Reliable Computing 7: 29–39, 2001.
2001 Kluwer Academic Publishers. Printed in the Netherlands.
On Kernel Inclusions
at Kaiserslautern, 67653 Kaiserslautern, Germany,
(Received: 4 September 1998; accepted: 30 January 2000)
Abstract. Kernel inclusions are used to derive a new and efﬁcient solution method for Fredholm
integral equations. Concepts from enclosure theory and interval analysis are combined and lead to
effective error bounds for a broad class of kernels.
A major drawback to the development of numerical methods with automatic error
estimation is a lack of appropriate theoretical tools for assessing the accuracy of a
computation. In this paper we describe a new concept for an effective a posteriori
error analysis for Fredholm integral equations of the second kind
W.l.o.g. we can restrict to the interval [a
There are only few papers already dealing with this aspect (cf. , , , ),
but in principle the error bounds proposed there, are based upon the approximation
of the given kernel by a degenerate kernel k
so that for the remainder
k − k
< 1 (1.2)
holds. We provide a different approach getting rid off this restrictive require-
In operator notation (1.1) is written as
x = g + K(x)
For simplicity we take the continuous functions with maximum norm as underlying
space and suppose that both kernel k(s
t) and driving term g(s), are continuous, but
all considerations presented here carry over to a L
The plan is as follows. First we review basic ideas from enclosure theory and
formulate then the main estimation result under the restriction that the solution of
(1.3) has a nonzero solution in [0
1]. Next we demonstrate that these requirements
can be met for general equations. We close with some examples.