A typical turning point problem consists of the linear differential equation
for a nonnegative integer n on with prescribed boundary values and a small positive parameter , i.e., . Limiting solutions, away from narrow so-called boundary and interior shock layers of rapid change, take the form
as for constants C, so satisfy the limiting reduced equation
Determining constants C and the location of layers is a nontrivial task, the subject of boundary layer resonance . It involves detailed asymptotic analysis and often uses special functions. The classical techniques of matched asymptotic expansions   and the boundary function method of Vasil’eva et al.  may break down, though the newer composite asymptotic expansions  seem to apply. Many experts have studied such problems over the last fifty years    for surveys. An important application to stochastic differential equations in described in the 2017 SIAM von Neumann lecture by Matkowsky . Computing solutions to such problems remains a challenge, although Trefethen et al.  succeed for some examples using the program Chebfun.
A simple, but still rich, the related problem concerns the asymptotic solution of the two-point problem 
with the special boundary values
and a smooth coefficient . Its unique exact solution is
the solution y will increase monotonically with x. The asymptotic value of in (6) is the area under the curve
for . Sophisticated techniques to obtain the asymptotic evaluation of integrals can be found in Olver , Wong  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. The first papers by Pearson in 1968 stressed a numerical approach. In the intervening fifty years, software has improved tremendously, though 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. Inspired by this, in this paper we consider the asymptotic solution of two-point boundary value problems (4)-(5). In our examples, we’ll find the constant “outer” limits 0, 0.5, and 1.
Case 1: .
Here, decays exponentially as , so for any fixed , the numerator and denominator of (6) are both and the ratio (6) is asymptotically one. Since , there is an initial boundary layer region of thickness involving nonuniform convergence of y. Here, we’re using the big O order symbol.
As an example, take , and 0.001, and plot the solution (6). One gets
The constant limiting solution for as , satisfies the reduced equation away from . We plot the solution for three small values in Figure 1.
The limit of is discontinuous at , signaling nonuniform convergence.
Case 2: .
Now grows exponentially large as . This causes y to be asymptotically zero for any and a terminal boundary layer of nonuniform convergence to occur near .
As an example, take and plot
For , the limiting solution as is trivial. The limiting terminal layer will have thickness.
Figure 1. from (9).
2. Turning Points
Case 3: .
Now there’s a simple turning point at and
We write the ratio (6) as
The exact solution is
where is the error function . It satisfies , it is odd, it increases monotonically, and it tends to ±1 as .
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 . Clearly, , there’s antisymmetry about , and an thick region of nonuniform convergence about the midpoint.
Plotting the solution (12) for , we get Figure 2.
Case 4: .
We rescale I to get
a function that peaks in an interval about and is asymptotically negligible elsewhere. This implies that a shock layer of nonuniform convergence occurs about the turning point. The exact solution (6) is
Figure 2. from (12).
For and , we get Figure 3.
Not surprisingly, the asymptotic solution is essentially a translation of that for . For ,
For , there is an analogous terminal layer. For or , the boundary layer is , i.e. thinner.
Case 5: .
We have a third order turning point at . Again, the rescaled integral peaks at , causing y to jump there. The shock layer is now
thick. The exact solution is
where is the incomplete gamma function.
Evaluating for and , we get Figure 5.
To steepen the shock layer, we must take much smaller.
We change the sign of A for the next three examples.
Case 6: .
Rewriting (6) as
Figure 3. from (14) with .
Figure 4. (a) from (15) in the initial layer; (b) from (15).
Figure 5. from (16) for .
the exact solution is
We note that the solution could be expressed in terms of Dawson’s integral . The integrands in (17) peak symmetrically at and 1,
being asymptotically negligible elsewhere. Moreover, . Indeed for , and twin boundary layers occur near both endpoints. For , we have Figure 6.
Case 7: For , we have
Its exact solution is
The integrand of (19) is asymptotically negligible for , but asymptotically large for . This implies that
so there is as -thick terminal layer.
As an example, consider
We have Figure 7 for picture of .
Figure 6. from (18).
Figure 7. for .
Case 8: For , the integrand in (19) decays for . Thus
and there is an thick initial layer (see Figure 8).
For appropriate , we’d expect that the shock layer moves across the interval. We’re now using the little o order symbol, which admittedly isn’t very explicit.
Case 9: For , we have simple turning points at and . Moreover, peaks at and and is asymptotically negligible elsewhere. The sizes of the contributions to the integral differ, however. The area under I near is , but that near is , i.e., larger. Thus, the ratio (6) is for and for .
Computing for , we get Figure 9.
This relies on the following figures. We’ve increased in Figure 10 to show the relative contributions. Normalizing to get , we get the solution in Figure 9. And Figure 11 shows the picture of integral for with .
Figure 8. for .
Figure 9. .
Figure 10. .
Figure 11. .
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).
This research is supported by the Natural Science Foundation of Shanghai Institute of Technology, Research Fund nos. ZQ2018-22, 391100190016027.
 Sharma, K.K., Rai, P. and Patidar, K.C. (2013) A Review on Singularly Perturbed Differential Equations with Turning Points and Interior Layers. Applied Mathematics and Computation, 219, 10575-10609.