Introduction
We consider in this work singularly perturbed second-order boundary value problems
of the form
(0.1) ey" = f(t,y,y',e) , 0 t 1 ,
(0.2)
a ^ y O n e ) + a2(e)y'(0,e) = A(e)
b
1
(e)y(l,e) + b
2
(e)y'(l,e) = B(e)
where e is a small positive parameter and f is, in general, a nonlinear continuously dif-
ferentiable function. Our aim is to provide sufficient conditions for the existence of a
solution of theboundary value problem (0.1), (0.2)and to then deduce the asymptoticbe-
havior of this solution as e 0 . Theprincipal assumptions are that an approximate
solution exists on [0,1]and that the function f satisfies a growth condition with respect
to the variable y'. Theprecise meanings of "approximate solution" and "growth condition"
will be provided later; fornow the following heuristic reasoning is sufficient. Ifwe
formally set e = 0 in theboundary value problem (0.1), (0.2), we obtain a first order
differential equation and twoboundary conditions. The solution, u, of this "reduced"
equation and a boundary condition at one end of the interval [0,1]canthen be used to
approximate the solution y of the full problem (0.1), (0.2). This approximation breaks
down near the endpoint atwhich u fails to satisfy the other boundary condition. Indeed,
such behavior is plausible if f is not toononlinear. To fix these ideas, consider the
simple model problem
(0.3)
(0.4)
ey" +
yf
= 0 , 0 t 1 ,
y(0,e) = 0 , y(l,e) = 1 .
The exact solution is y = y(t,e)
-l-1
l
1-e 1-e•te
-1
Thus y is uniformly close to
the constant value 1 except near t = 0,where y decreases rapidly to satisfy the boundary
condition y(0,e) = 0. The solution y is said to possess a boundary layer (a convenient
term borrowed from fluid mechanics) at t = 0. A plausible candidate for a reduced solu-
tion (i.e., approximate solution) is then the constant function u = 1,which isthe
Received by the Editors November 7, 1974.
This work was supported by theNational Science Foundation under a traineeship atthe
University of Southern California and under Grant No. NSF-GP-37069X at Courant Institute.
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