Comparative Analysis of Some New Runge-Kutta Type Techniques on the Solution of First Order Initial Value Problem in Ordinary Differential Equations

The derivation of numerical methods to deal with differential equations framed from real life problems has been on the rise of which great deal of attention have been drawn towards Runge-kutta methods. In recent times, researchers have explored the derivation of Runge-Kutta methods by introducing higher order derivative (up to the second order) in the terms of Runge-Kutta methods. We have also seen how other types of ‘mean’ are used as a substitute to the more usually applied arithmetic mean in the derivation Runge Kutta methods. However, in this paper some new Runge-Kutta type methods which border on the use of other types of ‘mean’ such as harmonic mean, geometric mean or heronian mean with higher derivatives up to the second derivative on a single explicit Runge-Kutta methods which were previously done on different explicit Runge-Kutta methods are constructed, analyzed, implemented and compared. The qualitative features of the methods including the local truncation error, consistency, convergence and stability of the new methods were comparatively analyzed, investigated and established. We demonstrated the validity of the comparisons with four numerical examples. The obtained results were compared with some numerical methods and the exact solutions of the proposed problems.