Simple Singular Perturbation Problems with Turning Points

The paper considers the asymptotic solution of two-point boundary value problems ey” + A(x)y’ = 0, 0 ≤ x ≤ 1, when 0 1, A(x) is smooth with isolated zeros, y(0) = 0 and y(1) = 1. By using perturbation method, the limit asymptotic solutions of various cases are obtained. We provide a reliable and direct method for solving similar problems. The limiting solutions are constants in this paper, except in narrow boundary and interior layers of nonuniform convergence. These provide simple examples of boundary layer resonance.


Introduction
A typical turning point problem consists of the linear differential equation 0, y xy ny ε ′′ for a nonnegative integer n on 1 1 x − ≤ ≤ with prescribed boundary values ( ) 1 y ± and a small positive parameter ε , i.e., 0 1 ε <  . Limiting solutions, away from narrow so-called boundary and interior shock layers of rapid change, take the form as 0 ε → for constants C, so satisfy the limiting reduced equation for 0 ε → . Sophisticated techniques to obtain the asymptotic evaluation of integrals can be found in Olver [12], Wong [13] and elsewhere. Simple arguments often provide the limiting ratio (6), often after rescaling I.
The variety of limiting behaviors to singularly perturbed linear two-point boundary value problems with turning points has not been clearly described. consider the asymptotic solution of two-point boundary value problems (4)-(5).
Case 1: , I s ε decays exponentially as 0 ε → , so for any fixed 0 x > , the numerator and denominator of (6) are both ( ) O ε and the ratio (6) is asymptotically one. Since ( ) 0, 0 y ε = , there is an initial boundary layer region of ( ) O ε thickness involving nonuniform convergence of y. Here, we're using the big O order symbol.
As an example, take ( ) 1 The constant limiting solution We plot the solution for three small ε values in Figure 1.
The limit of ( ) , y x ε is discontinuous at 0 x = , signaling nonuniform convergence.
Case 2: , I x ε grows exponentially large as 0 ε → . This causes y to be asymptotically zero for any 1 x < and a terminal boundary layer of nonuniform convergence to occur near 1 x = .
As an example, take ( ) 1 A x ≡ − and plot ( )

Turning Points
Case 3: We write the ratio (6) as The exact solution is ( ) is the error function [14]. It satisfies it is odd, it increases monotonically, and it tends to ±1 as x → ±∞ .
Since the integrands of (11) peak at the turning point and are asymptotically negligible elsewhere, we will have ( ) The numerator and denominator of (11) are both ( ) O ε . Clearly,    x y x To steepen the shock layer, we must take ε much smaller. We change the sign of A for the next three examples.

Case 6:
( ) Rewriting   1 We note that the solution could be expressed in terms of Dawson's integral The integrand of (19) is asymptotically negligible for 2 s α < , but asymptotically large for 2 s α > . This implies that As an example, consider     This relies on the following figures. We've increased ε in Figure 10 to show the relative contributions. Normalizing to get ( ) 1 1 y = , we get the solution in Figure 9. And Figure 11 shows the picture of integral for ( ) , I x ε with 3 10 ε − = .

Conclusion
We have not been exhaustive, but we have certainly demonstrated a wide variety of asymptotic solutions to turning point problems of the form (4) - (5). They mimic the asymptotics of the more general boundary layer resonance problem.
When the problem of turning points becomes complicated, numerical methods will become unreliable. Finding the limiting solution is extremely ill conditioned as Trefethen recently observed. Due to the serious instability of direct numerical methods, the examples found in scattered literature are usually less detailed. In this paper, we only give asymptotic solutions for a class of singularly perturbed with a turning point. Indeed, the techniques developed here might be expected to apply to that problem. Readers are encouraged to study other limiting possibilities for (4) -(5).