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This study introduces the reader to the theory of approximating thesolution(s) of a non-linear, second order, ordinary differential equation (ODE)with piecewise polynomial functions by using the collocation method. It thenfocuses on the application of the method to generate collocation approximationsfor the Van der Pol equation with initial conditions and with varying equationparameter, {\mu}. It is shown that the collocation method in its original formis impractical for generating these approximations due to the numerical costsof producing such an approximation, particularly as the stiffness of theequation increases with parameter, {\mu}. An adaptation to the method, termedsegmented collocation, is proposed, reliant upon on the specific structure andconvergence for initial value problems, and shown theoretically and numericallyto be capable of generating equivalent approximations for much superior costs.Further study is proposed; to compare the segmented method to other numericalmethods traditionally employed with equations of this type such as theRunge-Kutta family; and to develop and test the method in order to ascertainits complete usefulness and competitive standing in the context ofapproximating second order, ODEs with initial conditions.

On the numerical solution of the Van der Pol equation using collocation