Fast computer algorithms for solving differential equations

摘要

The invention consists of means which computes the numerical solution of an interconnected system of first order differential equations in a single computational pass. The method in effect treats the digital computer as a close approximation to an analog computer whose solutions are instantaneous. This is in contradistinction to the prior art regarding digital computer solutions of first order differential equations that utilize repetitive computational passes over the same time interval to obtain numerical solutions, and which further generates estimates of future values extrapolated from past values of state variables. The invention inserts an auxiliary function existing between, and at, the times marking the computational intervals. The invention rests on three postulates regarding the auxiliary function: a first postulate expands on the Euler formulation of the solution of a differential equation by including the unknown sought after state variable in an expanded Euler formulation. a second postulate introduces an integratable auxiliary equation existing in the computational interval that is bounded by successive sample times. Parameters of the auxiliary functions are determined by using boundary values of the system state equations, a third postulate separates the system auxiliary equation into at least a first and a second part. A first part is the set of independent solutions of each first order differential equation within a system of first order differential equations; a second part incorporates the interconnections between the first order solutions of the state variables, and. a fourth postulate adds to the above by choosing the state equation for each independent first order differential equation, in a system of differential equations, as the integratable auxiliary equation; furthermore choosing the solution of the chosen auxiliary equations as the definite integral of each auxiliary equation; and furthermore obtaining the overall system state via simultaneous solution of the resulting system of algebraic