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P. Lima, L. Morgado, M. Schöbinger, E. Weinmüller:
"A Novel Computational Approach to Singular Free Boundary Problems in Ordinary Differential Equations";
Applied Numerical Mathematics, 114 (2017), 97 - 107.

We study the numerical solution of a singular free boundary problem for a second order nonlinear ordinary differential equation, where the differential operator is the degenerate m-Laplacian. A typical difficulty arising in free boundary problems is that the analytical solution may become non-smooth at one bound-ary or at both boundaries of the interval of integration. A numerical method proposed in [18] consists of two steps. First, a smoothing variable transformation is applied to the analytical problem in order to im-prove the smoothness of its solution. Then, the problem is discretized by means of a finite difference scheme. In the present paper, we consider an alternative numerical approach. We first transform the original problem
into a special parameter dependent problem sometimes referred to as an `eigenvalue problem´. By applying a smoothing variable transformation to the resulting equation, we obtain a new problem whose solution is
smoother, and so the open domain Matlab collocation code bvpsuite
[16] can be successfully applied for its numerical approximation.

Degenerate Laplacian, singular free boundary problem, smoothing variable substitution, collocation methods

http://dx.doi.org/10.1016/j.apnum.2016.09.017