Osculatory extrapolation and a new method for the numerical integration of differential equations

Abstract

This chapter presents the problem of numerical integration of differential equations and the techniques available for the derivation of numerical algorithms. It presents a unified and direct development of many of the equations of interest in the solution of initial value ordinary differential equations (ODES). In addition to the first-order scalar equation, it is possible to consider a set of simultaneous first-order equations or an equivalent high-order single equation. To lead into the development of certain equations of major importance in numerically solving ODES, a number of interpolation formulas are defined in the chapter. These two formulas are termed “Newton's forward formula” and “Newton's backward formula” respectively. Each is obtained by fitting a polynomial to the sequence of points (yn) at equally spaced (xn).