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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