Code for singular boundary/eigenvalue problems in ODEs

We deal with the numerical solution of boundary value problems with either a singularity of the first kind or an essential singularity. The search for an efficient method to solve Problems with a singularity of the first kind is strongly motivated by numerous applications from physics, mechanics, or ecology. Problems with an essential singularity are of particular importance when problems posed an an infinite interval are transformed to a finite interval. Such Problems arise for example in fluid dynamics or foundation engineering. For Problems with a singularity of the first kind we have designed a Matlab standard Code, where the approximate solution is computed using collocation methods. These numerical schemes have been shown to be robust with respect to the singularity and yield high order solutions. Moreover, we incorporated a new error estimation procedure which was only recently shown to be asympotically correct for an important class of singular problems. This estimate of the global error is used for a mesh selection strategy intended for the equidistribution of the global error. Regular problems (without singularity) can also be efficiently solved by our code. The project goals related to problems with a singularity of the first kind are the following: We intend to improve our solution routine for the n0nlinear algebraic equations used to compute the collocation solution. Moreover, we aim at a generalization of the proof of the asymptotical correctness of our error estimate for the most general Problem class. A modification of the present grid selection strategy (intended for solution structures with steep slopes) and the theoretical justification of this procedure is also a project aim. For Problems with an essential singularity, we have experimental evidence that collocation methods Show favorable convergence properties. We cannot use our Mailab solver for this Problem class, however, since our error estimation routine fails in this case. Our aim is to prove theoretically that collocation indeed retains its classical convergence properties. We also intend to propose an estimation procedure for the global error and provide theoretical justification for this routine. Together with a grid selection algorithm, this should result in a new Mailab code suitable for the efficient solution of Problems with an essential singularity. Moreover, we intend to incorporate eigenvalue problems into our solution routine.