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{\large{\bf{A NEW APPROACH TO THE NUMERICAL SOLUTION OF PARABOLIC PROBLEMS BACKWARD IN TIME}}}
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J.M.Marb\'an$^{(1)}$ and C.Palencia$^{(1)}$
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\noindent
\hskip -0.5cm
 {\small{(1)}} Dpto. Matem\'atica Aplicada y Computaci\'on.\\
     Facultad de Ciencias.\\
     Universidad de Valladolid.\\
     47005 Valladolid, Spain.\\

 Let $A:{\cal{D}}(A)\subset X\rightarrow X$ be a linear operator on a Banach space such that it is the infinitesimal generator of an analytic semigroup.  It is well-known that the Cauchy problem for the {\it{backward}} equation:
\begin{equation}
u'(t)=Au(t) \hskip 1cm (0\leq t\leq T)
\label{PC}
\end{equation}
 with the value of $u$ prescribed at $t=T$,
\begin{equation}
u(T)=u_T,
\label{ID}
\end{equation}
is, in general, not properly posed. However, this difficulty can be somehow overcome
if we restrict ourselves to seek only for solutions satisfying the a priory bound
\begin{equation}
\|u(t)\|\leq M  \hskip 1cm (0\leq t\leq T),
\label{RC}
\end{equation}

\noindent
 so that (\ref{PC}) together with (\ref{ID}) and (\ref{RC}) becomes well-posed (see [1]).\\

\hskip -0.3cm
 There is a large amount of work on the numerical solution of this kind of problems, but most of
it is done for $X$ being a Hilbert space (usually an $L_2$-like one). We present a method wich does
not require such a restriction on $X$. In fact, we are able to obtain rigurous estimates of the error
between the exact and the approximate solutions in the supremum norm. Numerical examples can be provided,
as well as some comparisons of efficiency with other methods developed so far such as [2].

 \begin{thebibliography}{2}

 \bibitem{1}{\sc H.O. Fattorini}
  {\it \lq\lq The Cauchy Problem,"}
  \par Encyclopedia of Mathematics and its Applications, v.18, 1983, Addison-Wesley Pub..

 \bibitem{2}{\sc Karen A.Ames and James F.Epperson}
  {\it \lq\lq A Kernel-Based Method for the Approximate Solution of Backward Parabolic Problems,"}
 \par SIAM J.Numer.Anal., vol.34, No.4, pp. 1357-1390, August 1997.

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