Perturbed Collocation Hybrid Methods with Fixed Stepsize for Nonlinear Systems of Initial Value Problems

Abstract

The perturbed collocation methods are effective numerical methods for solving stiff system of ordinary differential equations. In this study, we develop new numerical methods based on perturbation of the collocation process and discuss some applications and illustrative examples for system of ordinary differential equations. This class incorporates some off-grid points into the numerical schemes, extending the implicit linear multistep methods developed hybrid type. Absolute and residual error analysis are used to qualitatively and objectively examine the stability, convergence, and accuracy of the suggested procedures. These methods provide high-accuracy approximations of system of equations across the integration interval, for example, when compared to the conventional methods. Effective utilization of the new integrators' applications yields physical interpretations of what the complex systems of ordinary differential equations represent in natural occurrences, that is, the genuine representation of the systems in real life. These are first demonstrated as phase plots with strange and novel characteristics. The surface phase plots that are obtained represent segments or subsets of a system's phase space and frequently illustrate facts observed in the actual world. The discrepancies between the exact solutions and the numerical solutions which are presented in tabular and graphical forms can be used to evaluate how accurate the numerical solutions are. The results in Tables and Figures obtained support our conclusion, highlighting the revolutionary potential of the perturbed collocation methods in furthering the numerical approximation of system of ordinary differential equations in the physical world.