Attainable Order of Hybrid Methods from Polynomial Nodes for Non-Linear Singularly Perturbed Initial Value Problems

Abstract

We develop hybrid methods utilizing collocation at polynomial nodes for the numerical integration of singularly perturbed non-linear initial-value problems. The present methods are intended for solving nonlinear singularly perturbed initial value problems without linearization and provide third and fourth-order convergence results. We use piecewise-uniform meshes which resolve the difficulties arising from the steep gradient of the solution in the initial layer. Linear stability of these methods are studied. Numerical experiments are carried out to verify the efficiency and accuracy of the methods. The new hybrid integrators are effectively employed to achieve accurate representation of singularly perturbed systems, providing physical interpretation of what they represent in natural phenomena. These are illustrated through phase plots that exhibit unusual and novel behaviors. The resulting surface phase plot curves represent portions of the phase space of a perturbed system, frequently illustrating real world observed phenomena. These are depicted graphically as surface phase plot curves shown in Figures, while numerical values are presented side by side in Tables.