Citation
Kevorkian, Jirair Kevork (1961) The Uniformly Valid Asymptotic Approximations to the Solutions of Certain NonLinear Ordinary Differential Equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/K8NE5X16. https://resolver.caltech.edu/CaltechETD:etd12222005092728
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 nondimensional 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 nonlinear 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 nondimensional 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.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  (Aeronautics and Mathematics) 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Aeronautics 
Minor Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Group:  GALCIT 
Thesis Committee: 

Defense Date:  1 January 1961 
Record Number:  CaltechETD:etd12222005092728 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd12222005092728 
DOI:  10.7907/K8NE5X16 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  5116 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  22 Dec 2005 
Last Modified:  20 Nov 2023 22:57 
Thesis Files

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