The Singly Implicit Runge-Kutta Methods for nonlinear singular initial value problems

Résumé

The use of collocation process for constructing numerical methods for solving ordinary differential equations has been attractive for nonlinear singular initial value problems. In this paper, a class of singly implicit Runge-Kutta methods has been developed which can efficiently solve non-linear singular initial value problems in ordinary differential equations. The methods are derived
based on collocation at the polynomial nodes which produce dense output within the integration interval. Preliminary numerical calculations using the new methods clearly show improved performance, efficiency and effectiveness compared to some methods with strong algebraic stability properties. In order to capture the system behavior for long period, graphic surface curves of phase
plots were generated, which show very strange and new behaviors. The obtained surface phase plots are portions of the phase space of the systems, often depicting real world observed facts.