The Uniformly Valid Asymptotic Approximations to the Solutions of Certain Non-Linear Ordinary Differential Equations

Citation

Kevorkian, Jirair Kevork (1961) The Uniformly Valid Asymptotic Approximations to the Solutions of Certain Non-Linear Ordinary Differential Equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/K8NE-5X16. https://resolver.caltech.edu/CaltechETD:etd-12222005-092728

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. This work deals with the application of an expansion procedure in terms of two independent time variables for the uniform asymptotic representation of solutions representing certain mechanical systems. The method is first applied to systems governed by the equation [...] where [...] is a small parameter, and f has the character of a damping (i. e. y is a bounded function of t for all t [...] 0). It is shown that the physical problems which can be brought to the above non-dimensional form possess two characteristic time scales, one associated with the oscillatory behavior of the solution, while the other measures the time interval in which the effects of the non-linear term become apparent. The dependence of the solution on these time scales is not simple, in the sense that an asymptotic representation of the exact solution which is valid for large times cannot be obtained by a limit process in which a non-dimensional time variable is held fixed. This fact has motivated the introduction of an expansion procedure in functions of two time variables, and it is shown that with the use of certain simple boundedness criteria a uniform asymptotic representation can be derived. In addition to the above mentioned class of problems a variety of examples possessing certain boundedness properties is studied by this method, including, for example, the Mathieu equation. The main emphasis of this paper is on the constructive rather than general approach to the solutions of specific examples. These examples are introduced in turn to illustrate the underlying ideas of the method, whose main advantage is its simplicity especially for computing the higher approximations.

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