Morgado and Lima deal with the numerical solution of a class of free boundary value problems for a special kind of multi-parametric, nonlinear, second-order ordinary differential equations (ODEs) on a half line. The goal is to find the right endpoint of an interval, so that there exists a positive solution of the equation, satisfying given boundary conditions. There is a singularity at the origin, and different types of singularities may also occur at both endpoints of the interval for several choices of the parameters.

This type of problem has applications in describing some phenomena of force-free magnetic fields in passive media and Tokamac plasma equilibria with magnetic islands. The behavior of the solution is studied in the neighborhood of the endpoints of the interval, by constructing asymptotic expansions of the solutions of appropriate singular Cauchy problems. The value of the right endpoint of the interval is estimated by using some general properties of the nonnegative solutions of quasilinear equations. Two numerical methods for solving the obtained boundary value problem are presented: the shooting method and the finite difference method. The shooting algorithm is applied so that the two asymptotic approximations of solutions of the Cauchy problems from the two endpoints are equal and smooth at the middle of the interval. The ordinary finite difference scheme of the second order of approximation is also used, by taking into account the asymptotic solutions near the endpoints of the interval. Numerical results illustrate the solutions obtained.