Numerical Solution of First and Higher Order IVPs Via a Single Continuous Block Method

This article focuses on the development and implementation of a single continuous collocation numerical scheme for solving first and higher-order ordinary differential equations (ODEs). By employing the interpolation and collocation technique on power series as basis function, we were able to come up with a continuous scheme from which block methods for effectively solving first and higher order ODEs were derived. This is better and faster than the traditional way of developing a continuous scheme for a specific order of ODE. The method's accuracy is determined to be of order seven, establishing its consistency. Results from the implementation of our method show its applicability on nonlinear equations and application problems from first, second, third, and fourth order ODEs that are of significant implications on various fields in physics, engineering, biology and mathematics. Furthermore, the numerical results generated by the method reveal its effectiveness and accuracy, and also its superiority over some methods that exist in literature.