\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 122, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/122\hfil A wavelet regularization method]
{A wavelet regularization method for an inverse heat conduction
problem with convection term}

\author[W. Cheng, Y.-Q. Zhang, C.-L. Fu \hfil EJDE-2013/122\hfilneg]
{Wei Cheng, Ying-Qi Zhang, Chu-Li Fu}  % in alphabetical order

\address{Wei Cheng \newline
College of Science, Henan University of Technology\\
Zhengzhou 450001, China}
\email{chwei670815@163.coma}

\address{Ying-Qi Zhang \newline
College of Science, Henan University of Technology\\
Zhengzhou 450001, China}
\email{zyq2018@126.com}

\address{Chu-Li Fu \newline
School of Mathematics and Statistics, Lanzhou University \\
Lanzhou 730000, China}
\email{fuchuli@lzu.edu.cn}

\thanks{Submitted February 5, 2013. Published May 17, 2013.}
\subjclass[2000]{65M30, 35R25, 35R30}
\keywords{Ill-posed problem; inverse heat
conduction; dual least squares method; \hfill\break\indent 
Shannon wavelet; regularization}

\begin{abstract}
 In this article, we consider an inverse heat conduction problem
 with convection, which is  ill-posed; i.e.,  the solution does 
 not depend continuously on the given data.
 A special projection dual least squares method generated
 by the family of Shannon wavelets is applied to formulate an
 approximate solution. Also an optimal-order estimate for the error
 between the  approximate solution and exact solution is obtained. 
\end{abstract}

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


\section{Introduction}

In many industrial applications it is needed to determine the
temperature on the surface of a body, where the surface  is
inaccessible for measurements \cite{b2}. In this case, it is
necessary to determine the surface temperature from a measured
temperature history at a fixed location inside the body. This
is called an inverse heat conduction problem (IHCP) and has been an
interesting subject recently. 
The standard problem is to determine the temperature $u$ in the 
sideways heat equation
\begin{equation} \label{e1.1}
\begin{gathered}
u_t=u_{xx}, \quad x>0,\,t>0,\\
u(x,0)=0,  \quad  x\geq0,\\
u(1,t)=g(t), \quad t\geq0,\\
u(x,t)\text{ remains bounded as $x\to \infty$},
\end{gathered}
\end{equation}
which has been considered by many authors; see  
\cite{c1,e1,e2,q1,r2,s1,w1} and the references therein.

In this article we consider a non-standard inverse heat conduction
problem: A heat conduction problem with
convection term in a quarter plane which appears in some applied
subjects \cite{b1,f2,x1,x2},
\begin{equation} \label{e1.2}
\begin{gathered}
u_t+u_x=u_{xx}, \quad x>0,\;t>0,\\
u(x,0)=0,  \quad x\geq0,\\
u(1,t)=g(t), \quad t\geq0,\\
u(x,t)\text{ remains bounded as $x\to \infty$},
\end{gathered}
\end{equation}
where the convection term $u_x$ relates to a fluid going through
the body \cite{b1}. We want  the temperature distribution in the
interval $[0,1)$ for problem \eqref{e1.2}. This problem is ill-posed 
problem in the sense that small perturbations in the data may cause 
dramatically large errors in the solution. Details can be seen in \cite{f2}.

Xiong and his colleagues  investigated  \eqref{e1.2} by the
central difference method in \cite{x1,x2}. Regi\'{n}ska \cite{r1}  solved
 \eqref{e1.1} in the interval $[0,1)$ by applying the wavelet dual
least squares method, which is based on the family of Meyer wavelets.
This regularization method has also been used for solving an unknown
source identification problem by Dou and Fu \cite{d1}. In this paper, we
solve \eqref{e1.2} in the interval $[0,1)$ by determining
the temperature distribution using a wavelet dual least squares method
generated by the family of Shannon wavelets.

To the best of our knowledge, so far most theoretical results
concerning the error estimates of regularization methods in the
literature are of H\"older type; i.e., the
approximate solution $\nu$ and the exact solution $u$ satisfy
\begin{equation*}
\|u(x,\cdot)-\nu(x,\cdot)\|\leq 2E^{1-x}\delta^x
\end{equation*}
where $E$ is an a priori bound on $u(0,t)$. However, from the
inequality mentioned above we know that when $x\to 0^+$ the
accuracy of the regularized solution becomes progressively lower. At
$x=0$, it merely implies that the error is bounded by $2E$; i.e.,
the convergence of the regularized solution at $x=0$ is not proved.
In this paper, we apply the wavelet dual least squares method to
stabilize the problem \eqref{e1.2}. Taking suitable regularization
parameter, we not only obtain the H\"older continuity with
$p=0$ in \eqref{e1.3} for $0<x<1$, but also get a logarithmic
H\"older convergence error estimate with $p>0$ for $0\leq
x<1$, especially gain the logarithmic type convergence estimate on
the boundary $x=0$. In a sense, this is an improvement of known
result in \cite{e2}, and as our aim here is to obtain only stability
estimate.

As we consider  \eqref{e1.2} in $L^2(\mathbb{R})$ with respect to
variable $t$, we extend $u(x,\cdot)$, $g(\cdot):=u(1,\cdot)$,
$f(\cdot):=u(0,\cdot)$, and other functions of variable $t$ appearing
in the paper to be zero for $t<0$. By a solution of  \eqref{e1.2} we
understand a function $u(x, t)$ satisfying \eqref{e1.2} in the classical
sense; and for every fixed $x\in[0,1)$, the functions  $u(x, \cdot)$
belongs to $L^2(\mathbb{R})$. Throughout the paper, we assume that
for the exact $g$, the solution $u$ exists and satisfies an a-priori
bound
\begin{equation}  \label{e1.3}
\|f(\cdot)\|_p:=\|u(0,\cdot)\|_p\leq E, \quad p\geq0,
\end{equation}
where $\|f(\cdot)\|_p$ is defined by
\begin{equation*}
\|f(\cdot)\|_p:=\Big(\int_{-\infty}^{\infty}(1+\xi^2)^p|
\hat{f}(\xi)|^2d\xi\Big)^{1/2}.
\end{equation*}
Since $g$ is measured by the thermocouple, there will be measurement
errors, and we would actually have as data some function 
$g_\delta \in L^2(\mathbb{R})$, for which
\begin{equation} \label{e1.4}
\|g_\delta(\cdot)-g(\cdot)\|\leq \delta,
\end{equation}
where the constant $\delta>0$  represents a bound on the measurement
error, and  $\|\cdot\|$ denotes the $L^2(\mathbb{R})$ norm and
$$
\hat{h}(\xi)=\frac{1}{\sqrt{2\pi}}\int^\infty_{-\infty} e^{-i\xi
t}{h}(t)\,dt
$$
is the Fourier transform of function $h(t)$. For the uniqueness of
solution, we require that $\|u(x, \cdot)\|$ be bounded \cite{f1}, which
implied that $u(x, \cdot)|_{x\to \infty}$ is bounded. The
solution of problem \eqref{e1.2} is given by its Fourier transform 
\cite{f2,x1}:
\begin{equation} \label{e1.5}
\hat{u}(x, \xi) = e^{(1-x)\theta(\xi)}\,\hat{g}(\xi) ,
\end{equation}
where
\begin{gather} \label{e1.6}
\begin{aligned}
\theta(\xi)&=\sqrt{i\xi+1/4}-1/2\\
&=(1/2)\big[\sqrt[4]{1+16\xi^2}(\cos(\beta/2)+i\sin(\beta/2))-1\big],
\quad \xi\in\mathbb{R}, 
\end{aligned}\\
\beta=\arg(1+4i\xi),\quad
\tan\beta=4\xi,\quad-\pi/2<\beta<\pi/2\quad
\xi\in\mathbb{R},\label{e1.8}\\
\cos(\beta/2)=\frac{\sqrt{\sqrt{1+16\xi^2}+1}}{\sqrt{2}\sqrt[4]{1+16\xi^2}},
\quad \xi\in\mathbb{R}, \label{e1.9}\\
\sin(\beta/2)=\sigma\frac{\sqrt{\sqrt{1+16\xi^2}-1}}{\sqrt{2}\sqrt[4]{1+16\xi^2}},
\quad \xi\in\mathbb{R},\quad\sigma=\operatorname{sign}(\xi).
\label{e1.10}
\end{gather}
It is easy to verify from \eqref{e1.5} and \eqref{e1.8} that
\begin{equation} \label{e1.11}
\hat{f}(\xi) = e^{\theta(\xi)}\hat{g}(\xi),\quad \xi \in \mathbb{R}.
\end{equation}
The following lemma will be used in our proofs.

\begin{lemma}[\cite{f2}] \label{lem1.1} 
Let $\theta(\xi)$ be given by \eqref{e1.6}, then
there holds 
\begin{equation} \label{e1.12}
e^{-x\sqrt{|\xi|/2}}\le|
e^{-x\theta(\xi)}|\le\sqrt{e} e^{-x\sqrt{|\xi|/2}},\quad
0\le x\le1,\; \xi \in \mathbb{R}.
\end{equation}
\end{lemma}

To formulate problem \eqref{e1.2} for $x\in[0,1)$ in terms of an
operator equation in the space $X=L^2(\mathbb{R})$, we define an
operator $K_x:u(x,\cdot)\mapsto g(\cdot)$, i.e.,
\begin{equation} \label{e1.13}
\forall\,u(x,\cdot)\in X,\quad K_xu(x,t)=g(t),\quad 0\le x<1.
\end{equation}
From \eqref{e1.5}, we obtain
\begin{equation} \label{e1.14}
\widehat{K_xu(x,\xi)}=e^{-(1-x)\theta(\xi)}\hat{u}(x,\xi)=\hat{g}(\xi)
\quad 0\le x<1.
\end{equation}
Denote $\widehat{K_xu(x,\xi)}:=\widehat{K_x}\hat{u}(x,\xi)$, and we
can see that $\hat{K}_x:L^2(\mathbb{R})\mapsto L^2(\mathbb{R})$
is a multiplication operator,
\begin{equation} \label{e1.15}
\widehat{K_x}\hat{u}(x,\xi)=e^{-(1-x)\theta(\xi)}\hat{u}(x,\xi).
\end{equation}

\begin{lemma} \label{lem1.2} 
Let $K^{*}_x$ be the adjoint to $K_x$, then
$K^{*}_x$ corresponds to the following problem where the left-hand
side $u_t$ of problem \eqref{e1.2} is replaced by $-U_t$, says
\begin{equation} \label{e1.16}
\begin{gathered}
-U_t+U_x=U_{xx}, \quad x>,\; t>0,\\
U(x,0)=0,  \quad x\geq0,\\
U(1,t)=g(t), \quad t\geq0,\\
U(x,t)\text {remains bounded as $x\to \infty$},
\end{gathered}
\end{equation}
and
\begin{equation} \label{e1.17}
\widehat{K^{*}_x} =e^{-(1-x)\overline{\theta(\xi)}} .
\end{equation}
\end{lemma}

\begin{proof}
By  \eqref{e1.15} and the relations
$$
\langle K_xu,\upsilon\rangle
=\langle \widehat{K_x}\hat{u},\hat{\upsilon}\rangle
=\langle\hat{u},\widehat{K_x}^{*}\hat{\upsilon}\rangle
=\langle u,K^{*}_x\upsilon\rangle
=\langle \hat{u},\widehat{K^{*}_x}\hat{\upsilon}\rangle,
$$
 we have the adjoint operator $K^{*}_x$ of $K_x$ in frequency domain is
\begin{equation*}
\widehat{K^{*}_x}=\widehat{K_x}^{*}=e^{-(1-x)\overline{\theta(\xi)}}.
\end{equation*}
On the other hand, Problem \eqref{e1.16} can be formulated, in
frequency space, as follows:
\begin{equation} \label{e1.18}
\begin{gathered}
-i\xi\hat{U}+\hat{U}_x=U_{xx}, \quad x>,\;\xi\in\mathbb{R},\\
\hat{U}(x,0)=0,  \quad x\geq0,\\
\hat{U}(1,\xi)=g(\xi), \quad \xi\in\mathbb{R},\\
\hat{U}(x,\xi)\text{ remains bounded as $x\to \infty$}.
\end{gathered}
\end{equation}
Problem \eqref{e1.2} can be formulated, in  the frequency space as
\begin{equation} \label{e1.19}
\begin{gathered}
i\xi\hat{u}+\hat{u}_x=u_{xx}, \quad x>,\; \xi\in\mathbb{R},\\
\hat{u}(x,0)=0,  \quad x\geq0,\\
\hat{u}(1,\xi)=g(\xi), \quad \xi\in\mathbb{R},\\
\hat{u}(x,\xi)\text{ remains bounded as $x\to \infty$}
\end{gathered}
\end{equation}
Taking the conjugate operator for problem \eqref{e1.19}, we realize that
$\hat{U}(x,\xi)=\overline{\hat{u}(x,\xi)}$. Then, with \eqref{e1.5}, we
conclude that
\begin{equation*}
\hat{U}(x,\xi)=\overline{\hat{u}(x,\xi)}
=e^{(1-x)\overline{\theta(\xi)}}\hat{g}(\xi);
\end{equation*}
i.e.,
\begin{equation} \label{e1.20}
\hat{g}(\xi)=e^{-(1-x)\overline{\theta(\xi)}}\,\hat{U}(x,\xi)
=\widehat{K^{*}_x}\hat{U}(x,\xi):
=\widehat{K^{*}_xU}.
\end{equation}
This completes the proof.
\end{proof}

\section{Wavelet dual least squares method}

In this section we stabilize the non-standard
inverse heat conduction problem \eqref{e1.2} in the interval $0\leq x<1$ under
condition \eqref{e1.3} by a wavelet dual least squares method.

\subsection{Dual least squares method}

For an operator equation $Ku=g$, 
$K: X=L^2(\mathbb{R})\mapsto X=L^2(\mathbb{R}) $, a general projection 
method  is generated by two subspace families $\{V_j\}$ and $\{Y_j\}$ 
of $X$ and the approximate solution $u_j\in V_j$ is defined to be the 
solution of the  problem
\begin{equation} \label{e2.1}
 \langle Ku_j, y\rangle=\langle g,y\rangle,\quad \forall\,y\in
Y_j,
\end{equation}
where $\langle \cdot,\cdot\rangle$ denotes the inner product in $X$.
 If $V_j \subset R(K^*)$ and
subspaces $Y_j$ are chosen in such a way that
\begin{equation*}
K^*Y_j=V_j.
\end{equation*}
Then we obtain a special case of projection method known as the dual
least squares method. If $\{\psi_\lambda\}_{\lambda \in
\tilde{I}_j}$ is an orthogonal basis of $V_j$ and $y_\lambda$ is the
solution of the equation
\begin{equation} \label{e2.2}
 K^*y_\lambda=k_\lambda \psi_\lambda, \quad  \|y_\lambda\|=1,
\end{equation}
the approximate solution is explicitly given by the expression
\begin{equation} \label{e2.3}
 u_j=\sum_{\lambda \in \tilde{I}_j}\langle
g,y_\lambda\rangle\frac{1}{k_\lambda}\psi_\lambda.
\end{equation}

\subsection{Shannon wavelets}

The Shannon scaling function is $\phi=\frac{\sin(\pi t)}{\pi t}$ and
its Fourier transform is
\begin{equation} \label{e2.4}
 \hat{\phi}(\xi)=\begin{cases}
 1, &  |\xi|\le \pi,\\
0, & \text{otherwise}.
\end{cases}
\end{equation}
The corresponding wavelet function $\psi$ is given by its Fourier
transform
\begin{equation} \label{e2.5}
\hat{\psi}(\xi)=\begin{cases}
e^{-i\frac{\xi}{2}},&  \pi\le|\xi|\le 2\pi,\\
0,&  \text{otherwise}.
\end{cases}
\end{equation}
Let us list some notation:
$\phi_{j,k}(t):=2^{j/2}\phi(2^jt-k)$,
$\psi_{j,k}(t):=2^{j/2}\psi(2^jt-k)$,
 $j,k\in \mathbb{Z}$,
  $\Psi_{-1,k}:=\phi_{0,k}$ and
$\Psi_{l,k}:=\psi_{l,k}$ for $l\ge 0$, the index set
\begin{equation} \label{e2.6}
\begin{gathered}
 \tilde{I}=\{\{j,k\}:j,\,k\in \mathbb{Z}\}\subset\mathbb{Z}^2,\\
\tilde{I}_J=\{\{j,k\}:j=-1,0,\ldots, J-1;\,k\in
\mathbb{Z}\}\subset\mathbb{Z}^2.
\end{gathered}
\end{equation}
Due to the equality
$V_J=V_{J-1}\oplus W_{J-1}=V_{J-2}\oplus W_{J-2}\oplus
W_{J-1}=\ldots=V_0\oplus W_1\oplus \ldots\oplus W_{J-1}$, we
can define the subspaces 
\begin{equation} \label{e2.7}
V_J=\overline{\operatorname{span}\{\Psi_\lambda\}_{\lambda \in \tilde{I}_J}}.
\end{equation}
We define an orthogonal projection $P_J: L^2(\mathbb{R})\to V_J$:
\begin{equation} \label{e2.8}
P_J\varphi=\sum_{\lambda \in \tilde{I}_J}\langle
\varphi,\Psi_\lambda\rangle\Psi_\lambda,\quad \forall\, \varphi\in
L^2(\mathbb{R}),
\end{equation}
according to \eqref{e2.3} we easily conclude $u_J=P_Ju$. From the point of
view of an application to the problem \eqref{e1.2}, the important property
of Shannon wavelets is the compactness of their support in the
frequency space. Indeed, since
\begin{equation} \label{e2.9}
\hat{\psi}_{j,k}(\xi)=2^{-j/2}e^{-i2^{-j}k\xi}\hat{\psi}(2^{-j}\xi),\,\,
\hat{\phi}_{j,k}(\xi)=2^{-j/2}e^{-i2^{-j}k\xi}\hat{\phi}(2^{-j}\xi),
\end{equation}
it follows that for any $k\in\mathbb{Z}$.
\begin{equation} \label{e2.10}
\operatorname{supp}(\hat{\psi}_{j,k})=\{\xi:\pi 2^j\le|\xi|
 \le \pi 2^{j+1}\},\quad
\operatorname{supp}(\hat{\phi}_{j,k})=\{\xi:|\xi|\le \pi 2^j\}.
\end{equation}
From \eqref{e2.8}, $P_J$ can be seen as a low pass filter. The frequencies
with greater than $\pi 2^{J+1} $ are filtered away.

\begin{theorem} \label{thm2.1} 
 If $u(x,t)$ is the solution of \eqref{e1.2}
satisfying the condition $\|u(0,\cdot)\|_p \leq E$, then for any
fixed $x\in[0,1)$,
\begin{equation} \label{e2.11}
\|u(x,\cdot)-P_Ju(x,\cdot)\|\leq \sqrt{e}\,
(2^{J+1})^{-p}\,e^{-x\sqrt{\frac{1}{2} \pi2^J}} E.
\end{equation}
\end{theorem}

\begin{proof} 
From \eqref{e2.8}, we have
\begin{gather*}
u(x,\cdot)=\sum_{\lambda}\langle u(x,\cdot),
\Psi_\lambda\rangle \Psi_\lambda, \\
P_Ju(x,\cdot)=\sum_{\lambda\in \tilde{I}_J}\langle
u(x,\cdot), \Psi_\lambda\rangle \Psi_\lambda.
\end{gather*}
Due to Parseval relation and \eqref{e1.5} \eqref{e1.11} 
\eqref{e1.12} \eqref{e1.3}, we obtain
\begin{align*}
&\|u(x,\cdot)-P_Ju(x,\cdot)\|\\
&=\|\hat{u}(x,\cdot)-\widehat{P_Ju}(x,\cdot)\|\\
&=\|\sum_{\lambda \in \tilde{I}} \langle \hat{u},
\hat{\Psi}_\lambda \rangle \hat{\Psi}_\lambda- \sum_{\lambda
\in \tilde{I}_J} \langle \hat{u}, \hat{\Psi}_\lambda \rangle
\hat{\Psi}_\lambda \|\\
&=\|\sum_{\lambda \in \tilde{I}_{j\geq J+1}}\langle\hat{u},
\hat{\Psi}_\lambda \rangle \hat{\Psi}_\lambda\|\\
&= \|\sum_{\lambda \in \tilde{I}_{j\geq J+1}} \langle
e^{(1-x)\theta(\xi)}\,
\hat{g}(\cdot), \hat{\Psi}_\lambda \rangle \hat{\Psi}_\lambda\|\\
&=\|\sum_{\lambda \in \tilde{I}_{j\geq J+1}} \langle
e^{-x\theta(\xi)}\,
\hat{f}(\cdot), \hat{\Psi}_\lambda \rangle \hat{\Psi}_\lambda\|\\
&\leq \sup_{\pi 2^J\leq |\xi|\leq \pi 2^{J+1}}
\big[|\xi|^{-p}|e^{-x\theta(\xi)}|\big]
\big\|\sum_{\lambda \in \tilde{I}_{j\geq J+1}}
 \langle (1+(\cdot)^2)^{p/2} \hat{f}(\cdot),\hat{\Psi}_\lambda 
 \rangle \hat{\Psi}_\lambda\big\|\\
&\leq \sup_{\pi 2^J\leq |\xi| \leq  \pi 2^{J+1}}
\sqrt{e}|\xi|^{-p} e^{-x\sqrt{|\xi|/2}} E \\
&\leq \sqrt{e} (2^{J+1})^{-p} e^{-x\sqrt{ \pi2^J/2}} E.
\end{align*}
The proof is complete.
\end{proof}

\subsection{Subspaces $Y_j$}

In this section, we present some properties of the subspaces $Y_j$.
According to $K^*Y_j=V_j$, the subspaces $Y_j$ are spanned by
$\rho_\lambda$, $\lambda \in \tilde{I}_J$, where
\begin{equation} \label{e2.12}
 K^* \rho_\lambda=\Psi_\lambda,\quad  
k_\lambda=\|\rho_\lambda\|^{-1},\quad
y_\lambda= \frac{\rho_\lambda}{\|\rho_\lambda\|}=k_\lambda\rho_\lambda.
\end{equation}
The value $\rho_\lambda$ can be determined by solving the  parabolic
equation (see Lemma 1.2)
\begin{equation} \label{e2.13}
\begin{gathered}
-U_t+U_x=U_{xx}, \quad x>,\,t>0,\\
U(x,0)=0,  \quad  x\geq0,\\
U(1,t)=\Psi_{j,k}(t), \quad  t\geq0,\\
U(x,t)\text{ remains bounded as $x\to \infty$}.
\end{gathered}
\end{equation}
Because $\operatorname{supp}\hat{\psi}_{j,k}$ is compact, the solution exists
for any $t\in (0,\infty)$. Similarly the solution of the adjoint
equation is unique. So for a given $\Psi_\lambda$, $\rho_\lambda$
can be uniquely determined according to \eqref{e2.13}, and
\begin{equation} \label{e2.14}
\hat{\rho}_\lambda=e^{(1-x)\overline{\theta(\xi)}}
 \hat{\Psi}_\lambda(\xi)\;\Leftrightarrow\;
\hat{y}_{\lambda}=e^{(1-x)\overline{\theta(\xi)}}
k_\lambda\hat{\Psi}_{\lambda}(\xi),\;\lambda=\{j,k\}.
\end{equation}

The approximate solution for noisy data $g_\delta$ is explicitly
given by
\begin{equation} \label{e2.15}
 P_Ju^\delta (x,t)=u_J^\delta
=\sum_{\lambda\in \tilde{I}_J}
\langle u^\delta,\, \Psi_\lambda \rangle \Psi_\lambda
=\sum_{\lambda\in \tilde{I}_J} \langle
g_\delta,\, y_\lambda \rangle\frac{1}{k_\lambda} \Psi_\lambda.
\end{equation}
We call it the wavelet dual least squares approximation solution of
problem \eqref{e1.2} in the interval $0\leq x<1$.

\section{Error estimates}

In this section we estimating the error
$\|P_Ju^\delta-P_J u\|$.

\begin{theorem} \label{thm3.1} 
 If $g_\delta$ is noisy data satisfying 
$\|g(\cdot)-g_\delta(\cdot)\| \leq \delta$, then for any
fixed $x\in[0,1)$,
\begin{equation} \label{e3.1}
 \|P_Ju^\delta-P_J u\|\leq c_4 e^{(r-r_1)\sqrt{ \pi 2^J/2}}\delta.
\end{equation}
\end{theorem}

\begin{proof}
 From \eqref{e2.14}, we obtain
$\hat{y}_\lambda=e^{(1-x)\overline{\theta(\xi)}}\,k_\lambda
\hat{\Psi}_\lambda$. Note that $P_J u^\delta$ given by \eqref{e2.15}, 
$P_J u$ given by \eqref{e2.3} and \eqref{e1.12}, for $0\leq x<1$, we have
\begin{align*}
\|P_Ju^\delta (x,\cdot)-P_Ju(x,\cdot)\|
&=\|\sum_{\lambda \in \tilde{I}_J} \langle g_\delta-g,\,y_\lambda \rangle
\frac{1}{k_\lambda} \Psi_\lambda\|\\
&=\|\sum_{\lambda \in \tilde{I}_J} \langle
\hat{g}_\delta-\hat{g},\,\hat{y}_\lambda \rangle \frac{1}{k_\lambda}
\hat{\Psi}_\lambda\|\\
&=\|\sum_{\lambda \in \tilde{I}_J} \langle
\hat{g}_\delta-\hat{g},\,e^{(1-x)\overline{\theta(\xi)}}\,k_\lambda
\hat{\Psi}_\lambda\rangle
\frac{1}{k_\lambda}\hat{\Psi}_\lambda\|\\
&\leq \sup_{\pi 2^{J-1}\leq |\xi|\leq  \pi
2^J}|e^{(1-x)\overline{\theta(\xi)}}|\cdot
\big\|\sum_{\lambda \in \tilde{I}_J} \langle
\hat{g}_\delta-\hat{g},\,\hat{\Psi}_\lambda \rangle
\hat{\Psi}_\lambda\big\| \\
&\leq \sup_{\pi 2^{J-1}\leq |\xi|\leq  \pi 2^J}\big|
e^{(1-x)\theta(\xi)}\big|\cdot
\|\hat{P}_J(\hat{g}_{\delta}-\hat{g})\| \\
&\leq \sup_{\pi 2^{J-1}\leq |\xi|\leq  \pi
2^J}\big|e^{(1-x)\theta(\xi)}\big|\cdot \delta\\
&\leq \sup_{\pi 2^{J-1}\leq |\xi|\leq  \pi 2^J}e^{(1-x)\sqrt{|\xi|/2}}\delta\\
&\leq e^{(1-x)\sqrt{\pi 2^J/2}}\delta.
\end{align*}
This completes the proof. 
\end{proof}

We now give the following result which is the main conclusion of this article.

\begin{theorem} \label{thm3.2} 
 Let $u$ be the exact solution of
\eqref{e1.2} and let $P_Ju^\delta$ be given by \eqref{e2.15}. 
Let the measured data $g_\delta(t)$,  satisfy the condition 
\eqref{e1.4} at $x=1$,  and the a priori condition \eqref{e1.3} hold. 
If we select the regularization parameter
\begin{equation} \label{e3.2}
J=\log_2\Big[ \frac{2}{\pi}\Big(
\ln\big(\frac{E}{\delta}(\ln\frac{E}{\delta})^{-2p}\big)\Big)^2\Big],
\end{equation}
then for any fixed $x\in[0,1)$,
\begin{equation} \label{e3.3}
\|u(x,\cdot)-P_Ju^{\delta}(x,\cdot)\|  \leq
E^{1-x}\delta^x\big(\ln\frac{E}
{\delta}\big)^{-2p(1-x)}\big(\sqrt{e}+1+o(1)\big)
\quad\text{as }\delta\to0.
\end{equation}
\end{theorem}

 \begin{proof}
 Combining  Theorem 3.1 with Theorem 2.1, and noting the
choice  \eqref{e3.2} of $J$, we have
\begin{align*}
|u(x,\cdot)-P_Ju^{\delta}(x,\cdot)\|\\
&\leq \sqrt{e} (2^{J+1})^{-p}\,e^{-x\sqrt{\frac{1}{2} \pi2^J}}
E+e^{(1-x)\sqrt{\pi 2^J/2}}\delta\\
&\leq E\sqrt{e} \Big(
\ln\big(\frac{E}{\delta}\big(\ln\frac{E}{\delta}\big)^{-2p}
\big)\Big)^{-2p}
\Big(\frac{E}{\delta}\big(\ln\frac{E}{\delta}\big)^{-2p}
\Big)^{-x}\\
& +
\delta\Big(\frac{E}{\delta}\big(\ln\frac{E}{\delta}\big)^{-2p}
\Big)^{1-x}\\
&\le E^{1-x}\delta^x\big(\ln\frac{E}{\delta}\big)^{-2p(1-x)}
\Big\{
 \frac{\sqrt{e} ( \ln\frac{E}{\delta})^{2p}}
{\Big(\ln\big(\frac{E}{\delta}
\big(\ln\frac{E}{\delta}\big)^{-2p}\big)\Big)^{2p}}+1\Big\}.
\end{align*}
Note that
\begin{equation*}
\frac{\ln\frac{E}{\delta}}
{\ln\left(\frac{E}{\delta}\big(\ln\frac{E}{\delta}\big)^
{-2p}\right)}=\frac{\ln\frac{E}{\delta}} {\ln\frac{E}{\delta}-2p
\ln\big(\ln\frac{E}{\delta}\big)}\to 1\quad\text{as }\delta\to0;
\end{equation*}
therefore,  for $\delta\to0$,
\begin{equation*}
\|u(x,\cdot)-P_Ju^{\delta}(x,\cdot)\| \le
E^{1-x}\delta^x\big(\ln\frac{E}
{\delta}\big)^{-2p(1-x)}\,(\sqrt{e}+1+o(1)).
\end{equation*}
The proof is complete.
\end{proof}

\begin{remark} \label{rmk3.3} \rm
  (i) When $p=0$ and $0<x<1$, estimate \eqref{e3.3} is a
H\"{o}lder stability estimate given by
\begin{equation} \label{e3.4}
\|u(x,\cdot)-P_Ju^{\delta}(x,\cdot)\|\le
(\sqrt{e}+1)E^{1-x}\delta^x.
\end{equation}

(ii) When $p>0$ and $0\leq x<1$, estimate \eqref{e3.3} is a logarithmical
H\"{o}lder stability estimate.

(iii) When $p>0$ and $x=0$, estimate \eqref{e3.3} becomes
\begin{equation} \label{e3.5}
\|u(0,\cdot)- P_J u^{\delta} (0,\cdot)\|
u^{\delta}(x,\cdot)\|
\le E \big(\ln\frac{E}{\delta}\big)^{-2p}(\sqrt{e}+1+o(1))\to0
\quad\text{as }\delta\to0.
\end{equation}
We can see this  estimate  is a logarithmical stability
estimate similar to the convergence estimate in \cite{n1}.
\end{remark}

\begin{remark} \label{rmk3.4} \rm
 In general, the a-priori bound $E$ is unknown
in practice. In this case, with
\begin{equation} \label{e3.6}
J=\log_2\Big[ \frac{2}{\pi}
\Big(\ln\big(\frac{1}{\delta}(\ln\frac{1}{\delta})^{-2p}
\big)\Big)^2\Big],
\end{equation}
we have
\[
\|u(x,\cdot)-P_Ju^{\delta}(x,\cdot)\|\leq \delta^x
\big(\ln\frac{1} {\delta}\big)^{-2p\,(1-x)}(\sqrt{e}\,E+1+o(1))
\quad\\text{as }\delta\to0,
\]
where $E$ is only a bounded positive constant and it is not
necessary known, exactly.
\end{remark}

\subsection*{Acknowledgments}

The authors  like thank  the anonymous referees for
their valuable comments and suggestions. 
This work is supported by grant 11171136 from the
National Natural Science Foundation of China,
grants 132300410231 and 132300410013 from the
Natural Science Foundation of Henan Province of China,
grant 2009B110007 from the Fundamental Research Fund for
Natural Science of Education, Department of Henan Province of China,
and grants 11JCYJ16 and 10XZP006 from the Foundations of Henan University of
Technology.

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