Reliable Computing 7: 485–495, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
The Preliminary Enclosing of the ODE Solutions
on the Base of the Cauchy-Duhamel Identity
GREGORY G. MENSHIKOV
Faculty of Applied Mathematics—Control Processes, Saint Petersburg State University,
Bibliotechnaya Pl. 2, Stary Petershof, Saint Petersburg, 198904, Russia,
(Received: 12 September 2000; accepted: 29 March 2001)
Abstract. An iterative interval algorithm of a preliminary enclosing of the integral curve in the next
step of an integration process is described. It is based on the well-known Cauchy-Duhamel identity.
The conditions of the ending of the process are derived. The criterion of existence and uniqueness of
the solution is discussed, too.
So for the equation
= ay + g(x) (1.1)
with continuous g(x) and initial point x
, the Cauchy-Duhamel identity has the
Vice versa, the equation (1.2) implies (1.1).
It seems that M. K. Gavurin  and G. Dahlquist  were the ﬁrst who proposed
to take the identity (1.2) as the base of numerical integration.
Their idea was that non-linear equation may be written extracting the linear
= ay + g(x
Then identity (1.2) takes the form:
The advantage is due to the linear component which is eliminated from the
process of the discretization the right hand of this equation; quite a little remains
under the integral.