3}$
\end{center}
\caption{Graph of $u(\cdot,1)$ and $u^\delta_\alpha(\cdot,1)$}\label{fig2}
\end{figure}

\begin{table}[ht]
\tabcolsep 2mm
\caption{The relative root mean square errors at
$y=0.6$ for various noisy levels}
\begin{center}
\begin{tabular}{|lllll|}
\hline
$\varepsilon$ & 0.00001 & 0.0001 & 0.001 & 0.01 \\ \hline
$\alpha$    & 0.0032 & 0.01  & 0.0316& 0.1 \\
$\epsilon(u)$  &0.0118 & 0.0345&  0.0914& 0.2098\\ \hline
\end{tabular}
\end{center}
\end{table}\label{ab1}

\begin{table}[ht]
\tabcolsep 2mm
\caption{The relative root mean square errors at
$y=1$ for various noisy levels}
\begin{center}
\begin{tabular}{|lllll|}
\hline
$\varepsilon$ & 0.00001 & 0.0001 & 0.001 & 0.01 \\ \hline
$\alpha$ &0.0032 & 0.01&  0.0316& 0.1\\
$\epsilon(u)$  &0.0269 &0.0727&  0.1721& 0.3424\\
\hline
\end{tabular}
\end{center}
\end{table}\label{ab2}

\subsection*{Conclusions}%\label{sec6}
  In this article, a modified quasi-boundary value regularization
method is used to solve a Cauchy
problem for the elliptic equation with variable coefficients.  The
convergence estimates for $0<y<1$ and $y=1$ have been obtained under
two different a-priori bound assumptions for the exact solution.
Some numerical results show that our proposed regularization method
is feasible.

\subsection*{Acknowledgements}%\label{sec7}
The author would like to thank the anonymous reviewers for their
constructive comments and valuable suggestions that improve
the quality of our article.  The work described in this article was
supported by grant 10971089 from  the NSF of China.


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\end{document}