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# Accession Number:

## AD0680015

# Title:

## A MULTIPLE TIME SCALES APPROACH TO THE ANALYSIS OF LINEAR SYSTEMS

# Descriptive Note:

## Final technical rept.

# Corporate Author:

## PRINCETON UNIV NJ

# Report Date:

## 1968-10-01

# Pagination or Media Count:

##
204.0

# Abstract:

## An investigation is made of uniform approximations to the solutions of linear differential equations with variable coefficients. The ordinary differential equations are replaced by an appropriate set of partial differential equations that determine the unknown function in terms of a set of independent time scales. The time scales are determined so as to obtain uniformly valid approximations. The partial differential equations, in conjunction with the requirement of uniformity of the approximation in a given interval, determine the time scales through a set of clock functions k sub i, which may depend on the interval of interest. It is essential for the success of the approximation that the clock functions be nonlinear functions of time, in addition to being complex quantities. The constant coefficient case arises as a natural limit. Thus the present approach generalizes earlier time scale analyses. With this generalization we recover for second order systems the Liouville-Green or WKBJ approximation. The difference between the present approach and the PLK method is emphasized with examples.

# Distribution Statement:

## APPROVED FOR PUBLIC RELEASE

#