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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 109, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2009/109\hfil A quasi-boundary value method]
{A quasi-boundary value method for regularizing
nonlinear ill-posed problems}
\author[ D. D. Trong, P. H. Quan, N. H. Tuan\hfil EJDE-2009/109\hfilneg]
{Dang Duc Trong, Pham Hoang Quan, Nguyen Huy Tuan} % in alphabetical order
\address{Dang Duc Trong \newline
Department of Mathematics, Ho Chi Minh City National University \\
227 Nguyen Van Cu, Q. 5, HoChiMinh City, Vietnam}
\email{ddtrong@mathdep.hcmuns.edu.vn}
\address{Pham Hoang Quan \newline
Department of Mathematics, Sai Gon University\\
273 An Duong Vuong , Ho Chi Minh city, Vietnam}
\email{tquan@pmail.vnn.vn}
\address{Nguyen Huy Tuan \newline
Department of Mathematics and Informatics,
Ton Duc Thang University \\
98, Ngo Tat To, Binh Thanh district, Ho Chi Minh city, Vietnam}
\email{tuanhuy\_bs@yahoo.com}
\thanks{Submitted June 19, 2009. Published September 10, 2009.}
\subjclass[2000]{35K05, 35K99, 47J06, 47H10}
\keywords{Backward heat problem; nonlinearly Ill-posed problem,
\hfill\break\indent
quasi-boundary value methods; quasi-reversibility methods,
contraction principle}
\begin{abstract}
In this article, a modified quasi-boudary regularization method
for solving nonlinear backward heat equation is given.
Sharp error estimates for the approximate solutions,
and numerical examples to illustrate the effectiveness
our method are provided. This work extends to the nonlinear
case earlier results by the authors \cite{t2,t3} and by
Clark and Oppenheimer \cite{c1}.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks
\section{Introduction}
For $T$ be a positive number, we consider the problem of finding
a function $u(x,t)$, the temperature, such that
\begin{gather}
u_t-u_{xx} = f(x,t,u(x,t)),\quad (x,t)\in (0,\pi)\times (0,T),
\label{eq1}\\
u(0,t)= u(\pi,t)=0,\quad t\in (0,T), \label{eq2}\\
u(x,T)= g(x),\quad x\in (0,\pi), \label{eq3}
\end{gather}
where $g(x), f(x,t,z)$ are given functions. This problem is called
backward heat problem, backward Cauchy problem, and
final value problem.
As is known, the nonlinear problem is severely ill-posed;
i.e., solutions do not always exist, and in the case of existence,
these do not depend continuously on the given data.
In fact, from small noise contaminated physical measurements,
the corresponding solutions have large errors.
It makes difficult to numerical calculations. Hence,
a regularization is in order. In the mathematical literature
various methods have been proposed for solving backward Cauchy problems.
We can notably mention the method of quasi-solution
(QS-method) by Tikhonov, the method of quasi-reversibility
(QR method) by Lattes and Lions, the quasi boundary value method
(Q.B.V method) and the C-regularized semigroups technique.
In the method of quasi-reversibility, the main idea consists
in replacing operator $A$ by $A_\epsilon=g_\epsilon(A)$, where $A[u]$
is the left-hand side of \eqref{eq1}.
In the original method, Lattes and Lions \cite{l1} proposed
$g_\epsilon(A)=A-\epsilon A^2$, to obtain well-posed problem
in the backward direction. Then, using the information from
the solution of the perturbed problem and solving the original
problem, we get another well-posed problem and this solution
sometimes can be taken to be the approximate solution
of the ill-posed problem.
Difficulties may arise when using the method quasi-reversibility
discussed above. The essential difficulty is that the order
of the operator is replaced by an operator of second order,
which produces serious difficulties on the numerical implementation,
in addition, the error $c(\epsilon)$ introduced by small change in
the final value $g$ is of the order $e^{T/\epsilon}$.
In 1983, Showalter \cite{s1} presented a method called
the quasi-boundary value (QBV) method to regularize that linear
homogeneous problem which gave a stability estimate better
than the one in the previous method. The main idea of the method is
of adding an appropriate ``corrector'' into the final data.
Using this method, Clark and Oppenheimer \cite{c1}, and Denche-Bessila,
\cite{d1}, regularized the backward problem by
replacing the final condition by
\[ %4
u(T)+\epsilon u(0)=g
\]
and
\[ %5
u(T)-\epsilon u'(0)=g
\]
respectively.
To the author's knowledge, so far there are many papers on the
linear homogeneous case of the backward problem, but we only
find a few papers on the nonhomogeneous case, and especially,
the nonlinear case of their is very scarce.
In \cite{t1}, we used the Quasi-reversibility method to regularize
a 1-D linear nonhomogeneous backward problem. Very recently,
in \cite{q1}, the methods of integral equations and of Fourier
transform have been used to solved a 1-D problem in an
unbounded region.
For recent articles considering the nonlinear backward-parabolic heat,
we refer the reader to \cite{t3,t4}.
In \cite{t2}, the authors used the QBV method to regularize the latter
problem. However, in \cite{t2}, the authors showed that the error
between the approximate problem and the exact solution is
\[
\|u(.,t)-u^\epsilon(.,t)\| \le \sqrt{M}\exp
\big( \frac{3k^2T(T-t)}{2}\big) \epsilon^{t/T}.
\]
In \cite{t4}, the error is also of similar form,
$$
\|u(t) - u^\epsilon (t)\| \leq M \beta(\epsilon) ^{t/T}.
$$
It is easy to see that two errors above
are not near to zero, if $\epsilon$ fixed and $t$ tend to zero.
Hence, the convergence of the approximate solution is very
slow when $t$ is in a neighborhood of zero. Moreover,
the regularization error in $t=0$ is not given.
In the present paper, we shall regularize \eqref{eq1}-\eqref{eq3}
using a modified quasi-boundary method given in \cite{t3}.
This regularization method is rather simple and convenient
for dealing with some ill-posed problems. The nonlinear backward
problem is approximated by the following one dimensional problem
\begin{gather}
u_t^{\epsilon}-u_{xx}^{\epsilon}=\sum_{k=1}^\infty
\frac{e^{-T k^2}}{\epsilon k^2+e^{-Tk^2}} f_k(u^{\epsilon})(t)
\sin (kx), \quad (x,t)\in(0,\pi)\times (0,T), \label{eq6}\\
u^{\epsilon}(0,t)=u^{\epsilon}(\pi,t)=0, \quad t \in[0,T], \label{eq7}\\
u^{\epsilon}(x,T)=\sum_{k=1}^\infty
\frac{e^{-T k^2}}{\epsilon k^2+e^{-Tk^2}} g_k \sin (kx),\quad
x\in[0,\pi], \label{eq8}
\end{gather}
where $\epsilon \in (0,eT)$,
\begin{gather*}
g_k=\frac{2}{\pi}\langle g(x),\sin kx\rangle = \frac{2}{\pi}\int_0^{\pi}g(x)
\sin (kx)dx, \\
f_k(u)(t)=\frac{2}{\pi}\langle f(x,t,u(x,t)),\sin kx \rangle
=\frac{2}{\pi}\int_0^{\pi}f(x,t,u(x,t))\sin kx dx
\end{gather*}
and $\langle \cdot,\cdot\rangle$ is the inner product in $L^2(0,\pi)$.
The paper is organized as follows.
In Theorem \ref{thm2.1} and \ref{thm2.2}, we shall show that
\eqref{eq6}-\eqref{eq8} is
well-posed and that the unique solution $u^{\epsilon}(x,t)$ of it
satisfies the equality
\begin{equation} \label{eq9}
u^{\epsilon}(x,t)=\sum_{k=1}^\infty
\big({\epsilon k^2+e^{-T k^2}}\big)^{-1}
\Big({e^{-t k^2}}g_k -\int_t^T {e^{(s-t-T) k^2}}f_k(u^{\epsilon})
(s)ds\Big)\sin k x.
\end{equation}
Then, in theorem \ref{thm2.3} and \ref{thm2.4}, we estimate the
error between an exact solution $u$ of \eqref{eq1}-\eqref{eq3} and
the approximation solution $u^\epsilon$ of \eqref{eq6}-\eqref{eq8}.
In fact, we shall prove that
\begin{equation} \label{eq10}
\| u^{\epsilon}(.,t)-u(.,t)\|\le
H \epsilon^{t/T-1}\big(\ln(T/\epsilon)\big)^{\frac{t}{T}-1}
\end{equation}
where $\|\cdot\|$ is the norm of $L^2(0,\pi)$ and $H$ is the
term depend on $u$.
Note that the above results are improvements of
some results in \cite{q1,t1,t2,t3,t4}. In fact,
in most of the previous results, the errors often have the form
$C\epsilon^{t/T}$. This is one of their disadvantages in which
$t$ is zero. It is easy to see that from \eqref{eq10},
the convergence of the approximate solution at $t=0$ is also proved.
The notation about the usefulness and advantages of this method
can be founded in Remark 1 and Remark 2.
Finally, a numerical experiment will be given in Section 4,
which proves the efficiency of our method.
\section{Main results}
For clarity of notation, we denote the solution of
\eqref{eq1}-\eqref{eq3} by $u(x,t)$, and the solution of the
problem \eqref{eq6}-\eqref{eq8} by $u^\epsilon(x,t)$.
Let $\epsilon $ be a positive number such that $0<\epsilon Lees, M. H. Protter;
\emph{Unique continuation for parabolic differential equations
and inequalities },Duke Math.J. \textbf{28}, (1961),369-382.
\bibitem{l3} N. T. Long, A. P. Ngoc. Ding;
\emph{Approximation of a parabolic non-linear evolution equation
backwards in time}, Inv. Problems, \textbf{10} (1994), 905-914.
\bibitem{m1} I. V. Mel'nikova, Q. Zheng and J. Zheng;
\emph{Regularization of weakly ill-posed Cauchy problem}, J.
Inv. Ill-posed Problems, Vol. \textbf{10} (2002), No. 5, 385-393.
\bibitem{m2} I. V. Mel'nikova, S. V. Bochkareva;
\emph{C-semigroups and regularization of an ill-posed Cauchy
problem}, Dok. Akad. Nauk., \textbf{329} (1993), 270-273.
\bibitem{m3} I. V. Mel'nikova, A. I. Filinkov; \emph{The Cauchy
problem. Three approaches}, Monograph and Surveys in Pure and
Applied Mathematics, \textbf{120}, London-New York: Chapman \&
Hall, 2001.
\bibitem{m4} K. Miller;
\emph{Stabilized quasi-reversibility and other
nearly-best-possible methods for non-well posed problems},
Symposium on Non-Well Posed Problems and Logarithmic Convexity,
Lecture Notes in Mathematics, \textbf{316} (1973),
Springer-Verlag, Berlin , 161-176.
\bibitem{p1} L. E. Payne;
\emph{Some general remarks on improperly posed problems for
partial differential equations}, Symposium on Non-Well Posed
Problems and Logarithmic Convexity, Lecture Notes in Mathematics,
\textbf{316} (1973), Springer-Verlag, Berlin, 1-30.
\bibitem{p2} L. E. Payne;
\emph{Imprperely Posed Problems in Partial Differential
Equations}, SIAM, Philadelphia, PA, 1975.
\bibitem{p3} A. Pazy;
\emph{Semigroups of linear operators and application to partial
differential equations}, Springer-Verlag, 1983.
\bibitem{p4} S. Piskarev;
\emph{Estimates for the rate of convergence in the solution of
ill-posed problems for evolution equations}, Izv. Akad. Nauk SSSR
Ser. Mat., \textbf{51} (1987), 676-687.
\bibitem{q1} P. H. Quan, D. D. Trong;
\emph{ A nonlinearly backward
heat problem: uniqueness, regularization and error estimate},
Applicable Analysis, Vol. 85, Nos. 6-7, June-July 2006, pp. 641-657.
\bibitem{r1} M. Renardy, W. J. Hursa and J. A. Nohel;
\emph{Mathematical Problems in Viscoelasticity}, Wiley, New
York, 1987.
\bibitem{s1} R. E. Showalter;
\emph{The final value problem for evolution equations}, J. Math.
Anal. Appl, \textbf{47} (1974), 563-572.
\bibitem{s2} R. E. Showalter;
\emph{Cauchy problem for hyper-parabolic partial differential
equations}, in Trends in the Theory and Practice of Non-Linear
Analysis, Elsevier 1983.
\bibitem{s3} R. E. Showalter;
\emph{Quasi-reversibility of first and second order parabolic
evolution equations}, Improperly posed boundary value problems
(Conf., Univ. New Mexico, Albuquerque, N. M., 1974), pp. 76-84.
Res. Notes in Math., $n^0$ 1, Pitman, London, 1975.
\bibitem {t1} D. D. Trong, N. H. Tuan;
\emph{Regularization and error estimates for nonhomogeneous
backward heat problems},
Electron. J. Diff. Equ., Vol. 2006 , No. 04, 2006, pp. 1-10.
\bibitem{t2} D. D. Trong, P. H. Quan, T. V. Khanh, N. H. Tuan;
\emph{A nonlinear case of the 1-D backward heat problem:
Regularization and error estimate}, Zeitschrift
Analysis und ihre Anwendungen, Volume 26, Issue 2, 2007, pp. 231-245.
\bibitem{t3} D. D. Trong, N. H. Tuan;
\emph{A nonhomogeneous backward heat problem: Regularization and
error estimates}, Electron. J. Diff. Equ., Vol. 2008 , No. 33,
pp. 1-14.
\bibitem{t4} D. D. Trong, N. H. Tuan;
\emph{Stabilized quasi-reversibility method for a class of
nonlinear ill-posed problems }, Electron. J. Diff. Equ., Vol. 2008 ,
No. 84, pp. 1-12.
\bibitem{t5} Dang Duc Trong, Nguyen Huy Tuan;
\emph{Regularization and error estimate for the
nonlinear backward heat problem using a
method of integral equation.},
Nonlinear Analysis, Volume 71, Issue 9, 1 November 2009,
Pages 4167-4176.
\end{thebibliography}
\end{document}