An Asymptotic Study of a Glass of Ordinary Linear Differential Equations of the Second Order

Citation

Owens, Robert Hunter (1952) An Asymptotic Study of a Glass of Ordinary Linear Differential Equations of the Second Order. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/81ED-D440. https://resolver.caltech.edu/CaltechTHESIS:02082018-101404674

Abstract

The asymptotic behavior, with respect to the large parameter µ, of two solutions of d²y/dx² + µ² [c² - x² + f(x, µ^-1)] y = 0 is given where f(x, µ^-1) = 0(µ^-1) is subjected to suitable hypotheses and all variables are real. These solutions are approximated to by the parabolic cylinder functions D_v(±√2µ t) with v = µ a²/2 - 1/2 and t = Ø(x,µ^-1). The function Ø(x,µ^-1) and the quantity a² are constructed as a part of the analysis in which the parameter µ is restricted in such a way that v is bounded away from the positive integers. The relative error of the approximating functions is uniform for all x.

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