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Singular Perturbation Analysis of Boundary-Value Problems for Differential-DifferenceEquations III. Turning Point ProblemsAuthor(s): Charles G. Lange and Robert M. MiuraSource: SIAM Journal on Applied Mathematics, Vol. 45, No. 5 (Oct., 1985), pp. 708-734Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2101624Accessed: 27/02/2009 08:04

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SIAM J. APPL. MATH. ? 1985 Society for Industrial and Applied Mathematics Vol. 45, No. 5, October 1985 002

SINGULAR PERTURBATION ANALYSIS OF BOUNDARY-VALUE PROBLEMS FOR DIFFERENTIAL-DIFFERENCE EQUATIONS III.

TURNING POINT PROBLEMS*

CHARLES G. LANGEt AND ROBERT M. MIURAt

Abstract. This paper continues a study of a class of boundary-value problems for linear second-order differential-difference equations in which the second-order derivative is multiplied by a small parameter (SIAM J. Appl. Math., 42 (1982), pp. 502-531; 45 (1985), pp. 687-707). The previous papers focused on problems involving boundary and interior layer phenomena, rapid oscillations, and resonance behavior. The problems studied here have solutions which exhibit turning point behavior, i.e., transition regions between rapid oscillations and exponential behavior. The presence of the shift terms can induce large amplitudes and multiphase behavior over parts of the interval. A combination of exact solutions, singular perturbation methods, and numerical computations are used in these studies.

1. Introduction. Our previous singular perturbation analyses of boundary-value problems (BVP's) for differential-difference equations (DDE's) [5], [6] (this issue, pp. 687-707) have demonstrated that the solutions exhibit a variety of interesting phenomena. In this paper, we continue this study on classes of DDE's with a turning point. We will again see a variety of new phenomena which are not present in standard turning point problems. For references to potential applications of these results, see [5] and [6].

The singular perturbation problems studied here are linear second-order DDE's with the highest derivative term multiplied by ?2 (e -* 0), and have a single simple turning point in the interval (0, 1). We treat these problems using a (modified) WKB method for the rapidly oscillating portions of the solution and matched asymptotic expansions.

1.1. Statement of the problem. In this paper we study two classes of BVP's for the DDE

(1.1) e2y"(x; v)+q(x)y(x; E)+ a(x)y'(x-1; E)+pB(x)y(x-1; E) = i(x),

on 0 < x < 1, 0 < e << 1, subject to the interval and boundary conditions,

(1.2) y(x; ?) = +(x) on -1 c x '0, y(l; ) = y,

respectively, where q(x), a(x), /3(x), +i(x), and +(x) are smooth functions and y is a constant. For the function y(x; e) to constitute a "smooth" solution of (1.1)-(1.2), we shall require that y(x; ?) satisfy the conditions (1.2), be continuous on [0, 1], and be continuously differentiable on (0, 1). For simplicity, throughout this paper we restrict the length of the interval to 1 < 1 < 2. We assume there is a single simple turning point at x = 5 E (0, 1- 1), specifically

(1.3) q(x) =(-x)p(x).

Then the two classes of problems to be treated here are: 1) p(x)> 0, p(() = 1 for 0 < x < 1, and 2) p(x) < 0, p(f) = -1 for 0 < x < 1. If the turning point x = f is between

* Received by the editors January 31, 1984 and in revised form December 1, 1984. This work was supported in part by the National Science Foundation under grant MCS81-04258 and by the Natural Sciences and Engineering Research Council of Canada under Grant A-4559.

t Department of Mathematics, University of California, Los Angeles, California 90024. t Department of Mathematics and Institute of Applied Mathematics, University of British Columbia,

Vancouver, British Columbia, Canada V6T 1Y4.

708

PERTURBATION OF DIFFERENTIAL-DIFFERENCE EQUATIONS III 709

x = 1 and x = 1, then the analysis is straightforward and the qualitative behavior of the solution is similar to that for turning point problems without shifts.

In ? 2, we elaborate further on our discussion in [6] of the occurrence of "singular" parameter values at which the solutions become "infinite." Here our discussion centers on the birth of zeros in the solution as e -> 0.

Section 3 contains exact solutions to a special class of BVP's for homogeneous DDE problems with "constant" coefficients and a simple turning point.

In ?? 4 and 5, we treat BVP's for DDE's with a simple turning point in which the coefficients are variable. The exact solutions obtained in ? 3 are used as a guide to our singular perturbation analyses of these variable coefficient problems. The case with the oscillatory part of the solution to the left of the turning point is studied in ? 4 (and because of the difference terms in (1.1), there will also be oscillatory behavior to the right of x = 1). The case with the oscillatory part of the solution to the right of the turning point is studied in ? 5. In the latter case, we exclude the possible occurrence of resonance phenomena which was studied in [6].

2. Singular parameter values (eigenvalues). In [6] we showed that there are values of e at which there do not exist solutions to the BVP's for the DDE's. The analytical explanation for this general phenomenon is extremely simple. These values of e correspond to the "eigenvalues" of the associated homogeneous adjoint boundary-value problem for (1.1) and (1.2) given by

(2.1) ?2V,,(X; E) + q(x) v(x; ?) -[a (x+ 1)v(x + 1; ?)]'8 +(X + 1)v(x + 1; ?) = 0,

on 0 < x < 1, subject to the boundary and interval conditions

(2.2) v(0; e)=0, (v(x;E)=0 on[1,1+1),

see Hale [3]. As is well known, BVP (1.1)-(1.2) will not have a solution at these eigenvalues unless a solvability (orthogonality) condition is satisfied.

Our search for such singular parameter values was motivated originally by an attempt to answer the question, "how do the zeros of the solution become generated?" In computational studies, we found the zeros of the solution were generated quite abruptly, i.e., very small changes in e generated new zeros in the solution as well as moved the old zeros and drastically changed the qualitative shape of the solution. These changes are illustrated in Fig. 1 where we have plotted numerical solutions of a DDE with a turning point for three close but distinct values of E. For clarity only the portions of the solution between x = 0 and x = .75 are plotted.

To see how the zeros in the solution are generated as E is decreased, in Fig. 2 we give a series of a graphs showing the solutions at different values of E. In Fig. 2a, there are no zeros in the solution. Between Figs. 2a and 2b, the solution has been flipped about the x-axis and two zeros have been created. A second flip of the solution about the x-axis between Figs. 2c and 2d simply moves the two zeros to new positions. A third flip about the x-axis between Figs. 2e and 2f, however, inserts two additional zeros into the solution. This continues as long as singular parameter values occur. It should be noted that the new zeros in the solution are generated near the turning point.

3. Exact solutions for "homogeneous constant coefficient" cases. To illustrate the types of behavior which can occur, in this section we find the exact solution for the "homogeneous constant coefficient" case, by which we mean that p(x), a(x), ,8(x), and +(x) are constants denoted by p, a, ,, and 4, respectively, and qi(x) 0. Specifi- cally, we treat the problem

710 CHARLES G. LANGE AND ROBERT M. MIURA

20,000 y

10,000 c X .015125

0

\ / =~~~~~~~.01512

-10,000

-20,000-

0 .15 .30 .45 .60 .75

FIG. 1. Plots of solutions of E2y"(X; E)-(X-.25)y(x; E)+.5y(x-1; E) = 0 with y(x; E) = 1 on [-1, 0],

y(l.5; e) = 1 for three close values of E =.01512, .015125, .01513. The solutions are shown only on the interval

[0, .75].

c = .04 \ / =.026

(a) (b)

c E=.017 c =.014

(c) (d)

c = .012 e= .008

(e) (f)

FIG. 2. Generation of zeros in the solutions of E2y"(x;E)-(x-.25)y(x;E)+.5y(x-l;E)1 O with

y(x; E) = 1 on [-1, 0], y(l.5; E) = 1 as E decreases through the values indicated. The solutions are shown only

on the interval [0,.75].


with 0 << I - 1, 0 < E << 1, p = i 1, and the interval and boundary conditions

(3.2) y(x;?) = o on-1 _ -x ' O-, y(l; ?) = Y.

The two values of p allow us to solve both problems simultaneously. However, the leading-order results must be obtained separately. The more general asymptotic results obtained in ?? 4 and 5 for variable coefficient problems can be shown to reduce to the leading-order results given in this section when they are specialized to the homogeneous constant coefficient cases.

We split the domain 0 < x < 1 into the two intervals 0 < x < 1 and 1 < x < 1 denoted as regions A and B, respectively. In region A, we introduce a scaled independent variable and a new dependent variable to reduce (3.1) to a standard nonhomogeneous turning point problem. Let

X - ( 1 - y -Zee(+Ze (3.3) Z_a8 3-- )YA(;E e+s;E

where 8 = E2/3, so in region A, (3.1) becomes

(3.4) YA(Z; )- PZYA(Z; ?)=--

The general solution is given by

(3.5) YA(Z; e)= a, Ai (pz)+a2 Bi (pz)+ Gi (pz),

where Ai (z) and Bi (z) are the Airy functions and Gi (z) is a Scorer function which satisfies

(3.6) Gi" (z) - z Gi (z)= IT

For definiteness, we choose the Scorer function with the integral representation

(3.7) Gi (z)=-J sin ( s3+zs) ds

which has the asymptotic behavior [1]

1 Cos

[2 (_Z)3/2+M]

+

I, z

-0-a (3.8) Gi (z) I(Z)1/4 [3 (3 1 ) + Z

=oj (3z3Z

In region B, let YB(Z; E) y( +8Z; ?), SO (3.1) becomes

Y(Z; e) -PZYB(Z; ?)

(3.9) =-pdd2a[a1 Ai' (p(z-d))+a2 Bi' (p(z-d))+d13ir Gi' (p(z-d))I

- d,l3[al Ai (p(z - d)) + a2 Bi (p(z - d)) + d/3 +1r Gi (p(z - d))],

where, for convenience, we have set d-1/8 = 1/ ?2/3. In [7], we obtain exact particular

solutions to Airy equations forced by Airy and Scorer functions and their derivatives.

Using these results, summarized in the Appendix, we obtain the exact solution in


region B YB(Z; )=bIAi(Pz)+b2Bi(Pz)+d2af38OTrGi(Pz)

+ d{a[al Ai' (p(z - d)) + a2 Bi' (p(z - d))]

+p(a +83)/30ir[Gi (p(z - d)) - Gi (pz)]}

+p(a + 8)[al Ai (p(z - d)) + a2 Bi (p(z - d))].

To determine a,, a2, bl, and b2, we apply the continuity conditions at x = 0, 1 and the boundary condition at x = 1. For notational convenience, let Ao Ai (-pd(), A1 Ai (pd(l - 1 - a)), A2 Ai (pd(1 - a)), and A3 Ai (pd(l - a)). We use similar notation for Bi and Gi and denote derivatives with primes. The linear algebraic equations for these constants are

(3.11) AOaI + Boa2 = 4(1 - d/3irGo),

[daA'+p(a+,8)Ao-A2]a1 +[daB+p(a +,)Bo- B2]a2+ A2b+B2b2 (3.12)

2a2 2,+Bb

= d/38ir{-daGo+[1 +p(a + 38)]G2 -p(a +3)Go},

[d 2a(Ao - (a +8)AO+ pAJ]al + [d 2a(Bo- (a+ 8)BO+pB2]a2 -pA'bl -pBb2 (3.13)

2 2 2 2

= -dp8ard a Go+P+(p+a+f3)G! -(a+f3)Gj]

[daA +p(a + ,)AI]al +[daB +p(a + 8)BI]a2+ A3b, + B3b2 (3.14)

= y-di3Gir[daG' + p(a + ,)(G1 - G3)]

It is straightforward but tedious to solve (3.11)-(3.14) for a,, a2, bl, and b2 and we choose not to display them explicitly. Here we give only the leading-order results for p = 1 and p = -1 in both cases of a $ 0 and a = 0. For these evaluations, we use asymptotic expansions for the Airy and Scorer functions (Abramowitz and Stegun [1]). For p = 1 and a $ 0, the coefficients are approximated by

(3.15) a, - -d/3 4i;r cos L2 ((d)3/2+ sin L2 ((d)3/2+ ],

(3.16) a2 d512a,13kr[(1 - 5)]1/4 exp { -2 [(1- ()d]3/2} 2 sin [2 (sd)3/2+Mj,

(3.17) b, -d 512 a,87r[(1 - 5) ]1/4 exp 2[( - ( d ]2/3 sin 2

( d )3/2 + 'Tr

3 -J' 3 b 4J

(3.18) b2 - d1/4Nir(lT - -)1/4Ly - 3k ( + a exp { -j[(1 t)d]3/2}2

For p = 1 and a = 0, the coefficients are approximated by

(3.19) a --df348>r cos L2 (std)3/2+ i]/sin L2 ((d)3/2?+],

(3.20) a2 d,B </ 4exp {- [(1- ()d]3/2} 2(1 -_)/4 sin [ (sd)3/2+Mj,

(3.21) b d2<t7rrS114 exp - [(1 - g)d]3/2} sin [z (gd)3/2+A},


(3.22) b2 - d l/S I )/4 [7 - p

2o-1 ) exp 2

[(1 - })]1

It should be noted that for both a $ 0 and a = 0, the coefficients of Bi (z) in both regions A and B given by a2 and b2, respectively, are exponentially small. The terms a2 Bi (z) and b2 Bi (z) become important only near x= 1 and x = 1, respectively. On the other hand, b1, the coefficient of Ai (z) in region B, is exponentially large and the term b, Ai (z) is important only near x = 1.

We further note that a,, a2, and b, are undefined for those values of e where

(3.23) 2

(gd)3/2+- = nir, n = 1, 2, 3. 3 4

More correctly, the solution is undefined for those values of e which make the determinant of the coefficient matrix of (3.11)-(3.14) equal to zero. As noted earlier in ? 2, the nonexistence of the solution for values of E bounded away from zero is associated with the existence of a nontrivial solution to the corresponding homogeneous adjoint problem. When the determinant of the coefficient matrix is close to zero, (3.15)-(3.22) are no longer valid and a better approximation is necessary. We do not carry out these calculations here.

For p = -1 and a $ 0, the coefficients are approximated by

(3.24) a, - d8iT sin [(1 - - 1 )d]3/2 + }/cos [ )d]3/2 +{

(3-25) a2~ -d'

4+Sd'ar1P(

exp

L-2 (~d)3/2],

b(2d6 -7 [(1 )1/4 cos [(1 - ()d ]3/2 +} (3.26) (

- ()1/4 sin { [(1- i )d]312+ }]

b2~ -d7/4av'ir(i -~ )[v(1 - )1/4 cos {2 [(1- ()d]3/2+ i}

(3.27)

+ ( sin {2[(1 - ()d]3/2 +}1]

For p = -1 and a = 0, the coefficients are approximated by

a, - b,

-dP7,rT(cos {2 [(1- ~t)d]3/2+24}

(sin {4)d]3 4s{ 1

(sin [(l - )d] 6 +-s () in - [(l Xl


We note that for both a $ 0 and a = 0, the coefficient of Bi (-z) given by a2 in region A is exponentially small and the term a2 Bi (-z) becomes important only near x = 0. For a $ 0, the coefficient a1 is undefined when

(3.31) 3 [(I - 1 - e)d] 4= (2n + 1) 2' n= O,1, 2,*,

and for a = 0, the coefficients a, and b, are undefined when

(3.32) sin {-[(1- f)d]3/2+-} - 1(1 '-l)1 sin {-[(- 1 -e)d]3/2+ } = 0.

More correctly, as noted above, these coefficients are undefined when the determinant of the coefficient matrix of (3.11)-(3.14) equals zero.

4. Turning point problems for p(x) > 0. We find that a combination of WKB and limit process expansion techniques provides an effective approach to constructing asymptotic approximations of solutions of BVP (1.1)-(1.2). An important ingredient in such an approach is a suitable decomposition of the interval [0, 1] into regions where the structure of the solution has distinct features, e.g., layer regions, regions of rapid oscillations, and outer regions. Before engaging in the rather detailed analysis which this problem requires, we shall attempt to provide some insight into the decomposition which we have chosen for the case p(x) > 0.

4.1. Qualitative behavior of the solutions. The qualitative features of the solution of BVP (1.1)-(1.2) for the case p(x) > 0 are illustrated in Fig. 3 (where a(x) $ 0) and Fig. 4 (where a (x) 0) for the particular choices of the functions and parameters. Except for the larger amplitudes in Fig. 3, the two graphs reflect the same general behavior. Since q(x) > 0 for 0? x < = .25 and q(x) < 0 for 6 < x' 1, the homogeneous solution of DDE (1.1) will involve rapid oscillations and exponential behavior to the left and right, respectively, of the narrow turning point region at x = 4. On the other hand, if a, ,, X, and 4' are smooth, a particular solution of (1.1) will have 0(1) smoothness in e on (0, 1). The solution does, indeed, appear to be quite smooth on (g, 1) suggesting that this is an outer region.

900 y

600

300

0

-300

-600

-900 x 0 ^ .3 . .9 1.2 1.5

FIG.3. Plot ofthesolution ofE2y"(X; E )-(X-.25)y(x; E) +.25y'(x- ;E) + y(x- ;E)= O with y(x; E)= 1 on [-1, 0], y(l.5; E) = O, and E = 0.005. The large amplitude oscillationsfrom the left of x= 1 to near x = 1.25 are induced by the forcing term a(x)y'(x-1; E).


75 -

50-

25-

-75- , , - * I X

0 .3 .6 .9 1.2 1.5

FIG. 4. Plot of the solution of E2y"(X; E)-(x-.25)y(x; E) +y(x-1; E) = 0 with y(x; E) = I on [-1, 0], y(1.5; E) = 0, and E = 0.005. The large amplitude oscillations on [0, .25) and (1, 1.25) are of the same orders of magnitude because there is no a (x)y'(x -1; E) term.

Through the shift terms the oscillatory homogeneous solution in (0, a) induces an oscillatory particular solution in (1, 1+ e). Similarly, the combination of a turning point region at x = e and the shift terms generates a transition region at x = 1 + 4. In (1 + 4, 1) the shift terms have not created any dramatic effects as this appears to be an outer region. Finally, there seems to be an interior layer "centered" at x = 1 and a boundary layer at x = 1. The former arises because of the continuity requirements and the latter because of the boundary condition.

4.2. Asymptotic analysis. As a first step in constructing an asymptotic approximation of a smooth solution of BVP (1.1)-(1.2) for the case p(x) > 0, we decompose the interval 0 ' x' I into eight regions labelled I, II, * * *, VIII as depicted in Fig. 5. The motivation for this particular decomposition follows from our interpretation of the exact solution described in ? 3 and the numerical solution shown in Figs. 3 and 4. We designate the solution in the mth region as ym. It often proves convenient to express Ym as

(4.1) y H + Ym

where yH and yP correspond to a homogeneous and particular solution in region m, respectively. The actual determination of the approximation proceeds in a stepwise manner with the orders of magnitude of the solution in certain regions often having to be ascertained simultaneously via matching and continuity conditions. Although we have carried out the calculations to several terms in each region to verify that matching does occur and to check that no other features of the problem are overlooked, we shall be content here simply to record enough terms to permit the determination of the leading-order behavior of the solution in both the cases a(x) 0 0 and a(x) 0.

First we consider region I, 0 ? x < 4, where the governing equation (1.1) becomes

(4.2) E 21(x; E)+q(x)y1(x; E)=f(x),

with

(4.3) f(x) - (x) - a (x) OU - 1) -.8 (x) (x- 1)


I 1 I II IV v VI Vil viii

, | 4 | | ~~~0(1)<

(1/23) o (1/ E 3/2) O E/ 4/ 3)

0(i/v) I (1) O(1/E3/2) III 0(1)

smooth ~~rapid rapid oscillations smoth oscillations smooth I

0 1 1+Q

FIG. 5. Orders of magnitude of the solution to turning point problems for p(x) > 0 and a (x) w 0.

Since q(x) > 0 in region I we anticipate that the modified WKB approach used in [6] should apply.

The particular solution yp is given by

(4.4) yf(X;E)= ( + O(E2). (4.4) Yi ~~~~~q(x)

The homogeneous solution Y1 involves rapid oscillations of unknown amplitude which we express as

(4.5) Y1(X; E)~ ajAj(e)) Y1(x; e)+ c.c., j=o

where the aj are arbitrary complex constants, {Aj(e)}J0 is an asymptotic sequence as E - 0, and c.c. denotes the complex conjugate. The function Y1(x; e) has the WKB expansion

(4.6) Yl(x; ?) = ~E(x1/ [ + i8 QO(X) + O(e2)] (4.6) Y, (X; - q(x)114 L 8 t!XJJJ

where we introduce the notation

(4.7) Er(X) sexp L- r ) dsj

and

Cx r5q(s2 qI s (4.8) Qr(x) J n4 q(S)14 q (S)3/] ds2

The aj and Aj will be determined by matching with the solution in region II. Application of the continuity condition at x = 0 given by (1.2) yields

fo 2 i \ EC(O) + (2)+CC (4.9) Y1(O;E)=0<o=-+0(E + {( ajAj( e) q~14 [1+40(

where o= 0 (0), fo=f(O), and qo= q(O).


In the turning point region II, centered at x = e, we introduce

(4.10) x - (8=E2/3)

and designate the solution y(x; e) as

(4.11) Y2(x2; E)=y(e+8x2; E).

(The choice of scaling in (4.10) is standard in turning point problems for ODE's [2].) In terms of the variables x2 and Y2 the DDE (1.1) becomes

(4.12) Y2(X2; E)-X2p( f + 8x2)y2(x2; E)=f( + 8X2),

where we have used (1.3). For sufficiently smooth functions p and f it is natural to seek a series solution for a particular solution of (4.12) in the form

(4.13) y1x;e-- Z kx8k as e ->0. (4.13) ~~Yp2(X2; E) -

E Y2, k(X2)8 sEoO 8 k=O

Upon substituting this expansion into (4.12), expanding p and q for jx21 << 1, and equating coefficients of 8 to zero, we obtain the sequence of equations

(4.14) y 20(x2) -x2y20(x2) = f~, k 3

Y2,k(X2) - x2y2 k(x2) =f(2)X2 + E P(j) 2 Y2 kJ-(X2), k = 1, 2, j=~1

where f-f(f) f( )fk)(g and p(J) p(j)(e) (recall that p(f) = 1). Since the order of magnitude of Y2 is not clear at this stage, we express the

homogeneous solution of (4.12) as

(4.15) Y2'(X2; E) - E bjQfj(E) Y2(X2; E),

j=o

where the bj are arbitrary real constants and {Qfj(e)}j0 is an asymptotic sequence as E -*0. Moreover, we assume that Y2(x2; e) has a series expansion

(4.16) Y2(X2; E) Z Y2,k(X2)8, k=O

where Y2, k satisfies- (4.14) with Y2,k replaced by Y2, k and with all terms involving f set equal to zero.

We take

(4.17) Y2,o(x2) = Ai (x2),

which decays exponentially as x2 -*ox. To be completely general, we should express Y2,0 as a linear combination of Ai and Bi. However, examination of the exact solution


in ? 3 indicates that the Bi term will be exponentially small (in E) compared to the Ai term. Not including a Bi term in (4.17) considerably simplifies the analysis and, as we show, leads to a consistent approximate solution.

Using the results in the Appendix it is straightforward to obtain the leading terms in the expansions for y2H and y2p. We find

Y2(X2; (= E bijf(e )) [Ai (X2) + (x Ai' (x2) - x2 Ai (x2)) + 0(82)]

--Gi (x2) -ff+ (-_TX2 Gi' (x2) + 'Ix2 Gi (x2)+3) + 0(8), 8 ~~~5 2

where f=' p= p( Matching of the expansion for Yi as x - with the expansion for Y2 as x2- -00

provides information about the unknown order functions and coefficients in each expansion. Since the matching formalism is well known, we shall be content simply to indicate its flavor for this particular class of problems.

First we replace x in Yi by e+ 8X2 and expand on the basis of 1 << -x2 with -8x2 <" 1. Note that

(4.19) q(x) = -8x2[1 + 8Px2+ 0O((8x2)2)],

so that

(4.20) Ee(x) = exp [ (-X2)3/2] [ 1 8(-x2)51/2+ 0(82(-x2)7/2)]

With these results we readily obtain

y1(e+ 8X2; E) = - _ + OX2)

{E Aj(E) 1 + 0(8X2) -2 8(-x2) + 0(82( )7/2) + a3- 1/6 j -21/4 [1 -2 X

j=O C _2

(4.21) [e 21 6 4 R)

i[ 65?/2 + /x + - 4e/)

+ 0(\8X2) + O(E 2) + c.c.}

with

R() 5q'(X)2 q"(x) 5 ' (4.22) R 4q(x)=1/2- q(X)3/2 -4(x_ -)/2 8(X -f)3/

where the last two terms on the right-hand side subtract out enough of the singularity present in the first two terms as x - e- to insure that R(x) is integrable on [0, e]. Next we make use of the asymptotics of Airy and Scorer functions (see [1] and (3.8)) to


obtain the corresponding expansion for Y2 from (4.18)

y2(x2; ?) ( , bjQ j( )){[v _ )1/4 +{0 2_X2))4(

+ le41 (-X2)94 ( + i) + O((-x)3/4)) +(8)

exp [3-(-x2)3/2] + cc.} (4.23) _ ,s%{ [2vis+0((-xY-7/4)] exp (e-x2)3/p2] + c.c.}

+ Pf { 1[ ) () + (-32]

* exp [3-7(-x2)3/2] +c cc} + 0().

Upon comparing (4.21) and (4.23) it becomes apparent that it is only possible to achieve a consistent matching of the leading-order oscillatory terms in Yi and Y2 if the following condition is satisfied

+fep A(e)1- ir4(+i

(4.24) (Re a0-i Im a0) p1/6 = bQ()2/;e/2;

where we recall that bo is real. Equating real and imaginary parts, it follows that (without loss of generality),

(4.25) A0(e)=-; and Q() 23

Athird equation involving Re a9 Tm a, and bo follows from satisfaction of the continuity condition (.9) to 0(1/4 E), namely

(4.26) Re aCo(6) + Im a0S0(SX ) =0 ,


Solving, we obtain

expi ) -S()

(4.28) boo= c ag 2C0( ) + S ()p

(4.29) Re ao= b4_f_____ ___


(4.30) Im ao = ,"7; Co(f)

Note that singular parameter values of e occur if Co(e) + So(e) =0. Satisfaction of the continuity condition (4.9) to 0(1) and further matching of the

expansions for Yi and Y2 reveals that

(4.31) A1(e) = 1 and fQ1(e) = I/EI/,

and

(4.32) b i

qo/ ( f

(4.33) Re a,= - = Im al,

which completes the determination of Yi and Y2 through 0(1). In fact, we could continue this evaluation of order functions and coefficients to arbitrarily high order in e. In particular, it is not difficult to show that

(4.34) Aj(e) = e(j-l)/2 ?Qj(e) = e-1/6A(), j = 0, 1,

It is clear that our decision not to include the Bi function in (4.17) allows us to completely determine the solution (to within exponentially small terms in E) in regions I and II independently of the form of the solution in the remaining regions.

Region III, e < x < 1, is an outer region (since q(x) < 0) with governing equation given by (4.2)-(4.3) with Yi replaced by Y3. The expansion for Y3 takes the form of an asymptotic power series

f(x) 2. (4.35) y3(X; e)= + 0(E). q(x)

In order to match the expansions in regions II and III we need the asymptotic H behavior of Y2 as x2 -* 00. Since Ai (x2) becomes exponentially small in this limit, Y2

plays no role in the matching. To evaluate the contribution of yp, we make use of (3.8) from which it follows that

(4.36) Y2(X2;?)= - E + 0(1) for 1 << x2with8x2<< 1. 8X2

It is apparent that this expansion matches to leading order with that in (4.35) in the limit x-* f.

Based on the form of the solution in region I we anticipate that the shift terms in (1.1) will induce large amplitude rapid oscillations in the region 1 < x < 1+ +. We shall show that it is not possible for the large amplitude solution in 1 < x < 1 + e to satisfy the continuity requirements at x = 1 with y3(x; e). To gain the necessary flexibility we must insert an interior-layer region to the left of x = 1 (a similar situation occurs in [5]).

Region IV, 0< 1 - x << 1, is an interior-layer region where we set

(4.37) ,X y4(x4; E)-y(l + EX4; E).

For this region, as well, the governing equation is given by (4.2)-(4.3) in the form

(4.38) y4(x4; e) + q(l + EX4)y4(X4; E) =f(I + EX4),


where we recall that q(1) < 0. We assume that y4 has the following series expansion in powers of Ic:

1 X (4.39) Y4(X4; E)- ~ EY4(x4)Ej' as E-*O.

j=0

The motivation for this choice will be provided in our discussion of region V. Upon substituting (4.39) into (4.38), expanding q and f for I Ex4I << 1, and equating

the coefficients of powers of Ic to zero, we obtain through 0(1)

Y4(X4; E 3/2+ A C24 A-q q]

E ~ ~ ~ +3 4 q,/+q) x (X)+q+() (4.40) 'c 2 x\

4\Vq1 q,/

where co, cl, , are arbitrary real constants. In (4.40) we have omitted terms from the homogeneous solution of (4.38) which grow exponentially as x4 -> -00. It is then clear that this expansion matches with the leading-order term in region III. The co, cl, * * , are determined from the continuity conditions at x = 1 (i.e., X4 = 0).

In region V, 1 < x < 1 + e, according to (1.1) the solution y5(x; E) satisfies

(4.41) E2 Y(x; E)+q(X)Y5(X; E)=+O(x)- a(x)y(x-1; E)-3(x)yj1(x-1; E).

Actually region V incorporates two distinct features. The first is the appearance of rapid oscillations on the whole region due to the forcing terms in (4.41) and which can be treated easily by the WKB method. Substituting (4.4)-(4.8) and (4.34) into (4.41) and employing the results in the Appendix of [6], we find that

YS (X; E)

q()-(x-) ){iE3x l)q=) aiej/[1 8 QO(X0)+0e )

E4)q'( X-1 ) /2[+I/ 4 = (

(4.42) - q(x) iq(x -1){ {q -ij/2[21q(x -)(a q(x 1) )

a(x)q(x - 1)[1Q+o (E)] ] +c

~~~~~~~~~2(- /4q(x) - q(x -1] J)Ej=

q(x)-q(x-1){ K/q(x - 1)/ j=O

+ +'(x) - a(x)(f(x- -)/q(x- 1))'-f(x)f(x - l)/q(x -1)+ ( ) C q(x)

We observe that y5P is completely determined since Yi is completely determined. Moreover, y. describes rapid oscillations of amplitude 0(1/83/2) if 4(x)l 0 (or (1/41/2) if (X)50). It becomes apparent that the continuity conditions at xx=1

given by (1.2) can be satisfied only if y4 and yq are 0(1/83/2) if a(x) #0 (or 0(1/81/2) if aY(x)--a).


The second feature is an interior layer of width O(E) to the right of x = 1. This layer is associated with y5H which has exponential behavior on an O(E) scale since q(x) < 0. Relying on our interpretation of the asymptotic behavior of the exact solution in ? 3, we exclude exponentially growing terms (for (x - 1)/c >> 1) in y5H. This leaves y5H with only exponentially decaying terms which can be derived by the WKB method. However we prefer to express y5H in the simpler form of a limit process expansion valid near x = 1. This representation turns out to be accurate to within exponentially small (in E) errors uniformly on region V.

With this background information it readily follows that the expansion for y5His given by

H5(X; E) = +{_ + l+ [-qI do x

(-)I -+x-)]

where d0, d1, , are arbitrary real constants. Since y5' decays exponentially as (x -1)/c -o 00, only the terms in y are important when matching with the expansions in region VI.

Satisfaction of the continuity conditions at x = 1 leads to a complete determination of y- and y. We will returnh to his problem after our discussion of the remaining two regions.

A narrow transition region VI, centered at x = {+ 1, must be included to account for the fact that near x = e + 1 the shift terms in (1. 1) reflect the behavior of the solution in the turning point region centered at x= 4 We define new variables (with 8 = 2/3

(4.44) 8X6 =

, Y6(X6; E) = y(l + + SX6; ).

In terms of these variables the DDE (1.1) becomes

8Y6(X6; ?) + q(I + e+8X6)y6(x6; ?)

( ) (1 ++ x6)y(X6; E) _ X ( 1 + + e+8X6)y2 (X6; E)-

Since q(6+ 1) <0, yYH will be a linear combination of two exponential terms: one which grows as x6 -> -co and the other which grows as x6 - oc0. The former behavior prevents matching with the assumed form of y5, while the latter leads to difficulty in satisfying the boundary condition at x = 1. Thus to within an exponentially small (in E) error, we take y6H equal to zero.

An expansion for yp (and thus, for Y6) follows by expanding the coefficients in (4.45) for I8x61 << 1 and applying the obvious recursion formula. The result is

Y6(X6; I) =-a (I + OY2(X6; ?) + +(i ) q(1+) 1 8

(4.46) -a'(1 + )X6Y2f(X6; c) + a(1 + e)Y2'(x6; c)

+a (1 + e)q(1l+e)X6Y2(X6; c)-3(1 +e)Y2(X6; ?)+O(6)}

Since Y2 is completely determined, it follows that Y6 is completely determined. Moreover, the matching of y, and Y2 insures the matching of y5 and Y6. The size of Y6 is particularly


noteworthy in the case when a(1 + e) ? 0 (compare with Y5 in (4.42) and (4.43)). The effect of the differentiated shift term is not quite as dramatic in the transition region VI as it is in region V because Yi varies on a shorter scale than does Y2.

The solution in outer region VII, e + 1 < x < 1, can be found as a regular perturbation expansion and is given by

(4.47) Y7(x; e ) = q(x) k q (x - 1) q(x)q(x - 1) (

which matches with the solution in region VI. In general, y7 does not satisfy the boundary condition at x =1 and hence it is necessary to introduce a boundary layer region VIII.

Finally, in the boundary-layer region VIII near x = 1, we define the new variables

(4.48) X8 = x

Y8X8 -)YIE8 )

The equation for y, can take one of two possible forms according to whether 1 < 2 or I = 2. The resulting solutions in these two cases show dramatically different behavior (contrast Figs. 3 and 6). If 1= 2, then the equation for y8 has forcing terms involving

1000 - Y

-1000

-2000

-3000 -

0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0

FIG.6. Plotofthesolutionof 2y"(X; E)-(X-_.25)y(x; e)+.25y'(x-1; e)+y(x-1; s)=Owithy(x; e)= I on [-1, 0], y(2; e) = 0, and e = 0.005. The solution on [0, 1.5) is nearly identical to that in Fig. 3 where I = 1.5. The spike nearx = 2 is induced by the boundary layer to the left ofx = 1 through the a (x)y'(x - 1, e) term.

the known boundary-layer solution y5, and, as a consequence, Y8 would describe a "spike" of height O(1/e5/2) if a(2) ? 0 and 0(1/E3/2) if aC(2) = 0. While it is straightforward to obtain the solution in this case, we choose not to present the details here. On the other hand, if I < 2, then the equation for Y8 has only smooth forcing functions,


including y4, and is given by

Y8(X8; )+ q(l+ X8)y8(X8; ?)

(4.49) =1(1+ EX8) -a(1+EX8) kq(X) |=_+x ](

- (9 + I+ 0X) 1+ O(E2)

We expand Y8 in a power series in 8,

(4.50) Y8(X8; ?)- E Y8,j(X8)?8. j=o

Solving (4.49) as a regular perturbation for Y8,0, we obtain

(4.51) y8(x8;f)=( ( A) exp fqx8)+-+O(e), q, q,

where we have applied the boundary condition Y8(, 8) = y and defined

(4.52) f a f(x) | _ I

411 (x) xll q- qV x=1-1 -,

It remains to satisfy the continuity conditions at x = 1 given by (1.2). With our decomposition this translates into the two conditions

(4-53) 140 ?) =5(;?, e4(0, ?) = y5(0; ?),

where y4 iS given by (4.40) and Y5 by (4.42)-(4.43). We treat two separate cases; namely, a(x) #0 and a (x) 0. For the case, a (x) #0, satisfaction of the first condition in (4.53) to O(8-3/2) and of the second to o(8-5/2), leads to

(4.54) Co= qo _ Re ao[ Co() - So()l + Im aO[ So( + Co()},

(4.55) do = aqo _ qRe aO[ Co(() + So(()] + Im aO[ Sof ) - Cof {)]} q - qo I l Il O'b -bJ O 1qll0~Co4'j

where Re ao and Im ao are given in (4.29) and (4.30), respectively. Identical equations are obtained for cl and d, except that ao is replaced with a, (given in (4.33)). In Fig. 5 we indicate the orders of magnitude of the solution in each of the eight regions for a 0.

For the second case, a (x) 0; since yp = 0(1/JE), we take for y4 and y5,

(4.56) Y4(x; 8)=)exp e 1) + (J)

y5(x; 8) =(do+ di) exp (v/ q x) + F(x)

(4.57) A_ (x_ E{(x-1) [ +ao


where

(4.58) F(x) 4'(x) - a(x)(f(x - 1)/q(x - 1))'-f 3(x)f(x - 1)/q(x - 1) q(x)

Applying the continuity conditions at x 1 to leading order, we obtain

Co =4_ f [ 1

SO(f) + Co(()

do=~ ) 01/4 Re ao S0()+- C0(f)i (4.60) r---(q-qoqo

_ qjJ

-Im ao[ C(1)+S0(e) qjj

and cl and d, are given by identical expressions except ao is replaced by a1 and 4F(1) is added to both. In Fig. 7 we indicate the orders of magnitude of the solution in each of the regions for a 0.

Finally, we note that in the special case when the functions p, a, /3, 4 are constants and 4, = 0 the above asymptotic results for both the cases a $ 0 and a = 0 agree with the exact results obtained in ? 3.

I II III IV V VI VII VIII

E I 1I I I o(l)

I I I ,(OW 2/3) 0(1/4) I,/o(W/E 23)

0(1/o ) 1 1 1 0(1) O I 0(lr1 OM

rapid oscillations j smooth rapid oscillations smooth

0 1 1 1+ Q

FIG. 7. Orders of magnitude of the solution to turning point problems for p(x) > 0 and a (x) 0.

5. Turning point problems for p(x) <0. This case is similar in many respects to that considered in ? 4. However, there are enough differences to merit a separate treatment. For example, recall that for p(x) > 0 it was possible to determine the complete expansion of the solution in the turning point region without knowing the solution on (1, 1). In contrast, for p(x) <0, even determination of the order of magnitude of the solution in the turning point region requires application of all boundary and continuity conditions. Another important difference is the possibility of resonance phenomena ([4], [6]) if q(x) = q(x - 1) for some x E (1 + (, 1) (resonance is not possible for p(x) > 0). At the appropriate point in our analysis we shall indicate where resonance phenomena could play a role and what the necessary modifications would be.

5.1. Qualitative behavior of the solutions. The qualitative features of the solution of BVP (1.1)-(1.3) for the case p(x) < 0 are illustrated in Fig. 8 (where a (x) # 0) and


1200 Y

800

400

-400

-800

-1200 ' , x

0 .2 .4 .6 .8 1.0 1.2 1,4 1.6 1.8 2.0

FIG. 8. Plot of the solution of 2y(X; E) + (x-.5)y(x;,?) +.5y'(x-1;) + y(x-1; e) = with y(x; ) =1 on [-1, 0], y(2; e) = 0, and e = 0.01. The large amplitude oscillations on (1.5, 2) are induced by the oscillations on (.5, 1) through the a (x)y'(x - 1; e) term. The multiphase behavior in the transition region around x = 1.5 results from the combination of different wavelength oscillations in the homogeneous and particular solutions.

Fig. 9 (where a (x) 0) for particular choices of the functions and parameters. Except for the larger amplitudes in Fig. 8, the two graphs reflect the same general behavior, Since q (x) < 0 for 0 _ x < e = .5 and q (x) > 0 for f < x 1, the homogeneous solution of DDE (1.1) will involve exponential behavior and rapid oscillations to the left and right, respectively, of the narrow turning point region centered at x = {. On the other hand, if a, p, X, and qf are smooth, a particular solution of (1.1) will have 0(1)

400 y

300

200

100

0A

-100

-200-

-300-

-400 x 0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0

FIG. 9. Plot of the solution of _2y"(x; )+(x-.5)y(x; e)+y(x-1; )=0 with y(x; e)= 1 on [-1,0], y(2; e) = 0, and e = 0.01. The multiphase behavior in (1.5, 2) resultsfrom the combination of different wavelength oscillations in the homogeneous and particular solutions.


smoothness in E on (0, 1). These features explain the presence of a boundary layer near x = 0 and an outer region on (0, {). The combination of the boundary layer near x = 0 and the shift terms induces a boundary layer near x = 1. (However, in our analysis it will not be necessary to introduce a separate boundary layer region near x = 1.) Similarly, the turning point region centered at x = f induces a transition region centered at x = 1 + f via the shift terms. Finally, we observe that through the shift terms the oscillatory homogeneous solution in (g, 1) generates an oscillatory particular solution in (1 + ~, 1), which when added to the oscillatory homogeneous solution, leads to the multiphase behavior depicted in the figures.

5.2. Asymptotic analysis. To construct an asymptotic approximation of a smooth solution of BVP (1.1)-(1.2) for the case p(x) < 0 we decompose the interval O'x'I into seven regions labelled I, I,- , VII as depicted in Fig. 10. The determination of the approximate solution proceeds in stepwise manner as in the case with p(x)> 0. Since the approach here parallels that in ? 4 we shall focus our attention on the distinguishing features of this case and record the basic results with a minimum of discussion and detail.

First we consider the boundary layer region I near x = 0 where we set

(5.1) ~~~~X1-_ Y1 (XI; E)y --(XI; E).

1 II III IV V Vy VI IV VI VII

o0(1) O/3 I oiu0(1O E )

| 0(1) 1 (/) 4 O(l/e) I o(l/3/2) |

rapid oscillations I I ~rapid I smooth rapid oscillations Lpid oscillations i oscillations l

0 1 +

FIG. 10. Orders of magnitude of the solution to turning point problems for p(x) < 0 and a(x) 0 0.

In terms of these new variables, the DDE (1.1) becomes

(5.2) y1(x1; E)+q(Ex1)y1(x1; E)=f(Ex1),

where f is defined in (4.3). In addition the solution Yi must satisfy the continuity condition at x = 0 given in (1.12) by

(5.3) Y1 E)- 0.

From the form of (5.2) and the results of ? 3 we assume that y, has a power series expansion in E in which all exponentially growing terms (for xl -> oo) must be excluded. Through terms of 0(1) we obtain

(5.4) Y1(XI; E) = (X )- ) exp(-A1/-qoxl)+-+ O(E).


The solution in the outer region II, 0 < x < e, can be found as a regular perturbation expansion and is given by

If(x) +OE (5-.5) Y2(X; E) =-q(x)

which obviously matches with the solution in region I. To this point the solution is completely determined.

In the turning point region III, centered at x = , we introduce the stretched variable

(5.6) x-- (8-E2/3)

and designate the solution y(x; e) as

(5.7) y3(X3; E)-Y(e+ 8X3; E)*

In terms of the new variables, the DDE (1.1) becomes

1 (5.8) Y 3(X3; ?)-X3p(e +X)y3(X3; E)=-f( +8X3),

where we have used (1.3). Note that (5.8) has exactly the same form as (4.12) except here p(x) < 0. As a consequence the structure of the solution in turning point region III is essentially the reflection about X3 = 0 of that occurring in the turning point region when p(x) >0. It follows that the expansion for y3 has basically the same form as (4.18) with x2 replaced by -X3, namely

y(;e)=(EbQ()[A(-x3) + (X Ai' (-X3) + X3 Ai (-X3)) + 0(82)]

(5.9)

- fffeGi (-x3) +ff+t (-X2 Gi' (- X3) - =X3 Gi (- X3) + 3) + 0 (8), 8 ~~~~5

where the bj are arbitrary real constants and {Qfj(E)}J=o is an asymptotic sequence as E 0 0. Again we note that exponentially growing terms (as x3 -o x) have been expressly excluded from (5.9).

Next we consider region IV, < x < 1, where we expect the solution to involve rapid oscillations. The governing equation is given by

(5.10) E Y'(x; E)+q(X)Y4(X; E)=f(X).

For sufficiently smooth functions q andf a particular solution of (5.10) can be expressed as

(5.11) y'(X; E)-=(x +O(E2).

In turn we assume that the homogeneous solution can be expanded as (following (4.4))

(5.12) yH= ( CjAj(e)) Y4(X; E) +C.C., j=0


where the cj are arbitrary complex constants and {Aj(E)}, o is an asymptotic sequence as E -*0. The function Y4(x; E) has the WKB expansion

(5.13) Y4(X; e) q(x) 14 L 8

where Ee(x) and Q1(x) are defined in (4.7) and (4.8), respectively. Matching of the expansion in region III for x3 -*X with that in region IV for

x -e + is carried out as described in ? 4 for the expansions in regions I and II. However, in the present case there is a crucial difference since the coefficients bj and the order functions ?lj remain unknown even after the matching. The expansion for y4 takes the form

Y4(X; ?) = O + 0(4)

(5.14) - /1r{(+)[{+lE(Q[1 + R(e)))+o(

)1E()+1


(5.15) R F)- R(s)ds-6(_)/-(_,,2

with R defined in (4.22). For future reference we note that to leading order

(5.16) ~ ~ ~~~ 0 + 0 ?) -, + ,,4 C (Ql () + (On,E( )xS)(1,c.c)

4 Re)+ s 0(j

and

(5.17) y4(1; ?)~f~N'1q s4 bjS(i, ) + ?11 4b0f10EC

where we set

(5.18) Cr(x )cos ( f (s) ds - )6 1 (- 4) s(? s ds +

as an extension of (4.27). To this point the relative orders of magnitude of the two terms on the right-hand side of (5.16) and, consequently, of (5.17), are not known.

It remains to construct expansions in regions V, VI, and VII which comprise the interval (1, 1). Since q(x)w> on this interval, the homogeneous solution of DDE (1.1) on (1, 1) can be expressed completely in terms of a single (WKB) expansion. We denote the homogeneous solution as yH(x; E) and expand it as


where the dj are arbitrary complex constants and {0j(E)}J- % is an asymptotic sequence as E -O. The function Y(x; E) has the WKB expansion (cf. (5.13))

(5.20) Y(x; ?) = E (x) [1 + 8QJ(X)+O(?2)] q (x) 1/4 L 8

To obtain a particular solution on (1, 1) we must consider each of the domains V, VI, and VII separately.

In region V, 1 < x < 1 + e, the governing equation is given by

(5.21) E2Y5(X; E) +q(X)Y5 (X; s) =+(X) - a (x)y( (- 1; E) -,(X)Y (X- 1; E).

To simplify matters, we express the function y(x - 1; E), which takes on values in either region I or II, as a composite expansion that is uniformly valid in region V. From (5.4) and (5.5) we obtain

(5.22) y(x-1; E) q(x 1) + ( qo )exp - -qo + 0(E).

Such a representation allows us to avoid introducing a boundary layer to the right of x = 1. Upon substituting (5.22) into (5.21) we obtain the particular solution

q(x)-SOq0 ̂ :X2q { /-1 'S(X; ?)=E ) (1 + 0 (E)) exp -/q J

+q(x) q(x)q e E

where we set

fx 1)1 q(x -1) (5.24) F(x)--O(x) - a(x)q(x -l)) -(x) q(x 1)'

and where we have simply recorded enough terms to give us the leading-order approximation when either a(x) 6 0 or a(x) 0. Clearly the exponential terms in (5.23) are negligible outside an O(E) neighborhood of x = 1.

In order to discuss the continuity conditions at x = 1, we will need

(5.25) Y5(l; ?) 4 Re do-/qo( _ q -E + ,

and

____1_ (Po -fo/ qo alq0 '13J-qo\ (5.26) Y (1; E) - Im do0o(E) + ( + E q, qo E2 E

Transition region VI centered at x = 1 + e, must be included to account for the fact that near x = 1 + e the shift terms in (1.1) reflect the behavior of the solution in the turning point region at x= {. In terms of the scaled variable

(5.27) _ x

Y6 (X6; E) satisfies the equation

(5.28) 36y"(X6; E) + q(1 + e + 6X6)Y6 (X6; E)

X) 2a(l + e+ 5X6) Y3(X6; E)-f(1 + e+ 6X6)Y3(X6; E) 36


An expansion for y' follows by expanding the coefficients in (5.28) for I8x6J << 1 and applying the obvious recursion formula. We obtain

Y6 (x6; e) = q(3+ X) { ( y3(x6; 0)(1 + 0(8))

(5.29)

+4((1 + e)-}(1 + )Y3(X6;-e)+ O(3)}

which matches with yp'(x; E). Finally, in the oscillatory region VII, 1 + e < x < 1, a particular solution satisfies

(520 &Y7(X; E) +q(x)y7(x; E) =+O(X)- a(x)y4(x -1; E)-(X)Y4(X -1 ; E),

where Y4 iS given in (5.14). Using the results from the Appendix in [6], we obtain

Y7'(X; e)

{ .(1-ia(x)q(x 1)1

3 e1/2q(x 1 i)/4 (1 + 0(e))) + (ZE bflj(e))

((1 + i)ax(x)q(x - 1)1/4 (1- _ i)E116f(X)

2Vi~ &5/6 (1 + 0I) _(1+0-)

*Ee(x -1) + c.c.} + (x)+ 0( E2).

We note that the matching of the expansions in regions III and IV guarantees the matching of yp and yA.

It is clear that the expansion in (5.31) becomes invalid at points where q(x)= q(x -1), x E (1 + f, 1). At such points the shift terms reflect the rapid oscillatory behavior in region IV and correspond to resonant forcing terms. Resonance phenomena in the context of BVP's for DDE's are discussed [6]. For example, if x = q E (1 + e, 1) is an isolated point where q(7) = q('q -1), then we would have to insert an 0(2?) "resonance" layer at x = i7 in region VII. The amplitude of the oscillations to the right of x= 77 would be enhanced. For simplicity we exclude the possible occurrence of resonance phenomena by assuming that q(x) $ q(x - 1), x E (1 + e, 1).

Now it remains to satisfy the boundary condition at x = I given in (1.2) and the continuity conditions at x = 1. Using (5.19)-(5.20) and (5.31) the boundary condition at x l becomes

= {(E, djj(e)) 1/4) [1+ Q+ ()+0(2)] +c.c.}

1 f{[lirf((1 i)alqt 1 +0(e))l (1+ O(

(5.32)~ ~ ~ ~~~- +( +EbQ,e) 02>/2- 5/ (lOe+0( > ) rlO() (/41 + 1 > 1 n 2 (5.32) 00 \/l1+i iaaqlil _________+

Ee( - ) +c.c +F1 O2.


We are now in a position to determine the unknown coefficients (bj and dj, j = 0,1,.**) and the unknown order functions (fQj(E) and 0j(E), j = 0,1, ). Two cases will be considered, namely, a (x) * 0 and a (x) 0. We note that we want nonitrivial bo (real) and do.

First we consider the case with a(x) * 0. From the boundary condition at x = I given by (5.32), we see that the possible dominant terms occur on the right-hand side and have the orders of magnitude: 0O(E), 1/3/2, and flo(E) 51/6. The condition that the solution y(x; E) be continuously differentiable at x = 1 translates into

(5.33) Y4(; ?) = Y5(; ?), y4'(1; ?) =y5'(1; ?),

where y4(1; E) and y5(1; E) are given by (5.16) and (5.25), respectively. The possible dominant terms in the first continuity condition have the orders of magnitude: E16fo1(?),

0o(E), and 1/ E. (Consideration of y'(1; E) leads to the same three terms, each multiplied by 1/E.)

We consider the various possibilities. If we assume that 00 is dominant in (5.32) or at least of comparable magnitude with E-3/2 and flo-5/6, then fo would be dominant in (5.33) and would lead to the requirement that do = 0. On the other hand, if Qios-5/6 is dominant in (5.32), then bo must equal zero unless its coefficient is zero which corresponds to an eigenvalue condition on E. A similar argument shows that E-3/2

cannot be the dominant term in (5.32) since in general we do not have fe = 0. Therefore, the only consistent balance is (without loss of generality)

(5.34) fo(E) 5/ = E -3/2 = no(?)-?2/3

with o(s) << E-3/2. From (5.32) we thus obtain (see (5.18))

(5.35) bo - - (l- 1,r/4)

(Note that the previously mentioned eigenvalue condition corresponds to the zeros of the denominator in (5.35).) Knowledge of flo(E) and bo determines the leading-order solution on 0 < x < 1.

Since 8a(E) = 0(1), the continuity conditions at x = 1 given by (5.33) require

(5.36) 00(? = ?

and

( * ) Re do - _ c} l~aj,-q0 q 1/4(00 _fol qo) (5.37) Re do= a1/q l4G0f/0 2(q1-qo)

(5.38) Im do= ajq0(400-f0/q0) 2q 1'4(q - qo)

The choice of 0O(E) = ?- leads to a difficulty in the boundary condition at x= 1 given by (5.32). There is no other term to balance this E?- term unless we choose

(5.39) fl(E) =

in which case

lIT al(q - qj-D)(f0/q0- 40)[J-q0 ql'4C1(l)+ q0Sj(l)/q /4] (5.40) b1= 141 aj,(q,aj_j) 14(ql - qo) Ce(1- 15 7T/4)


This completes our consideration of the a 0 0 case. In Fig. 10, we summarize the order of magnitudes of the solution in all the different regions.

For the case a (x) 0, a study of the boundary condition at x = 1 given by (5.32) and the continuity conditions at x = 1 given by (5.33) requires that

1 1 (5.41) fl?(E) = 2/31, o?(E) =

The unknown constants are then computed to be

(5.42) bo=!X4[ C (I1T/4) _3( q1' ) C8(l-1, 7/4)]

D qflF q1 q__-__l _ (5.43) Re do= --TSI (l)-8 q14 C '((l -) 2D q,_l q, -q,-l

(5.44) Im do= -2D L ?C1/- q q 1 q 1) 2D ql-l q, - ql-l

where

(5.45) D- ) _ )4

This completes the determination of the leading-order approximation of y(x; C). If we allow C to decrease towards zero, for discrete values of E we obtain D = 0. For these values of E, the above results are no longer valid and the condition D = 0 is simply the leading-order condition for the existence of an eigenvalue for the homogeneous problem, see the discussion in ? 2. In Fig. 11, we summarize the order of magnitudes of the solution in all the different regions.

I II III IV V VI VII

. I I I I | I i~I I I

I I l I o0(13) O(1/E'2/3)

0(1) 1 I O(1/.') I 0(1/r) I i O(l /v') I

smot l || multiphase

. I smooth t ? rapid oscillations rapid oscillations oscillations

x 0 1 l+e t

FIG. 11. Orders of magnitude of the solution to turning point problems for p(x) <0 and a(x) 0.

Finally, we note that in the special case when the functions p, a, /3, 4 are constants and qi =0 the above asymptotic results for both the cases a $ 0 and a = 0 agree with the exact results obtained in ? 3.

Appendix. Here we state a result from [7] on particular solutions of Airy equations forced by Airy and Scorer functions with shifted arguments and their derivatives. Airy and Scorer functions satisfy

(A.1) Y''(x)-xY,(x)= 1, 1=0,1


where 1=0 corresponds to the Airy functions Ai and Bi, and 1= 1 corresponds to Scorer functions Gi and Hi.

THEOREM. A particular solution of

(A.2) y "(x) - xy (x) = CY(n)(X-a,n=O ,2 ,a#O

where c is a constant and y(n)I 0 1 is the nth-order derivative of a solution of (A.1), is given by

(A.3) Y (x) =a[n Y()E -j)! j n =0, 1, 2,**.

Acknowledgments. We thank Mr. Robert Cheung and Mr. Mark Stewart who ably carried out the numerical computations for the figures.

REFERENCES

[1] M. ABRAMOWITZ AND I. A. STEGUN, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964.

[2] C. M. BENDER AND S. A. ORSZAG, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.

[3] J. K. HALE, Functional Differential Equations, Springer-Verlag, New York, 1971. [4] J. KEVORKIAN AND J. D. COLE, Perturbation Methods in Applied Mathematics, Springer-Verlag, New

York, 1981. [5] C. G. LANGE AND R. M. MiuRA, Singularperturbation analysis ofboundary-valueproblemsfordifferential-

difference equations, this Journal, 42 (1982), pp. 502-531. [6] , Singular perturbation analysis of boundary-value problems for differential-difference equations. II.

Rapid oscillations and resonances, this Journal, this issue, pp. 687-707. [7] R. M. MIURA AND C. G. LANGE, Particular solutions offorced generalized Airy equations, J. Math. Anal.

Appl., to appear.

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