Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge-Kutta formulae and upgrading of their order

Abstract

AbstractThe eight main contributions of the author to the field of approximate solutions of ordinary differential equations described herein are all application-oriented, with the purposes of simplification and the increase in efficiency and effectiveness of the Runge-Kutta processes generated. They range from the determination of an initial trial step-size to be adopted to expedite the approximation process through embedded Runge-Kutta algorithms to a more recent procedure for upgrading the order of Runge-Kutta processes. These contributions encompass all classes of differential equations of all orders, such as explicit, implicit, single or systems, and their treatment by Runge-Kutta processes of scalar or vector type (with the related equivalence conditions), of discrete or continuous kind, including the computer derivations of nonlinear algebraic equations associated with the Runge-Kutta processes. Specifically, the author developed the first fifth order Runge-Kutta formulae with fourth order embedded and the first C1 approximate solution through interpolation and Runge-Kutta formulae, which he improved by developing C1 embeddings with Runge-Kutta formulae without the use of interpolative techniques