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\title{On Application Of Interval Approach
       To Finding The Latent Singularity Points
       Of The Initial Value Problems}
\author{G. G. Men'shikov}

\pagestyle{myheadings}

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

\maketitle

\auffil{The author is with the 
Faculty of Applied Mathematics -- Control Processes,
Saint Petersburg State University, Russia.}


\section{The Problem of Finding The Singular Points}
Let us consider the differential equation of first order
\begin{eqnarray}
y'=f(x,y)
\end{eqnarray}
Suppose that the conditions of Picard-Lindeloef's theorem \cite{ref1} are
valid
in some neighbourhood of the initial point $(x_0,y_0)$.

Assume that the solution $y(x)$ 
that satisfies the initial condition $y(x_0)=y_0$ 
cannot be extended to all $x$. So, the maximal
possible extension of this solution is a solution $y(x)$ that is
defined on
some (unknown) interval $(a,x^*)$, where $x^*$ is a {\it singularity
point} for this solution.
The problem is: how to compute $x^*$? 

In this paper, we will consider 
the case when
\begin{eqnarray}
y(x)\rightarrow+\infty\ \ \ (x\rightarrow x^*\!-\!0)
\end{eqnarray}

\section{Orlov's Algorithm}
For the case (2), the problem of computing the singular point $x^*$ 
has been the subject of V. N. Orlov's research \cite{ref2}. He
constructed an auxiliary equation
\begin{eqnarray}
u'=g(x,u)
\end{eqnarray}
by means of a transformation
\begin{eqnarray}
u=1/y.
\end{eqnarray}
Since $u'=-y'/y^2=-f(x,y)/y^2$, we have $g(x,u)=-u^2f(x,1/u)$ (when
$u\ne 0$). 
The equation $(3)$ has the solution $u(x)=1/y(x)$ on
interval $(b,x^*)$, where $y(x)>0$, and $b$ is some point between $x_0$
and $x^*$.

Assume that the function $g(x,u)$ is extended in such a way that 
$g(x,0)$ exists, and the conditions
of Picard-Lindeloef's theorem are valid  in some rectangle containing the
point $(x^*,0)$. Then, $u(x)$ is well-defined for $x=x^*$, and $u(x^*)=0$.

Thus, to compute $x^*$, 
it is sufficient to find zeros of corresponding solution $u(x)$ of
the equation $(3)$. This problem becomes easier if $u(x)$ changes its sign
at the point $x^*$. The inequality $$u'(x^*)=g(x^*,0)\neq 0$$ is sufficient
to guarantee this change of sign.

V. N. Orlov applied this method to Riccati equation (that are used,
e.g., in {\it control theory}), and to first and second
equation of Painlev\'e. 

\section{Orlov's Method Does Not Give Guarenteed Bounds For The
Singularity Point}
Orlov's method is approximate, and does not give a
guaranteed estimates for $x^*$. 

In this paper, we describe 
a two-sided modification of this method, a 
modification that uses interval approach
and results in an interval that is {\it guaranteed} to contain $x^*$.

\section{Straightforward Interval Approach}
At first glance, it may seem that it is sufficient 
to solve the ``inverse" initial problem
$\ u(x_0)=1/y_0\ $, $\ u'=g(x,u)\ $ by one of interval methods. The
integrating
process passes through the values $x=x_0, x=x_1=x_0+\Delta x,
x=x_2=x_0+2\cdot\Delta x, \ldots$ and gives the
intervals $U(x_k),\ k=1,2,\ldots$, which include $u(x_k)$. Suppose
that
\begin{eqnarray*}
& U(x_k)>0 & (k=0,1,\ldots,m),  \\
& 0\in U(x_k) & (k=m+1,\ldots,p),  \\
& U(x_{p+1})<0 &
\end{eqnarray*}
Then the interval $(x_m,x_{p+1})$ contains the zero of $u(x)$, i.e. the
singular point $x^*$ of $y(x)$.

\section{Straightforward Interval Approach Does Not Always Work}
The difficulties appear when $y(x)$ has zeros on the interval 
$[x_0,b)$. The
described simple interval method is not suitable in this case.

\section{Application Of Co-Integration}
In this case, a more reasonable idea is to use the integration 
process proposed in \cite{ref3} and called ``co-integration". In this
process, we simultaneously solve both equations (1) and (3). 

For $k=0$, we start with $Y(x_0)=y_0$ and $U(x_0)=1/y(x_0)$. After we
have computed the intervals $Y(x_k)$ and $U(x_k)$ for some $k$, we do
the following:
\begin{itemize}
\item[1)] We run one step of 
two independent (and potentially parallel) interval processes
for solving equations $(1)$
and $(3)$:
\begin{itemize}
\item[$\bullet$]the first of these processes uses an interval
$Y(x_k)$ (that is guaranteed to contain the unknown actual value
$y(x_k)$) to compute the new interval $Y^0(x_{k+1})$ (that is guaranteed
to contain $y(x_{k+1})$);
\item[$\bullet$]the second process uses an interval
$U(x_k)$ (that is guaranteed to contain the unknown actual value
$u(x_k)$) to compute the new interval $U^0(x_{k+1})$ (that is guaranteed
to contain $u(x_{k+1})$);
\end{itemize}
\item[2)] 
Then, we apply 
the mutual correction by computing the following intersections:
\begin{eqnarray*}
Y(x_{k+1})=Y^0(x_{k+1})\cap (1/U^0(x_{k+1})),\\
U(x_{k+1})=U^0(x_{k+1})\cap (1/Y^0(x_{k+1})).
\end{eqnarray*}

\end{itemize}
In this process, an unbounded interval 
$U(x_{k+1})$ can
 appear because of the location
of zero near the point $x_{k+1}$, or because of 
the unperfection of underlying
processes (for instance, a too large value of $\Delta x$). Therefore,
in this method, we use
a special interval software to deal with unbounded intervals.

The method does not have any problem with the possible zeros of the
function $y(x)$.
Near these zeros, the interval 
$U$ has a tendency to be equal to $R=(-\infty,+\infty)$ and
thus, to be excluded. The mutual correction makes the computed
intervals narrower.

If simultaneously $Y(x_{k+1})=U(x_{k+1})=R$, then further integration has no
sense. Then, we must decrease $|\Delta x|$ and renew the process
either from
the initial point $x_0$, 
or from some intermediate point $x_k$. 

Decreasing $\Delta x$ This is also a reasonable idea 
for the case when
the resulting interval for $x^*$ is too wide.

The transformation (4) 
is good for certain equations (1), but not
for all of them. In general, 
each equation $(1)$ has its own appropriate transformation
$y\rightarrow u(y)$.

\begin{thebibliography}{99}
\bibitem{ref1} E. Kamke, {\bf Differentialgleichungen. \\Loesungsmethoden und
Loesungen. 1. Gewoenlichen Differentialgleichungen}, Teubner, Leipzig, 1959.
\bibitem{ref2} V. N. Orlov, {\bf The investigation Of Approximate Solution
With Mobile Poles Of Nonlinear Ordinary Differential Equations}, Ph.D.
Dissertation,
Minsk, 1989 (in Russian).
\bibitem{ref3} G. G. Men'shikov, ``Interval Co-Integration Of Differential
Equations Connected By A Substitution Of The Variable In Interval
Computation'', 1992, No. 4(6), 1992.
\end{thebibliography}

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