\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \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}, $$\label{e1.1} \begin{gathered} u_{xx}+a(y)u_{yy}+b(y)u_y+c(y)u=0, \quad x\in\mathbb{R},\; y\in(0,1)\\ u(x,0)=\varphi(x),\quad x\in\mathbb{R},\\ u_y(x,0)=0,\quad x\in\mathbb{R}, \end{gathered}$$ 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 $$\label{e2.1} \widehat{f}(\xi):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)e^{-i\xi x}dx, \quad \xi\in \mathbb{R}.$$ Let the exact data $\varphi\in L^2(\mathbb{R})$ and the measured data $\varphi^\delta\in L^2(\mathbb{R})$ satisfy $$\label{e2.2} \|\varphi^\delta-\varphi\|\leq\delta,$$ 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, $$\label{e2.3} \|u(\cdot,1)\|\leq E.$$ or $$\label{e2.4} \|u(\cdot,1)\|_p\leq E.$$ Here $\|u(\cdot,1)\|_p$ denotes the Sobolev space $H^p$-norm defined by $$\label{e2.5} \|u(\cdot,1)\|_p=\Big(\int_{-\infty}^{\infty}(1+\xi^2)^p |\widehat{u}(\cdot,1)|^2d\xi\Big)^{1/2}.$$ Now, we firstly consider the following Cauchy problem in the frequency domain, $$\label{e2.6} \begin{gathered} -\xi^2v(\xi,y)+a(y)v_{yy}(\xi,y)+b(y)v_y(\xi,y)+c(y)v(\xi,y)=0, \quad \xi\in\mathbb{R},\;y\in(0,1)\\ v(\xi,0)=1, \quad \xi\in\mathbb{R},\\ v_y(\xi,0)=0, \quad \xi\in\mathbb{R}. \end{gathered}$$ 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, $$\label{e2.9} A(y)=\int^y_0\frac{ds}{\sqrt{a(s)}},\;y\in[0,1].$$ \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 $$\label{e3.1} \begin{gathered} a(y)\widehat{u}_{yy}(\xi,y)+b(y)\widehat{u}_y(\xi,y) +c(y)\widehat{u}(\xi,y)-\xi^2\widehat{u}(\xi,y)=0, \quad \xi\in\mathbb{R},\;y\in(0,1)\\ \widehat{u}(\xi,0)=\widehat{\varphi}(\xi), \quad \xi\in\mathbb{R},\\ \widehat{u}_y(\xi,0)=0, \quad \xi\in\mathbb{R}. \end{gathered}$$ It can be shown that the solution of \eqref{e1.1} in the frequency domain is $$\label{e3.2} \widehat{u}(\xi,y)=v(\xi,y)\widehat{u}(\xi,0) =v(\xi,y)\widehat{\varphi}(\xi).$$ Then, the exact solution of \eqref{e1.1} is $$\label{e3.3} u(x,y)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}v(\xi,y) \widehat{\varphi}(\xi)e^{i\xi x}d\xi.$$ From Lemma \ref{lem2.1} and $v(\xi,1)\neq 0$, we have $$\label{e3.4} \widehat{\varphi}(\xi)=\widehat{u}(\xi,0) =\frac{\widehat{u}(\xi,1)}{v(\xi,1)},$$ 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 $$\label{e3.5} \begin{gathered} a(y)\widehat{u}_{yy}(\xi,y)+b(y)\widehat{u}(\xi,y) +c(y)\widehat{u}(\xi,y)-\xi^2\widehat{u}(\xi,y)=0, \quad \xi\in\mathbb{R},\;y\in(0,1)\\ \widehat{u}(\xi,0)+\alpha\widehat{u}(\xi,1) =\widehat{\varphi}_\delta(\xi), \quad \xi\in\mathbb{R},\\ \widehat{u}_y(\xi,0)=0, \quad \xi\in\mathbb{R}. \end{gathered}$$ Denoting $\widehat{u}^\delta_{\alpha1}(\xi,y)$ as the solution of \eqref{e3.5}, we obtain $$\label{e3.6} \widehat{u}^\delta_{\alpha1}(\xi,y)=\frac{v(\xi,y)}{1+\alpha v(\xi,1)}\widehat{\varphi}_\delta(\xi).$$ If $\widehat{\varphi}(\xi)>0$, $\widehat{u}(\xi,1)<0$, we consider the following Cauchy problem in the frequency domain $$\label{e3.7} \begin{gathered} a(y)\widehat{u}_{yy}(\xi,y)+b(y)\widehat{u}(\xi,y)+c(y) \widehat{u}(\xi,y)-\xi^2\widehat{u}(\xi,y)=0, \quad \xi\in\mathbb{R},\;y\in(0,1)\\ \widehat{u}(\xi,0)-\alpha\widehat{u}(\xi,1) =\widehat{\varphi}_\delta(\xi), \quad \xi\in\mathbb{R},\\ \widehat{u}_y(\xi,0)=0, \quad \xi\in\mathbb{R}. \end{gathered}$$ Denoting by $\widehat{u}^\delta_{\alpha2}(\xi,y)$ the solution of \eqref{e3.7}, we have $$\label{e3.8} \widehat{u}^\delta_{\alpha2}(\xi,y)=\frac{v(\xi,y)}{1-\alpha v(\xi,1)}\widehat{\varphi}_\delta(\xi).$$ 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: $$\label{e3.9} \widehat{u}^\delta_{\alpha}(\xi,y)=\frac{v(\xi,y)}{1+\alpha |v(\xi,1)|}\widehat{\varphi}_\delta(\xi).$$ 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 $$\label{e3.10} u^\delta_\alpha(x,y)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}\frac{v(\xi,y)}{1+\alpha |v(\xi,1)|}\widehat{\varphi}_\delta(\xi)e^{i\xi x}d\xi.$$ 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