We investigate the following class of nonlinear singular initial value problems:
z'(t)=M(t)/t z(t)+f(t,z(t)), t∈ (0,1], (1)
z ∈ C[0,1]. (3)
As a first step in the analysis we discuss analytical properties of (1)-(3) like existence and uniqueness of solutions bounded on [0,1]. We are particularly interested in describing the structure of the most general initial conditions which have to be imposed in order to satisfy (3). We also examine the smoothness properties of the solution z of (1)-(3). It turns out that smoothness of higher derivatives of z depends not only on smoothness of higher derivatives of M and f but also on the eigenvalue structure of the matrix M(0). Moreover, in order to describe the local behaviour of the solution near the singular point, we estimate z near t=0.

The second part of the analysis consists of the convergence theory for the implicit Euler method applied to approximate the solution of (1)-(3). The main motivation to consider a "low order method" as an alternative to a high order one is that for high order methods one often observes an order reduction, at least in the region close to the singularity. It may be a better idea to choose a low order method to obtain a basic solution and then enhance the order of such a solution using acceleration techniques such as Iterated Defect Correction (or extrapolation), see Part II of this presentation. We show that the implicit Euler method is stable and retains its classical convergence order O(h) even if the singularity is present. To prove this result we need to generalise the local stability and consistency concepts known from the classical analysis of regular problems. In particular, the convergence result is shown by explicitly inverting the associated discrete operators. Theoretical results are illustrated by numerical experiments.