where α is greater than 1. Here, f, g
are smooth functions of dimension
n and p, respectively. In general p
is smaller than n, and condition
(3) provides the additional n-p relations that
guarantee the well-posedness of the problem. Boundary value
problems with an essential singularity are frequently encountered
in applications. In particular, problems posed on infinite
intervals belong to the above problem class when they are suitably
transformed to a finite interval. Flow problems (Blasius equation,
von Karman swirling flow) and the classical electromagnetic
self-interaction problem, are sources for the models we are
interested in.
The purpose of the paper is to investigate a numerical approach
which may be successfully applied to obtain high-order solutions
for boundary value problems of the type
(1)-(3). The results are original and have
not been published elsewhere. Especially, we examine the empirical
convergence order of collocation methods at either equidistant or
Gaussian points. The motivation to apply these methods was their
satisfactory performance when solving boundary value problems with
a singularity of the first kind, where α = 1 in
(1). Therefore collocation has been implemented in our
MATLAB code sbvp designed for the latter class of
problems, together with an a posteriori global error estimate
based on the defect correction principle. This error estimate is
asymptotically correct when a singularity of the first kind is
present, but unfortunately, it does not work for problems with an
essential singularity. In this case however, a strategy based on
mesh halving seems to be a promising candidate which can provide
an asymptotically correct error estimate for the collocation
solution of (1)-(3).