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September/October 2013 Existence of solutions and asymptotic analysis of a class of singularly perturbed ODEs with boundary conditions
John Bryce McLeod, Susmita Sadhu
Adv. Differential Equations 18(9/10): 825-848 (September/October 2013). DOI: 10.57262/ade/1372777761

Abstract

We study the existence of solutions of a class of singularly perturbed BVPs $\varepsilon y'' +2y' +f(y) =0, ~y(0)=0, ~y(A)=0$ for some $A>0$ and $f>0$. Given an $f$ satisfying certain conditions, we will show that for each $\varepsilon>0$, there exists $A(\varepsilon)$ such that, if $0 < A < A(\varepsilon)$, then the problem has exactly two solutions. As an example we consider $f(y)=e^y$, where we show that $A(\varepsilon)>1$ for $\varepsilon>0$ sufficiently small. Both solutions have the same outer solution on $(0,1]$, but have different boundary behavior near $x=0$. One of them is bounded, the other unbounded as $\varepsilon \to 0$. A uniform expansion for the smaller solution has been obtained from matched asymptotic expansions. We prove rigorously that these asymptotics are correct up to $O(\varepsilon).$ We also prove that for a class of functions, $\underset{\varepsilon \to 0}{\lim} A(\varepsilon) = 2\int_0^{\infty} dy/f(y)$.

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John Bryce McLeod. Susmita Sadhu. "Existence of solutions and asymptotic analysis of a class of singularly perturbed ODEs with boundary conditions." Adv. Differential Equations 18 (9/10) 825 - 848, September/October 2013. https://doi.org/10.57262/ade/1372777761

Information

Published: September/October 2013
First available in Project Euclid: 2 July 2013

zbMATH: 1274.34166
MathSciNet: MR3100053
Digital Object Identifier: 10.57262/ade/1372777761

Subjects:
Primary: 34B08 , 34B15 , 34E05 , 34E15

Rights: Copyright © 2013 Khayyam Publishing, Inc.

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Vol.18 • No. 9/10 • September/October 2013
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