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A Novel computational approach to singular free boundary problems in ordinary differential equations

Lima, Pedro M.; Morgado, M. Luísa; Schöbinger, M. ; Weinmüller, E.B.

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 boundary 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 improve 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 [17] can be successfully applied for its numerical approximation.