Advancing Numerical Methods of Block Multi-Derivative Approaches for ODEs of Various Orders

This work presents numerical methods of block multi-derivative approaches for ordinary differential equations (ODEs) of Various Orders. The derivation of the methods is achieved by applying the techniques of interpolation and collocation to a power series polynomial, which is considered an approximate solution to the problems. Higher derivative terms are introduced to improve the accuracy of the method, giving room to modify the method for solving second and third-order initial value problems (IVPs) of ordinary differential equations (ODEs). Details conformation of the block method is presented, showing that the method is zero stable, consistent and convergent. The method is applied block-by-block to first, second and third-order initial value problems (IVPs) of ordinary differential equations. The application of the method to a real-life example also yields accurate results.