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\title{
A Posteriori Estimation of the
Solution of a System of Ordinary
Differential  Equations  with
the Help of Taylor Series}

\author{                                
V. S. Zyuzin and O. V. Mushtakova}

\pagestyle{myheadings}

\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }

\maketitle

\auffil{The authors are with the Department of Mechanics and Mathematics,
 Saratov State University,
 Astrakhanskaya 83,
 Saratov, 410071, Russia,
 e-mail zyuzin@scnit.saratov.su.}

\section{Description of a Problem}
Suppose that we have a system of ordinary differential equations, and
we know the initial conditions. The majority of the existing numerical
methods for solving this system (e.g., Ringe-Kutta method) 
provides us only with an {\it approximate} solution, but these method 
give no guaranteed estimate of the corresponding error. 

In \cite{1,2}, a class of methods (called {\it a posteriori} methods) 
was proposed for estimating errors of approximate solutions 
of systems of ordinary
differential equations (ODE) with known initial conditions. 
These methods consist of the following three steps:
\begin{itemize}
\item[1] An approximate solution must be found by some standard method.
\begin{itemize}
\item[]The authors of \cite{1,2} recommend to use Runge-Kutta method, or
other techniques based on replacing a differential equation with an
equation in finite differences.
\end{itemize}
\item[2] Then, we compute the {\it residual} for this approximate
solution, i.e., the 
difference between the left- and right-hand sides of the equation. 
\item[3] Finally, we use this residual to 
find two-sided boundaries for the solution of the system of ODE.
\end{itemize}
These methods give a guaranted estimate for the solution's error; in
other words, they give us an interval that is guaranteed to contain
the (unknown) exact solution. 

The main problem with this method is that it often ``overshoots'':
the resulting intervals are too wide. 

\section{Main Idea}
Our main idea is to use a {\it different} method for computing the
approximate solution: 
\begin{itemize}
\item instead of the Runge-Kutta method (or other
finite difference methods), 
\item we propose to use methods 
based on {\it Taylor series}. 
\end{itemize}

The idea of the Taylor-series based methods is well known:
\begin{itemize}
\item we express the (unknown) solution as a Taylor series with
unknown coefficients;
\item we substitute these expressions into the differential equations; as
a result, we get a system of polynomial equalities;
\item polynomials are equal iff all their cofficients are equal; so,
we arrive at an (algebraic) system of equations, in which the unknowns
are the coefficients of the desired Taylor expansions;
\item finally, we solve this system, and find the values of the
coefficients; substituting the resulting values of the coefficients
into the expressions for the 
Taylor series, we get the desired approximate solution.
\end{itemize}
This modification turns
out to be very efficient, especially in cases when in
addition to estimating the error of the approximate solution, 
we are also interested in the value of the residual.
\smallskip

\noindent{\it Comment.} Although Taylor series have been used for
solving differential equations, they have never been used as a part of
an a posteriori method.
\smallskip

\section{Test Problems}

We have used several test problems (for which exact solutions are known)
to compare our method with the method from \cite{1,2}. First, we
applied both methods to the following two {\it linear} systems:
\begin{itemize}
\item[1)]$y'=y+2,$ $y_1'=y_1+1,$   $y_2'=y_2,$
\item[]  $y(0)=1,$   $y_1(0)=0,$    $y_2(0)=1.$
\item[]For this system, the exact solution is known:
\item[]$y=3\cdot e^t-2,$   $y_1=e^t-1,$  $y_2=e^t.$
\item[]
\item[2)]  $y_1'=y_2,$   $y_1(0)=0,$
    $y_2'=-y_1$,  $y_2(0)=1$,
\item[]    $u_1'=u_2+1,$    $u_1(0)=0,$
    $u_2'=u_1+1,$    $u_2(0)=0,$
\item[] $v_1'=v_2,$      $v_1(0)=0,$
    $v_2'=v_1,$      $v_2(0)=0.$

\item[]For this system, the exact solution is:

\item[]$y_1(t)=\sin(t),$ $y_2(t)=\cos(t)$,  
\item[]$u_1(t)=e^t-1,$ $u_2(t)=e^t-1,$
\item[] $v_1(t)=C_1\cdot e^{-t}+C_2\cdot e^t,$
$v_2(t)=-C_1\cdot e^{-t}+C_2\cdot e^t$, 
\item[]where
\item[]$C_1=e^h/2\cdot (v_1(h)-v_2(h))$ and 
\item[]$C_2=e^{-h}/2\cdot (v_1(h)+v_2(h))$.
\end{itemize}    
Both algorithms were implemented in PASCAL-XSC
\cite{3}. For both examples, the error estimates that we obtained using
Taylor series method were strictly contained in the 
two-sided bounds obtained by using a method from \cite{1,2} (where the
approximate solution is obtained by using Runge-Kutta techniques). 

As a third test problem, we considered the following {\it non-linear}
system:
\begin{itemize}
\item[3)]
$y'=y^2$,     $y_1'=y_1^2+1$,  $y_2'=y_2^2$,
    \item[]$y(0)=-1$,   $y_1(0)=0,$      $y_2(0)=0.$
\item[]For this system, the exact solution is:
\item[]$y(t)=-1/(t+1),$    $y_1(t)=tg(t),$    $y_2(t)=0$.
\end{itemize}

\noindent{\it Comment.}
In the non-linear case, in contrast to the linear one, 
it is necessary to take into account
additional functions and a priori given constants from 
\cite{1,2}.

\section{Applications}
We have applied this method to 
airplane {\it inertial navigation} \cite{4}. As a result of this
applications, we obtained two-sided polynomial approximations that
contain the (unknown) exact solution of the corresponding system of
differential equations. The resulting error is $\le 10^{-8}$. 

We are planning to use this method for several other applications.

\begin{thebibliography}{99}

\bibitem{1}
B. S. Dobronets, ``Two-sided solution of ODEs via a posteriori error
estimates'', {\it Journal of Computation and Applied Mathematics},
1988, Vol. 23, pp. 53--61.

\bibitem{2}B. S. Dobronets and V. V. Shaydurov, {\bf 
Two-sided numerical methods,}
Nauka, Novosibirsk, 1990 (in Russian).

\bibitem{3} R. Klatte, U. Kulisch, M. Neaga,  D. Ratz, and  Ch. Ullrich,
{\bf PASCAL-XSC,  Sprachbeschrebung mit Biespielen},  Springer  Verlag,
    1991.

\bibitem{4} Ch. Broxmeyer, {\bf Inertial Navigation Systems},
McGraw-Hill, N.Y., Toronto, London, 1965.

\end{thebibliography}
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