Fourth-Derivative Hybrid Block Methods for Initial Value Problems in ODEs: A Comprehensive Study

Abstract

In this paper, we introduce a new and improved way to solve a common type of mathematical problem known as an initial value problem. We developed two new methods, called Fourth-Derivative Hybrid Block Methods (FDHBMs), which are both highly accurate and reliable. After testing our methods, we found they are very stable, meaning they can be trusted for complex, long-term calculations. When we compared them to existing techniques, our new methods performed significantly better. One of them, which we call FDHBM1, was particularly impressive; it’s so precise that its accuracy is near-perfect, and it’s much better at minimizing errors than existing methods. Ultimately, our work shows that this new approach is a powerful and dependable tool for solving these kinds of differential equations. The convergence and stability properties were also examined. It was discovered that the new method compared favorably with other existing methods in the selected literature.