Detecting and locating a singular point in the numerical solution of IVPs for ODEs

Many authors have worked on approaches for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) whose solutions contain one or more singular points within the interval of integration. Their approaches, however, assumed that the user knows in advance that the problem is singular. Hence they introduced new formulas to cope with this difficulty. In this paper, a new approach to detect and locate a singularity is suggested. This approach, which does not require the changing of the underlying formula, is comprised of two stages. The first is a preliminary singularity detection stage. The second stage is the confirmation stage which gathers more information about the existence and location of the singular point. We justify the first state and introduce three different techniques for confirming the existence of a singularity. The numerical results show that our approach is effective.