The negative gradient method extended to the computer programming of simultaneous systems of differential and finite equations

Abstract

In programming a system of simultaneous nonlinear equations on an analog computer it is most convenient to use the equations in their implicit form. In addition since the equations are nonlinear certain partial derivatives may change sign resulting in computer instability unless the equations are programmed by the negative gradient method. This paper considers programming nonlinear, explicitly time varying trajectory problems by coupling together a set of nonlinear positional equations and a set of nonlinear first order differential equations, each set being separately programmed by the negative gradient method. The equation of the first approximation to the perturbed motion is used to examine the convergence of the computer program to the solution of the given mathematical system.