Analysis of singular boundary value problems for an Emden-Fowler equation

Abstract

In this work we are concerned about a second order nonlinear ordinary differential equation. Our main purpose is to describe one-parameter families of solutions of this equation which satisfy certain boundary conditions. These one-parameter families of solutions are obtained in the form of asymptotic or convergent series. The series expansions are then used to approximate the solutions of two boundary value problems. We are specially interested in the cases where these problems are degenerate with respect to the unknown function and/or to the independent variable. Lower and upper solutions for each of the considered boundary value problems are obtained and, in certain particular cases, a closed formula for the exact solution is derived. Numerical results are presented and discussed.2000 Mathematics Subject Classification. 65L05. Key words and phrases. Singular boundary value problems, one-parameter families of solutions, convergent Lyapunov series, shooting methods.