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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Second-order differential equations with fractional transition points


Author: F. W. J. Olver
Journal: Trans. Amer. Math. Soc. 226 (1977), 227-241
MSC: Primary 34E20
DOI: https://doi.org/10.1090/S0002-9947-1977-0430445-7
MathSciNet review: 0430445
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Abstract: An investigation is made of the differential equation

$\displaystyle {d^2}w/d{x^2} = \{ {u^2}{(x - {x_0})^\lambda }f(u,x) + g(u,x)/{(x - {x_0})^2}\} w,$

in which u is a large real (or complex) parameter, $ \lambda $ is a real constant such that $ \lambda > -2$, x is a real (or complex) variable, and $ f(u,x)$ and $ g(u,x)$ are continuous (or analytic) functions of x in a real interval (or complex domain) containing $ {x_0}$. The interval (or domain) need not be bounded. Previous results of Langer and Riekstins giving approximate solutions in terms of Bessel functions of order $ 1/(\lambda + 2)$ are extended and error bounds supplied.

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DOI: https://doi.org/10.1090/S0002-9947-1977-0430445-7
Keywords: Asymptotic analysis, Bessel functions, error analysis, linear differential equations, transition point, turning point
Article copyright: © Copyright 1977 American Mathematical Society

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