Direct hybrid block methods for solving second and third-order ordinary differential equations in the presence of two higher derivatives

Some of the real-life problems that can be expressed as initial value or boundary value problems of higher-order ordinary differential equations (ODEs) may not have exact solutions. This situation necessitates the development of numerical methods for approximating the solution. One of the methods that is gaining popularity for solving higher-order ODEs is the hybrid block method (HBM). The method not only solves higher-order ODEs directly, but it also overcomes the zero-stability barrier that occurs in block methods. Furthermore, this method is capable of approximating the solutions at many points simultaneously. Literature reviews revealed that several of multi-step hybrid block methods (HBMs) have been derived based on generalised off -step point(s) without the presence of higher derivatives. In order to increase the accuracy, attempts have been made to include higher derivatives in the derivation of HBMs. However, they are limited to one-step HBMs and only consider one higher derivative in developing the methods. Hence, this study introduces new direct one and two-step HBMs based on generalised off-step point(s) in the presence of two higher derivatives for solving higher-order ODEs that have not been considered before.