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

Author: Owens, Robert Hunter

Year: 1952

Degree: Dissertation (Ph.D.)

Advisor: Erdélyi, Arthur

Committee Member: Unknown, Unknown

Option: Mathematics; Physics

DOI: 10.7907/81ED-D440

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 Dv(±√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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