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Applied Mathematical Sciences 1. John: Partial Differential Equations, 4th ed. 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential Equations, 2nd ed. 4. Percus: Combinatorial Methods.
5. von Mises/Friedrichs: Fluid Dynamics. 6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 8. Giacaglia: Perturbation Methods in Non-Linear Systems. 9. Friedrichs: Spectral Theory of Operators in Hilbert Space. 10. Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations. 11. Wolovich: Linear Multivariable Systems. 12. Berkovitz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential Equations. 14. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions.
15. Braun: Differential Equations and Their Applications, 3rd ed. 16. Lefschetz: Applications of Algebraic Topology. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I. 19. Marsdeni McCracken: The Hopf Bifurcation and its Applications. 20. Driver: Ordinary rnd Delay Differential Equations. 21. Courant/Friedrichs: Supersonic Flow and Shock Waves. 22. Rouche/Habets/Laloy: Stability Theory by Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II. 25. Davies: Integral Transforms and Their Applications. 26. Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems.
27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models-Rarity and Exponentiality. 29, de Veubeke: A Course in Elasticity. 30. Sniatycki: Geometric Quantization and Quantum Mechanics. 31. Reid: Sturmian Theory for Ordinary Differential Equations. 32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. 33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III. 34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics. 35. Carr: Applications of Centre Manifold Theory.
(continued on inside back cover)
K.W. Chang F.A. Howes
Nonlinear Singular Perturbation Phenomena: Theory and Applications
Springer-Verlag New York Berlin Heidelberg Tokyo
K.W. Chang
F.A. Howes
Department of Mathematics University of Calgary Calgary, Alberta Canada T2N 1N4
Department of Mathematics University of California Davis, California 95616 U.S.A.
Library of Congress Cataloging in Publication Data Chang, K.W. Nonlinear singular perturbation phenomena. (Applied mathematical sciences ; v. 56) Bibliography: p. Includes indexes. 1. Boundary value problems-Numerical solutions. 2. Singular perturbations (Mathematics) I. Howes, II. Title. III. Series: Frederick A. Applied mathematical sciences (Springer-Verlag New York Inc.); v. 56. QAI.A647 vol. 56 [QA379] 510 s (515.3'5] 84-14014
Printed and bound by R.R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America.
987654321 ISBN 0-387-96066-X Springer-Verlag New York Berlin Heidelberg Tokyo ISBN 3-540-96066-X Springer-Verlag Berlin Heidelberg New York Tokyo
Preface
Our purpose in writing this monograph is twofold.
On the one hand,
we want to collect in one place many of the recent results on the existence and asymptotic behavior of solutions of certain classes of singularly perturbed nonlinear boundary value problems.
On the other, we hope to
raise along the way a number of questions for further study, mostly questions we ourselves are unable to answer.
The presentation involves a
study of both scalar and vector boundary value problems for ordinary differential equations, by means of the consistent use of differential inequality techniques.
Our results for scalar boundary value problems
obeying some type of maximum principle are fairly complete; however, we have been unable to treat, under any circumstances, problems involving "resonant" behavior.
The linear theory for such problems is incredibly
complicated already, and at the present time there appears to be little hope for any kind of general nonlinear theory.
Our results for vector
boundary value problems, even those admitting higher dimensional maximum principles in the form of invariant regions, are also far from complete. We offer them with some trepidation, in the hope that they may stimulate further work in this challenging and important area of differential equations.
The research summarized here has been made possible by the support over the years of the National Science Foundation and the National Science and Engineering Research Council.
We offer each agency our sincerest
thanks for their generosity and consideration.
We also wish to thank our
colleagues and students who have shared their knowledge of and curiosity about singular perturbation theory with us, especially Bob O'Malley,
v
vi
Adelaida Vasil'eva and Wolfgang Wasow.
This monograph is but a small
token of our appreciation of their friendship and support.
K. W. Chang Calgary
F. A. Howes Davis
Contents Page
PREFACE CHAPTER I.
CHAPTER II.
v INTRODUCTION
1
Notes and Remarks
4
A'PRIORI BOUNDS AND EXISTENCE THEOREMS
6
Scalar Boundary Value Problems Vector Boundary Value Problems Notes and Remarks
13 17
SEMILINEAR SINGULAR PERTURBATION PROBLEMS
18
The Dirichlet Problem: Boundary Layer Phenomena 3.2. Robin Problems: Boundary Layer Phenomena Interior Layer Phenomena 3.3. Notes and Remarks
18 27 32 34
QUASILINEAR SINGULAR PERTURBATION PROBLEMS
37
The Dirichlet Problem: Boundary Layer Phenomena 4.2. Robin Problems: Boundary Layer Phenomena 4.3. Interior Layer Phenomena Notes and Remarks
37 49 55 59
QUADRATIC SINGULAR PERTURBATION PROBLEMS
61
Introduction The Dirichlet Problem: Boundary Layer Phenomena Robin Problems: Boundary Layer Phenomena 5.3. 5.4. Interior Layer Phenomena Notes and Remarks
61
SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS
91
Introduction A Dirichlet Problem Robin Problems: Boundary Layer Phenomena Interior Layer Phenomena A General Dirichlet Problem A General Robin Problem: Boundary and Interior Layer Phenomena 6.7. A Comment Notes and Remarks
91 93 95 96 98 101 104 105
SINGULARLY PERTURBED SYSTEMS
106
Introduction The Semilinear Dirichlet Problem The Semilinear Robin Problem The Quasilinear Dirichlet Problem Notes and Remarks
106 106 111 114 121
2.1. 2.2.
CHAPTER III.
6
3.1.
CHAPTER IV.
4.1.
CHAPTER V.
5.1. 5.2.
CHAPTER VI.
6.1. 6.2. 6.3. 6.4. 6.5. 6.6.
CHAPTER VII.
7.1. 7.2. 7.3. 7.4.
vii
61 76 83 90
viii
Page CHAPTER VIII.
EXAMPLES AND APPLICATIONS
123
Part I - SCALAR PROBLEMS
123
8.1. 8.2. 8.3.
8.4.
Examples of Semilinear Problems and Applications Examples of Quasilinear Problems and Applications Examples of Quadratic Problems and Applications Examples of Superquadratic Problems and An Application
Part II - VECTOR PROBLEMS B.S.
8.6.
Examples of Semilinear Systems and An Application Examples of Quasilinear Systems and An Application
123 132 141
155 161
161
165
REFERENCES
171
AUTHOR INDEX
177
SUBJECT INDEX
179
Chapter I
Introduction
We are mainly interested in quasilinear and nonlinear boundary value problems, to which some well-known methods, such as the methods of matched asymptotic expansions and two-variable expansions are not immediately applicable.
For example, let us consider the following boundary value
problem (cf. O'Malley [75], Chapter 5)
EYof = Y12,
0 < t < 1,
(A)
y(0,0 = l,y(l,c) = 0.
(B)
In general, it is not obvious that such a nonlinear boundary value problem will have a solution in of
e.
[0,1]
for all sufficiently small values
However, in this case, we can obtain by quadratures the following
exact solution in
[0,1]
y(t,c) = -c ln[t + e-1/c(1-t)]
which is defined for all positive values of An important feature of this solution tion of
(t,e), it behaves nonuniformly as
e.
y(t,e) t
and
is that, as a funcc
approach
0, that
is,
lim+ y(t,e) = 0
for each fixed
t > 0
(1.1)
lim+ y(t,c) = 1 t_0+
for each fixed
c > 0.
(1.2)
but
For decreasing values of
c, the solutions
Figure 1.1.
1
y(t,c)
are as shown in
2
I.
INTRODUCTION
y
0
1
Figure 1.1
From the graph or from the relations (1.1), (1.2), it is clear that, as
a -
to
t
val
0+,
the solution
y(t,e)
approaches
on each closed subinterval of Note that
[0,1].
y(t,e)
u(t) - 0
[6,1], for fixed
on
(0,13
0
uniformly with respect
but not on the whole inter-
(which is the limit of the solution
6 > 0) turns out to be the solution of the
corresponding reduced equation U'2 = 0
of the original equation (A) satisfying the right-hand boundary condition
(B) In order to illustrate the difficulties associated with the application of the methods of matched and two-variable asymptotic expansions, we assume, for the moment, that the exact solution of problem (A), (B) is not known, and so we proceed formally. methods is that the solution of (A), series in
The fundamental hypothesis of these (B) can be described by two power
c, known as the inner and outer expansions.
The outer expan-
sion represents the solution away from regions of nonuniform behavior and is simply a power series in
whose coefficients are functions of
a
t.
On the other hand, the coefficients of the inner expansion are functions
not only of
t
but also of a "stretched" variable
T = e
which can be
arbitrarily large as
a - 0+
(cf. [55], Chapter 2;
[75], Chapter 1) can be regarded as a rescaling
for a certain range of
t.
The variable
parameter which has the effect of enlarging the region of nonuniform behavior.
To fix these ideas, let us examine first the outer expansion
y0(t,e), that is, we substitute
T
Introduction
3
ao
enun(t)
YO(t,c) = u0(t) + eu1(t) + 0
into the differential equation like powers of
e.
ey" = y'2
and equate coefficients of
The first two terms of y0
are easily shown to sat-
isfy the equations u02
= 0
(1.3)
and
u"=u'2 1 1
(1.4)
The solutions of (1.3) are
u0(t) = constant, while the problem of
solving (1.4) is essentially equivalent to solving the original problem We note that
(A), (B). (1.4).
ul(t) = constant
If we require that
y0(t,e)
is one family of solutions of
satisfy one of the original boundary
conditions (B), then the obvious choices for the functions
u0
and
u1
are
u0(t) a 0,
ul(t)
0
(if
y0(l,e) = 0)
u0(t) E 1,
ul(t) = 0
(if
YO(01c) = 1).
y0(l,e) = 0
The first choice here (cf.
If the outer expansion
lows.
turns out to be the proper requirement
[75], Chapter 5) and it can be motivated geometrically as foly0(t,E)
were to satisfy the second choice
y0(0,e) = 1, then we would anticipate that 1im+ y(t,c) = 1, e-*0
In a neighborhood of
for
t
in
t = 1, y(t,e)
[0,1-6], where
d > 0.
must decrease rapidly from one to
zero in order to satisfy the other boundary condition the equation (A) requires that y" < 0.
y'
y(l,e) = 0.
Since
never change sign, we must have
This however is impossible since
ey" = y'2 > 0.
Consequently,
we must select the outer expansion which satisfies the boundary condition y0(l,c) = 0
at
t = 1.
In this case, the outer expansion
zero (up to terms of order 5)
y0(t,e) = -E In t
is also an outer expansion and Y0(I,E) = 0.
y0(t,e)
should be identically
e2); however, the function (cf. [75], Chapter
4
INTRODUCTION
I.
This function has a singularity at
t = 0
for
c > 0
and so one might
be tempted to reject it out of hand as an approximation to The surprising fact is that this function follows from the exact solution. t = 0
y0
y
on
(0,1].
is the outer expansion, as
Indeed, the singularity of
y0
at
is precisely what is needed to cancel the singularity of the inner
expansion there.
The construction of the inner expansion of the solution
y
is
equally fraught with difficulties, as it is not obvious at all what the correct stretched variable
t
should be.
standard change of variable (cf. t =
0
To see this we could make the
[55], Chapter 2;
and
4'(c)
-
0+
[75], Chapter 5)
as
and attempt to determine the asymptotic character of the terms of the transformed differential equation.
c - 0+, *(e)
by comparing
Clearly
ey" = y'2
is equivalent to C
d2y
1
*2(c)
`dt)
and so the change of variable accomplishes nothing.
Using various de-
vices, O'Malley ([75], Chapter 5) is able to construct an inner expansion which in fact has a singularity at larity of the function
y
there.
t - 0
that just cancels the singu-
His methods are nevertheless not ob-
vious a priori, and it is quite conceivable that more complicated problems of the form (A), (B) involving
y'2-nonlinearities could not be solved to
such a degree.
We can, however, solve this particular problem using our method, but rather than discussing this method now, we will defer it until Chapter V, when we deal with a much larger class of related boundary value problems. Notes and Remarks. 1.1.
The methods of matched asymptotic expansions and two-variable expansions have been very successful in solving a variety of difficult problems in engineering and applied science, and they continue to be two of the most powerful weapons in the arsenal of applied mathematicians.
Our discussion of the nonlinear problem (A),
(B) (and
indeed, the point of view taken in this monograph), is not meant to denigrate, in any way, the utility and importance of these stalwarts of asymptotic analysis.
Rather we wish to study certain types
of boundary value problems, for which precise results on the exist-
introduction
5
ence and asymptotic behavior of solutions can be derived from mathematical analysis. 1.2.
Alternative approaches to many of the problems discussed in this book can be found in the monographs of Wasow ([93], Chapter 10), Vasil'eva and Butuzov [88], O'Malley [7S], Habets [28], Habets and Laloy [31], Eckhaus [21] and Kevorkian and Cole [55], and in the survey articles of Vasil'eva [87], Erdelyi [23], O'Malley [73], Carrier [8] and Wasow [94].
In addition, these monographs and
papers contain a wealth of references.
Chapter II
A'priori Bounds and Existence Theorems
§2.1.
Scalar Boundary Value Problems Before discussing in detail the various classes of singularly per-
turbed boundary value problems, let us give an outline of the principal method of proof that we will use throughout.
This method employs the
theory of differential inequalities which was developed by M. Nagumo [66] It enables one to prove the existence
and later refined by Jackson [49).
of a solution, and at the same time, to estimate this solution in terms of the solutions of appropriate inequalities.
Such an approach has been
found to be very useful in a number of different applications (see, for It will be seen that for the general classes of
example, [5] and [83]).
problems which we will study in later chapters, this inequality technique leads elegantly (and easily) to some fairly general results about existence of solutions and their asymptotic behavior.
Many results which have
been obtained over the years by a variety of methods can now be obtained by this method, which we hope will also very clearly reveal the fundamental asymptotic processes at work.
Consider first the general Dirichlet problem x" = f(t,x,x'),
a < t < b, (DP)
x(a) = A,
in which
f
x(b) = B,
is a continuous function on
[a,b] xlR2.
The differential
inequality approach of Nagumo is based on the observation that if there exist smooth (say twice continuously differentiable or a(t)
and s(t)
possessing the following properties:
6
C(2)-) functions
2.1.
Scalar Boundary Value Problems
7
a(t) < B(t)
a(a) < A < B(a),
a(b) < B < 8(b) S" f(t,a,a'),
then the problem (DP) has a solution such that
a(t) < x(t) < S(t)
for
in
6(t)
x'.
C(2) ([a,b])
[a,b], provided that
does
f
More precisely, it is suffici-
satisfies what is known as a generalized Nagumo
f
condition with respect to every solution
of class
x = x(t) t
not grow "too fast" as a function of ent to require that
(P)
a
and
of
x = x(t)
there exists a constant
a(t) < x(t) 0.
glx(b) + g2x'(b) = B,
pi, qi
Note that if
satisfy
p2, q2 > 0, pi + p2 > 0
p2 = q2 = 0
and
and
pl = qi = 1, then the prob-
lem (RP) reduces to (DP), and so we are really interested in the case p2 + q2 > 0.
when
It turns out fortunately that the Nagumo theory for
(DP) can be extended with the obvious modifications to the problem (RP), as was observed by Heidel [36].
bounding functions
a
and
That is, if there exist piecewise
-C(2)
(a < s) which satisfy the above differen-
D
tial inequalities and the boundary inequalities
pla(a) - p2a'(a) < A -M + m(M/m) > 0 = c!!".
It follows from Theorem 2.1 that for each has a solution
y(t,e)
on
[a,b]
c > 0
the problem (2.1), (2.2)
satisfying
-M/m < y(t,c) < M/m.
The uniqueness of the solution follows from the maximum principle.
10
II.
A'PRIORI BOUNDS AND EXISTENCE THEOREMS
If we assume that
A > 0, B > 0
-M < f(t,0,0) < 0
in
and
[a,b],
then by the proof of Lemma 2.1, we will obtain the following more precise estimate of the solution: 0 < y(t,e) < M/m.
Similarly, if we assume that
A < 0, B < 0
and M > f(t,0,0) > 0
[a,b], then the solution of (2.1), (2.2) satisfies
in
M/m < y(t,e) < 0.
These results suggest the following modifications of Lemma 2.1. Lemma 2.2. t,y,y'
Assume that the function
and of class
[a,b] x]R2.
C(n) (n > 2)
Assume also that
is a positive constant and
m
3nf(t,y,0) > m > 0
with respect to
y
for
(t,y,y')
8yf(t,0,0) > 0
for
1 < j .S n-i
in
[a,b] x]R.
Then for each
the problem (2.1), (2.2) has a solution
y = y(t,e)
in
0 < y(t,e) < (n!m
(t,y)
[a,b]
e > 0,
satisfying
IM)I/n,
where M = max{ max
If(t,0,0)I, (mlAI/n!)n, (mIBI/n!)n}.
[a, b]
Remark. Proof:
We use the notation Define for
a(t) B 0
t
and
in
a
n
2nf/Byn.
for
f
[a,b]
S(t) _ (n!m
lM)1/n
Clearly, a < g, a(a) < A < $(a), a(b) < B < 0(b), and by virtue of our assumptions SB" < f(t,0,B')
on
A, B
and
f.
ca" > f(t,a,a')
Finally, we see that
since n-l
f(t,0,0) = f(t,0,0) +
E
j=1
in
A > 0, B > 0, f(t,0,0) < 0, and that there
such that for
is continuous with respect to
f
81f(t,0,0)131/j! y
+ 9nf(t,n,0)8n/n!
> -M + (m/n!)n! M/m > 0.
The conclusion of Lemma 2.2 follows by virtue of Theorem 2.1. If we make the change of dependent variable
y
-y
and apply
Lemma 2.2 to the transformed problem, we can obtain an analogous result
Scalar Boundaxy Value Problems
2.1.
for the case
A < 0, B < 0
and
11
f(t,0,0) > 0.
This is given in the next
lemma.
Assume that the function
Lemma 2.3.
and of class
t,y,y'
C(n) (n > 2) with respect to
Assume also that
[a,b] x1R2.
and
1 < j0(je) < n-1
(even), for
(t,y)
Then for each
integer.)
in
y = y(t,e)
such that
2
j0(je)
8y f(t,y,O) > m > 0
[a,b] x1R.
in
for
y
in
(t,y,y')
A < 0, B < 0, f(t,0,0) > 0, and that
there is a positive constant m for
is continuous with respect to
f
(< 0)
(< -m < 0) if n
is odd
denotes an odd (even)
j0(je)
(Here
f(t,0,0) > 0
c > 0, the problem (2.1), (2.2) has a solution
satisfying
[a,b]
-(n:m 1M)1/n < Y(t,t) < 0,
where M
is as defined in the conclusion of Lemma 2.2.
In the previous lemmas, we imposed strong conditions on the partial
f with respect to
derivatives of
In the lemma below we will relax
y.
our conditions on the partial derivatives of
f
with respect to
we will impose a condition on the partial derivative
y, but
and thereby
fy
obtain virtually the same result as in Lemma 2.1. Lemma 2.4. t,y,y'
Assume that the function
and of class
k > 0
for
is continuous with respect to
f
with respect to
for
y,y'
(t,y,y')
Assume also that there are positive constants
[a,b] x1R2.
such that
C(1)
If (t,y,0)I < R in
(t,y,y')
for
[a,b] x1R, and
in
(t,y)
Then for
[a,b] x1R2.
lem (2.1), (2.2) has a solution y = y(t,c)
in
and
k
If ,(t,y,y')I >
0 < e < k2/4t, the prob[a,b]
1)
(if
fy, < 0)
ly(t,c)I < YR 1(2eA(a-t) - 1)
(if
fy, > 0).
ly(t,e)I < YR-1(2ea(t-b) -
R
in
satisfying
or
Here for
+ 0(e)
A = -ft-
is a negative root of the polynomial
0 < c < £0 < k2/4R, Y = max{ max
cX2 + kA + R
If(t,0,0)I, 1IAI(2eX(a-b)-1),
[a, b]
RIBI}
if
fy, < 0, and
y = max{ max
If(t,0,0)I, LIAI,
[a,b]
RIBI (2eX(a-b) Proof:
- 1)
if y, > 0.
Suppose for definiteness that y , < -k < 0.
0 < £ < £0
and
t
in
[a,b]
Define for
12
A'PRIORI BOUNDS AND EXISTENCE THEOREMS
II.
a(t,c) =
-YR-1(2e (t-b)
- 1)
and YR-1(2e(t-b)
0(t,c) =
- 1).
Then, clearly a < 0, a(a) < A < $(a)
and
We will only verify that
choice of y.
.Q" < f(t,0,01)
the verification that
a(b) < B < 0(b)
ca" > f(t,a,a') in
by our (a,b), as
in
proceeds analogously.
(a,b)
Differentiating and substituting, we have ca" - f(t,a,a') = ca" - f(t,0,0) - {f(t,a,0) - f(t,0,0)} - {f(t,a,a') - f(t,a,0)} = call - f(t,0,0) - fy(t,g,0)a - y,(t,a,n)a' > -ea2YR-12e;k(t-b) - Y - kYR-12e)L(t-b) + Y kXYk-12ea(t-b) -
= 0,
The conclusion of the lemma in the case that
sat + kA + R = 0.
since
now follows from Theorem 2.1.
fy, < -k < 0
and
0 < s < s0
define for
a(t,c) =
-YR-1(2eA(a-t)
t
in
- 1),
If
fy, > k > 0
then we
[a,b]
s(t,e) = -a(t,c),
and proceed as above.
We consider finally a sufficient condition for the existence of a solution of the problem (2.1), (2.2) which includes the assumptions fy > 0
and
If
l
> 0
of Lemmas 2.1 and 2.3, respectively.
The next
lemma is due essentially to van Harten [33].
Assume that the function
Lemma 2.5. t,y,y'
and of class
[a,b] xlR2.
constant in
m
C(1)
f
is continuous with respect to
with respect to
y,y'
Assume also that there are a constant
[a,b] x1R2
and
v
(t,y,y')
0 < c < El.
y = y(t,c)
Then for in
[a,b]
in
and a positive
fy(t,y,0) + Vf y,(t,y,y') > m + ev2
such that
(2.2) has a solution
for
for
(t,y,y')
0 < c < ell the problem (2.1), satisfying
jy(t,c)I < m-1Net, where N = max{ max
If(t,0,0)e-vtl,
mIAje-av,
mlBle-bv}.
[a,b]
Proof:
The lemma follows by making the change of variable
y = zevt
and
Vector Boundary Value Problems
2.2.
13
applying Lemma 2.1 to the resulting problem for z(a,c) = Ae-av, z(b,c) = Be-bv, where vfy,(t,zevt,n)
-
z, namely
ez" = F(t,z,z',e),
F(t,z,z',e) = f(t,O,O)e-vt +
ev2}z + {fy,(t,zevt,n) - 2ev}z'.
We note that if
while if and
f > m > 0, then we may take v = 0 in Lemma 2.5, y Ify,I > k > 0, then we may take v such that sgn v = sgn fy, for
IvkI > R
t = supIfy(t,y,0)I, in order to derive the results of
Lemmas 2.1 and 2.4, respectively.
Lastly we wish to point out that analogous results hold for the Robin
problem a < t < b,
ey" = f(t,y,y'),
p1Y(a,e) - p2Y'(a,e) = A,
In particular, if
(2.3) (2.4)
g1Y(b,c) + g2Y'(b,e) = B.
f(t,y,y') = O(Iy'12)
as
(y'I
-i- -
for
(t,y)
in
[a,b] x1R, and if pl = ql = 1, then Lemmas 2.1 - 2.4
compact subsets of
hold verbatim for (2.3), (2.4), as the reader can easily verify.
Vector Boundary Value Problems
§2.2.
Analogous results also hold for vector boundary value problems.
The
existence and comparison theorems for vector problems can be regarded as higher dimensional forms of Nagumo's scalar theory.
Unfortunately, how-
ever, the assumptions which are imposed for vector problems are more difficult to verify in practice.
This is due, on the one hand, to our
limited experience in treating boundary value problems for systems of On the other hand, systems of differential equa-
differential equations.
tions are inherently more complicated than scalar equations, and so at best, we can only hope to mimic the scalar theory.
The results which
follow are taken mostly from the papers of Kelley [52], [53], although much of the early work was done by Hartman, [34], [35; Chapter 12] and others (see [5] and [83] for further references).
Consider then the boundary value problem x" = F(t,x,x'),
a < t < b, (DP)
x(a) = A,
where
x(b) = B,
x, A and
B
are vectors in IRN
vector function which is continuous on
and
F = (F1,...,FN)T is an
[a,b] x]R 2N
.
N-
It turns out that
the scalar Nagumo theory can be extended to (DP), provided that the vector function
F
satisfies a growth condition (Nagumo condition) with respect
14
to
II.
A'PRIORI BOUNDS AND EXISTENCE THEOREMS
In this vector setting, we say that
x'.
F
satisfies a Nagumo con-
dition if it satisfies one of the following two conditions, for in compact subsets of
and for all
z
in ]RN
(t,x)
(cf. [53]):
There exist positive, nondecreasing, continuous functions
(1)
on
[a,b] x 1I
such that each component
(0,co)
of
Pi, i = 1,...,N
F
$i
satisfies
IFi(t,x z)I 0
or
Y"(t) < (xT/I Ixl I)F(t,x,x'), whenever
y(t) = IIxII
tions are imposed on
and
y1(t) _ (xT/IIxII)x'.
F, then
y(t)
If appropriate assump-
can be~determined.
Suppose, on the other hand, we wish to obtain bounds on the individual components of the solution of (DP); then we obviously need more than one comparison function. P£(t,x) = x£ - s£(t)
where
a£ < S£, for
To this end, we define the and
2N
functions
P£+N(t,x) = -x£ + a£(t),
£ = 1,...,N.
well as the boundary inequalities
If these functions satisfy (2.5), as
16
A'PRIORI BOUNDS AND EXISTENCE THEOREMS
II.
at (a) < AR < 0t (a) ,
at (b) < B1 < at (b) ,
then we obtain two-sided bounds of the form
at(t) < xt(t) < St (t) t = 1,...,N, on
for
(cf. Theorem 2.1).
[a,b]
The reader can verify
that the functions above will satisfy (2.5), if at
and
satisfy,
at
respectively, the inequalities aR > FI(t,x,x')
when
xt = at(t),
xt'
gR < Ft(t,x,x')
when
x1 = st(t),
x't = 01(t),
= aR(t)
and
for all
xi (i 0 t) in
[ai(t),si(t)]
and all
x! (i 0 t)
in
IR.
By means of different types of invariant regions in Theorem 2.4, we can prove a priori existence and comparison results for the general singularly perturbed vector problem
Ey" = F(t,y,'), y(a,E) = A,
a < t < b,
y(b,E) = B,
and these can be regarded as analogous to Lemmas 2.1 - 2.5.
The analysis
is rather tedious, although straightforward, and we leave the precise formulation of these results to the reader.
The last existence and comparison theorem of this chapter deals with the following vector Robin problem a < t < b,
x" = H(t,x),
(RP)
Px(a) - x1(a) = A,
where
and H
x, A and
B
are
Qx(b) + x'(b) = B,
N-vectors, P, Q
are constant
(N x N)-matrices,
N-vector function defined and continuous on
is an
[a,b] x1RN
(We have limited ourselves to discussing only systems of singularly perturbed Robin problems whose right-hand sides do not depend on any derivatives, but of course we could have discussed more general systems.) If the matrices
P
and Q
are positive semidefinite in the sense that
there exist nonnegative scalars and
xTQx > gIIxII2, for any
x
p
and
q
such that
a result analogous to Theorem 2.3.
Since we will only seek bounds on
the norm of a solution of (RP), we call a set p(t,x) < 0}
xTPx > p1Ixi12
in IRN, then it is possible to prove
I = {(t,x)
in
[a,b] xe:
an invariant region for (RP) if the scalar function
the following three properties:
p
has
2.2.
Vector Boundary Value Problems
is of class
(1)
p
(2)
pp(a,x(a)) - pp(a,x(a)) < 0,
17
C(2) (1);
qp(b,x(b)) + pp(b,x(b)) < 0; (3)
p" > 0
F.)
Theorem 2.5.
whenever
I
p', p"
(The functions
in place of
in
p = 0
and
are as defined in (2.5), with the function
Then we have the following result (cf.
p(t,x(t)) < 0
in
x = x(t)
H(t,x)
[52], [57]).
Assume that there exists an invariant region
Then the Robin problem (RP) has a solution such that
p' = 0.
for (RP).
I
of class
C(2)([a,b])
[a,b].
This theorem will be used in Chapter VII to estimate the norms of solutions of singularly perturbed Robin problems.
There we will show,
under appropriate assumptions, how to construct functions of the form p(t,x,c) = IIx11 - y(t,e)
which satisfy the conditions (1) - (3).
Notes and Remarks 2.1.
Most of the differential inequality and invariant region results quoted in this chapter can be found in the monographs of Bernfeld and Lakshmikantham [5] and Schr6der [83].
These works also contain
many additional references to the relevant literature, as well as instructive illustrations of other applications of inequality techniques. 2.2.
M. Nagumo in a paper [67] published in 1939 was the first mathematician to apply differential inequalities in the study of a singular perturbation problem, namely the initial value problem f(t,y,y',e), 0 < t < T < co, y(0,e), y'(0,e)
ey" =
prescribed.
This paper
was overlooked until the Soviet mathematician N. I. Bris in [7] used Nagumo's results to study the singularly perturbed boundary problem g2y'(b,e)
ey" = f(t,y,y',E), Ply(a,c) - P2y'(a,e), gly(b,E) + prescribed, by means of a shooting technique.
Most, if
not all, of the present monograph is based on these two seminal papers; the reader would do well to consult them or the brief survey [45].
2.3.
It is possible to extend Theorem 2.5 to the Robin problem for the differential equation
x" = F(t,x,x')
and to more general invari-
ant regions of the type considered in Theorem 2.4; cf.
[52] or [57].
Chapter III
Semilinear Singular Perturbation Problems
The Dirichlet Problem:
§3.1.
Boundary Layer Phenomena
We consider first the semilinear Dirichlet problem ey" = h(t,y),
a < t < b,
y(a,c) = A,
y(b,c) = B,
(DP 1)
where
c
is a small positive parameter and prime denotes differentiation
with respect to are:
Some natural questions to ask regarding this problem
t.
Does the problem have a solution for all small values of
c?
Once
the existence of a solution has been established, how does the solution
behave as
c -+ 0+?
The answers to these questions depend greatly on the function (and also on the boundary values
A, B, if h
h
is a nonlinear function),
as we shall see by examining two simple linear equations in
(0,1)
cyil = y
(El)
cy1t = -Y
(E2)
and
subject to the boundary conditions y(O,e) = 1,
Y(l,e) = 2.
The general solution of (E 1) is
y = Y1(t,e) = cl
c2
and the general solution of (E2) is
y = y2(t,c) = cl cos(t/v) + c2 sin(t//),
18
The Dirichlet Problem:
3.1.
where
and
c1
c2
Boundary Layer Phenomena
are arbitrary constants.
19
Using the boundary conditions
to determine these constants, we find that the solutions are, respectively,
- el/re )-1{(e-1/v - 2)et/re Y1(t,e) _ (e-1/Y'
(2-el/re )e-t1r.-
+
2e-(l-t)/,re- +
and
y2(t,e) = cos(t/re) + {2 - cos(l/v)} Let us examine these solutions more closely. is defined for all
e > 0
y1(t,e) = 0
lira
For (E1), the solution
yl
and, moreover,
for
S < t 0 e'0
E-0 and
lim+ lira- y1(t,e) = 2 # 0 = lira
lim+ y1(t,e)
t->l- c0
t-rl
We note that by setting
(cf. Figure 3.1).
0 - h(t,u)
reduced equation
c = 0
whose solution is
in (E1) we obtain the
u - 0.
In the case of (E2), first of all we see that the solution is only defined if 2,...
Thus, for these particular values of
.
Suppose then
solution. Y2
re # (nx)-1, for
sin(1/ f) # 0, that is, if
a
is very small and
y2(t,e)
n = 1,
e, the problem (E2) has no re # (nw)-1.
The function
is a linear combination of two oscillatory functions of arbitrarily
large arguments, and therefore it is densely oscillatory with period 1 and with bounded amplitude (cf. Figure 3.2). for
u a 0
y2
Clearly, it is impossible
to satisfy a limiting relation such as (3.1) above, even though is again the solution of the reduced equation obtained from (E2)
by setting
e = 0.
What sets these two problems apart is of course the
difference in the sign of the coefficient of
y.
Let us now consider the third example in which function of
y
ey"" = Y2,
in
h
is a nonlinear
(0,1),
Y(O,e) = A,
y(l,c) = B.
(E3)
20
III.
SEMILINEAR SINGULAR PERTURBATION PROBLEMS
Figure 3.1
Figure 3.2
This example illustrates how the boundary values the existence and the behavior of solutions as
A and
B
can affect
From our con-
c + 0+.
sideration of the example (El), we expect that for sufficiently small e > 0, the solution of (E3) must remain close to zero, the unique solution of the reduced equation t = 0
and
t = 1.
y2 = 0, except possibly near the endpoints
However, in the neighborhood
of
t = 0
and
t = 1,
the solution must be convex (i.e., y" > 0), as dictated by the differc-ly2
ential equation or
y" =
> 0.
is negative, we would have
B
spectively; see Figure 3.3.
are nonnegative, we have
3.4 shows. tion
near
t = 0
t = 1, re-
On the other hand, if both
e.
y" > 0
For such values of A
y = y3(t,c)
y" < 0
or
In these cases, the problem has no solution
for sufficiently small values of B
A
Thus, if either the boundary value
near and
t = 0
and
A
and
t = 1, as Figure
B, the problem (E3) has a solu-
for all sufficiently small values of
e, and
N
y" 0,
we are able to deduce the sign restrictions on the allowable boundary values
A,B.
If we apply this observation and reasoning to the same problem for the differential equation
ey" = y3, we conclude that the problem has a
solution for any value of A
and
B.
Rather than dealing with specific cases, we study now the existence of solutions
y = y(t,c)
of the general problem (DP1) (and related ones)
which behave like the solutions of (E1) and (E3) in the sense that lim+ y(t,E) = u(t) Ey
where
u = u(t)
h(t,u) = 0,
in each closed subinterval of
(a,b),
(3.2)
is a certain solution of the reduced equation a < t < b.
(R1)
If the relation (3.2) holds, we can say that the reduced solution u(t)
is "stable" with respect to the original solution
y(t,E).
To be
III.
22
SEMILINEAR SINGULAR PERTURBATION PROBLEMS
precise, we wish to give explicit definitions of stability for the solution
u(t).
Let
be a solution of (R1) which is continuous in
u = u(t)
[a,b], and let us define the following three domains D0(u) = {(t,Y): a < t < b,
where
d(t)
DO(u)
U1(u), V2(u):
(Y - u(t)l < d(t)),
is a positive continuous function such that
d(t) s IA - u(a)l + 6,
for
d(t) =-d,
for a+6 u(a)
B > u(b), we define
and
0 < y - u(t) < d(t)},
D1(u) _ {(t,y): a < t < b,
and if A < u(a)
B < u(b), we define
and
D2(u) = {(t,y): a < t < b,
where
d(t)
b - 6/2 < t < b.
-d(t) < y - u(t) < 0},
is as above.
In the following definitions of stability for the solution we assume that the function
h(t,y)
has the stated number of continuous
partial derivatives with respect to that
q > 0
and
Definition 3.1. [a,b]
y
in
Di(u), i = 0,1
or
2, and
are integers.
n > 2
The function
u = u(t)
is said to be
if there exists a positive constant
ayh(t,u(t)) = 0
u(t),
a < t < b
for
m
and
(Iq)-stable in
such that 0 < j < 2q,
and
a2q+lh(t,Y) > m > 0 Definition 3.2. [a,b]
in
The function
u = u(t)
is said to be
if there exists a positive constant m ayh(t,u(t)) = 0
for
a2q+lh(t,u(t)) > m > 0 (A-u(a))ayq+2h(t,y) > 0 and
V0(u).
a < t < b for in
and
(Iq)-stable in
such that 0 < j f 2q,
a < t < b, DO(u) fl ([a,a+6] xlR),
The Dirichlet Problem:
3.1.
Boundary Layer Phenomena
(B-u(b))92q+2h(t,y)
in
> 0
23
V0(u) f1 ([b-d,b] xlR).
The above definitions are motivated by boundary value problems for the differential equation Definition 3.3. if
[a,b]
ey" = y2q+1
The function
(cf.
u = u(t)
u(a) < A, u(b) < B
(E I) above).
is said to be (IIn)-stable in
and if there exists a positive constant
m
such that
3 h(t,u(t)) > 0
and
a < t < b
for
1 < j< n-1,
and
anh(t,y) > m > 0 Definition 3.4. [a,b]
m
if
in
VI(u).
The function
u = u(t)
is said to be (11)-stable in
and if there exists a positive constant
u(a) < A, u(b) < B
such that 3yh(t,u(t)) > 0
and
a < t < b
for
1 < j < n-1,
anh(t,u(t)) > m > 0
for
3Y+lh(t,y) > 0
V1(u) fl [([a,a+d) U (b-6,b]) xIR].
a < t < b,
and in
The above definitions are motivated by boundary value problems for the differential equation Definition 3.5.
if
[a,b]
m
ey" = y2n
The function
u(a) > A, u(b) > B
(cf.
u = u(t)
(E3) above).
is said to be (111)-stable in
and if there exists a positive constant
such that
j0(je) h(t,u(t)) > 0
8y
where
j0(je)
if
[a,b]
m
for
a < t < b
and
1 < j0, je < n-1
denotes an odd (even) integer, and
anh(t,y) < -m < 0 Definition 3.6.
(< 0)
(> m > 0)
The function
u(a) > A, u(b) > B
in
u = u(t)
V2(u),
if n
is even (odd).
is said to be (III n)-stable in
and if there exists a positive constant
such that
2y
0e) h(t,u(t)) > 0
(< 0)
for
a < t < b
and
1 < j0,
je < n-1,
III.
24
SEMILINEAR SINGULAR PERTURBATION PROBLEMS
anh(t,u(t)) < -m < 0
(> m > 0)
3 h(t,y) > 0
in
a < t < b
for
if n
is
even (odd),
and ( (2gmm1
(t,a(t,e))
and
e, say
are all positive functions, we have + w22q+1 + r2q+l
and so call - h(t,a(t,e)) > au" - ewL - ew" + (2gm1).(w2q+1+w22q+l+r2q+1)
> -eIu"I + cy
(2q+1)!
by the definitions of wL, wR, and we obtain
call > h(t,a).
r.
Thus, by choosing y > lu"I(2q+1)!,
26
III.
The case that
u(t)
SEMILINEAR SINGULAR PERTURBATION PROBLEMS
(Iq)-stable can be treated analogously, by
is
carrying out the Taylor expansion to
terms; the details are
(2q+2)
left to the reader.
If the reduced equation (R1) has a (IIn)- or (IIn)-stable solution, we have the following result. Theorem 3.2.
Assume that the reduced equation (R1) has a (,In)- or (fn)-
the result analogous to Theorem 3.10 is valid provided that AIu'(a) > A, u(b) + w2u'(b) 2.B, u" < 0
in
u(a) -
(a,t0) U (t0,b)
and
u'(to) > u'(t0).
Notes and Remarks 3.1.
The theory of this chapter applies with little change to the more general problem
cy" - h(t,y,e), a < t < b, y(a,e) = A(c), y(b,c) _
B(e).
We need only require that
(t,y)
in
Di(u) (i = 0,1,2)
B(c) = B(0) + 0(1)
h(t,y,c) = h(t,y,O) + o(1)
and that
for all sufficiently small values of
for
and
A(c) = A(0) + 0(1) e.
Notes and Remarks
3.2.
35
The definitions of (Iq)-, (IIn)- and (III n)-stability were intro-
duced by Boglaev [6] and used by him to study the Dirichlet problem Earlier Bris proved Theorem 3.1 in the case of (lo)-stabil-
(DP1).
Among the other work done on the problem (DP1) we mention
ity.
only the papers of Tupchiev [86], Vasil'eva [87], Vasil'eva and Tupchiev [89], Carrier [8], Fife [24], O'Malley [76], Dorr, Parter and Shampine [20], Habets [29], Habets and Laloy [31], Flaherty and O'Malley [25] and Howes [39].
The Robin problems (RP1) and (RP2)
have also been considered by Habets and Laloy in the case of (In)stability. 3.3.
The stability requirements on the solution
u
of the reduced equa-
tion can be relaxed as follows (cf. Fife [24], Flaherty and O'Malley Namely it is enough in the case of (Iq)- or
[25] and Howes [39]).
(Iq)-stability that
8yh(t,u(t)) E 0 a2q+lh(t,u(t))
and for
for
0 < j < 2q,
> m > 0
in
[a,b],
u(a) # A,
h(a,s)ds > 0
for
t
u(a)
in
[A,u(a))
or
(u(a),A]
in
[B,u(b))
or
(u(b),B].
u(b) # B,
or, for
n
I (b)
h(b,s)ds > 0
for
rt
Similar relaxations apply to the cases of (IIn)-stability
(u(a) < A, u(b) < B) 3.4.
and (III)-stability (u(a) > A, u(b) > B).
We have not considered the occurrence of shock layer behavior,
that is, the situation in which a solution y = y(t,e)
of (DP1),
(RP1) or (RP 2) satisfies the limiting relation
lim y(t,e) _ e+0+
where
I ul(t),
a < t < t0,
u2(t),
t0 < t < b,
u1(t0) # u2(t0).
The functions
solutions of the reduced equation (R1).
ul
and
u2
are stable
These phenomena are
studied, for instance, by Vasil'eva [88], Fife [24], O'Malley [76] and Howes [39] to which the reader can refer for details.
36
3.5.
III.
SEMILINEAR SINGULAR PERTURBATION PROBLEMS
Oscillatory phenomena of the type exhibited by the solution of the problem (E2) are discussed for more general problems by Volosov [91] and O'Malley [76].
3.6.
We note that in the case of the Robin problem (RP2), (Iq)-, (IIn)and (III n)-stability are essentially equivalent to their "tilded" counterparts because function
3.7.
u(t), and
Ui(u) (i = 0,1,2) 6
is a
"6-tube" around the
can be taken arbitrarily small.
The theory developed in this chapter for the Robin problems (RP1)
and (RP2) applies with minor modification to the Neumann problem ey" = h(t,y), a < t < b, -y'(a,c) = A, y'(b,e) = B, and related problems.
Chapter IV
Quasilinear Singular Perturbation Problems
§4.1.
The Dirichlet Problem:
Boundary Layer Phenomena
We consider now the singularly perturbed quasilinear Dirichlet problem
ey" = f(t,y)y' + g(t,y) E F(t,y,y'),
a < t < b, (DP2)
Y(a.e) = A, If
y(b,e) = B.
f(t,y) t 0, a great variety of interesting phenomena can occur.
If
f(t,y) E 0, the problem (DP2) is identical to the problem (DP1) already discussed in the previous chapter.
Therefore at points
f(t,y) = 0, we require the function
F(t,y,y')
(t,y)
for which
to be stable with res-
pect to the y-variable (in the sense of Definitions 3.1-3.6), in order that the solutions of (DP2) may behave "reasonably".
There are, however,
qualitative differences between the two problems (DP1) and (DP2) which will be illustrated by the following simple examples. Consider first the linear problems ey" = ±y',
Y(0,c) = 0,
0 < t < 1, Y(l,e) = 1.
The solution of (E6) y(t,e) =
is easily found to be
(1-e-[-e-1/e
+
e-(1-t)/e
e-(1-t)/e,
and so lim
where
y(t,e) = 0
for
0 < t < 1 - 6 < 1,
d, 0 < 6 < 1, is a fixed constant.
37
(4.1)
That is, the solution only
38
QUASILINEAR SINGULAR PERTURBATION PROBLEMS
IV.
exhibits nonuniform convergence at the right endpoint ing function
The limit-
is the solution of the corresponding reduced equa-
u = 0
which satisfies
u' = 0
tion
t = 1.
u(0) = 0 = y(O,e).
On the other hand, the problem (E6) has the solution y(t,e) =
(e-1/e_1)-1(-l+e-t/e)
- 1 -
e-t/e
and so lim+ y(t,e) = 1
for
0 < d < t < 1.
(4.2)
That is, the solution exhibits nonuniform convergence at the left endpoint
Here
t = 0.
satisfying
u = 1
is a solution of the reduced equation
u(l) - 1 = y(l,e).
u' = 0
Thus the nature of the solutions of these
two problems depends critically on the sign of the coefficient of
y'.
It is interesting to note also that the relations (4.1) and (4.2) differ from (3.2) in that nonuniform convergence of the solution (that is, boundary layer behavior) occurs at one endpoint only.
Consider next the problem ey" _ -ty',
-1 < t < 1, (E7)
Y(-',E) = -1,
Y(l,e) = 1.
The exact solution is
Y(t,e) _ -1 + 2(J1
e-s2/2eds)-1 ft 1
e-s2/2eds, 1
from which it follows that -1
lim+ y(t,c) _ e+O
where
1
for
-1 < t < -d < 0,
for 0 < 6 < t < 1,
6, 0 < 6 < 1, is a fixed constant.
the solution of the reduced equation y(-l,e), while 1 = y(l,e).
tu' = 0
is the solution of
ul - -1
satisfying
tu' = 0
is
u1(-1) = -1 =
satisfying
u2(1) _
In this case, the solution exhibits uniform convergence at
both endpoints. left endpoint
u2 - 1
The function
This is not at all surprising, if we note that near the t = -1, the coefficients of
y'
in (E7) and in (E6) are
both positive, and therefore we do not anticipate nonuniform convergence of the solution at the left endpoint
t = -1.
endpoint
y'
t = 1, the coefficients of
Similarly, near the right
in (E7) and in (E6) are both
negative, and so we do not anticipate nonuniform convergence of the solu-
The Dirichlet Problem:
4.1.
tion at the right endpoint
Boundary Layer Phenomena
t = 1.
Instead, the solution exhibits non-
uniform convergence at an interior point continuously in the limit as
39
t = 0, where it switches (dis-
a - 0) from
ul == -1
to
u2
1.
As our third example we take the problem
-1 0
If a solution the domain
D(uL)
u = uL(t)
of (RL) exists in
D(uL) = {(t,y)
:
a < t < b, Iy-uL(t)I < dL(t)},
where the positive continuous function IB-uL(b)I + d
[a,b], then we define
by
for
b - 6/2 < t < b
Similarly, if a solution we define the domain
D(uR)
and
u = uR(t)
dL(t)
satisfies
dL(t)
d
for
dL(t)
a < t < b - d.
of (RR) exists in
[a,b]
then
by
D(uR) = {(t,y): a < t < b, iy-uR(t)I < dR(t)) where the positive continuous function IA-uR(a)I + 6
for
a < t < a + 6/2
and
dR(t)
satisfies
dR(t) E 6
for
dR(t)
a+6 < t < b.
4.1.
The Dirichiet Problem:
Boundary Layer Phenomena
Finally, for a solution
u = u(t)
41
of (R) we define the domain
D(u)
by
D(u) _ {(t,y): a < t < b,
Iy-u(t)I < d(t)},
where the positive continuous function for d(t)
a < t < a + 6/2, d(t) 6
for
satisfies
d(t)
IB-u(b)I + 6
for
d(t)
_-
b - 6/2 < t < b
IA-u(a)I + 6
and
In addition, we will also consider
a + 6 < t < b - 6.
paths of the form
uL(t), F
u(t),
u0(t) =
uR(t),
a 0
in
V(uL,u,uR) A ([a,tL] U [tR -
f(t,y) < 0
in
V(uL,u,uR) n ([t L, tL + 2) U [tR,b]).
Z,
tR])
and
If
tL = tR (= t0), then
V(uL,u,uR) = V(uL,uR); and the reduced path
uL(t),
a < t < t0,
uR(t),
t0 < t < b,
u0(t) _
is said to be weakly stable in
[a,b]
f(t,y) > 0
in
V(uL,uR) n [a,t0]
f(t,y) < 0
in
V(uL,uR) n [t0,b].
if
and
Definition 4.6. in
[a,b]
The reduced path
u = u1(t)
is said to be weakly stable
if there exists a small positive constant
6
such that
The Dirichlet Problem:
4.1.
Boundary Layer Phenomena
f(t,y) > 0
in
D(u,uR) fl [tR
f(t,y) < 0
in
D(uL,u) fl [ti, tL + Z].
43
2, tR]
-
and
Definition 4.7. in
The reduced path
u = u2(t)
is said to be weakly stable
if there exists a small positive constant
[a,b]
f(t,y) > 0
in
D(uL,u) fl [a,tL]
f(t,y) < 0
in
D(uL,u) fl [t L, tL + 6]
6
such that
and
With these definitions we can now discuss the of solutions of the Dirichlet problem (DP2). chapter that the functions of class
f
and
and y
with respect to
C(1)
Moreover, in the cases of that
f
q-
or
g
asymptotic behavior
We assume throughout this
are continuous in
t
and
y
and
in the domain under consideration.
n-stable reduced solutions, we assume
have the required number of continuous partial deriva-
g
tives with respect to
y.
Finally, in the conclusion of each theorem below, the constant
c
is a known positive constant depending on the reduced solution under consideration. Theorem 4.1. tion
Let the reduced problem (RR) have a strongly stable soluof class
u = uR(t)
such that for for
t
in
0 < e < e0
[a,b]
C(2)([a,b]).
Then there exists an
the problem (DP2) has a solution
e0 > 0 y = y(t,c)
which satisfies
(y(t,e) - uR(t)I < wL(t,e) + cc,
where wL(t,e) = IA - uR(a)) exp[X(t-a)] and
X = -k/e + i/k + 0(e).
Proof:
As in the proof of Theorem 3.1, we construct a bounding pair
of functions
(a,B)
which satisfy all the conditions of Theorem 2.1.
Let us first linearize (DP2) about the reduced solution setting
V(uR), we are further led to the linear non-homogeneous differential
equation ez" + kz' + Rz = -cult
R*
If
ea2 + kA + t = 0
0 < c < k2/4R., the corresponding auxiliary equation
has two negative roots A _ -k/c + 2./k + 0(c) _ -k/c + 0(1)
XI = -R/k + 0(c) = 0(1).
Indeed, the positive function
wL(t,e) = IA - uf(a)Iexp [a(t-a)]
is the
solution of the corresponding homogeneous differential equation
ew"+kw' +Lw= 0 such that wL'(a,c) = AIA - uR(a)I < 0,
wL(a,c) = IA - uR(a)I,
that is, the solution
Let y > IuRI. is such that
wL
decreases to the right.
r(t,c) = eyZ-(exp[-A1(b-t)] - 1)
Then the function
0 < r < cc
and
-cc < r' < 0
for some
c > 0, and it sat-
isfies
cr" + kr' + Rr = -cy,
r(b) = 0.
We now define the bounding pair a(t,c) = uR(t) - WL(t,c) - r(t,c), p(t,c) = uR(t) + wL(t,c) + r(t,c),
and proceed to show that they satisfy all the required inequalities of Theorem 2.1. e(b,c). for
Clearly, a < Q, a(a,c) < A < S(a,c)
and
a(b,e) < B 0, we have p
F(t,S,a') - C011 > (2q+i)! - eIuLI > 0
if we choose y > IuLI(2q+1)!.
On the other hand, if uL
ally weakly stable, then
f(t,E)wR 10 in
[b-6/2,b],
but in the rest of the interval
[a,b],
is only loc-
The Dirichlet Problem:
4.1.
Boundary Layer Phenomena
47
IB_uL(b)I(m/e)1/2exp[-(m/e)1/2(b-t)J,
I
if
q = 0,
if
q > i
wR IB-uL(b)IC1/2q oq-1[C1/2
=
so that, for sufficiently small
_ (cle, If(t,C)wR'I < `I l
for some
c1 > 0.
1/2q C
+
6(b-t)]-(q+l)/q,
c,
if
q=0
if
q > 1
Thus
F(t,6,6') - e6" > 0
y > (c1 + IuLI)(2q+1)!. The conclusion of the theorem
if we choose
follows from Theorem 2.1.
Assume that the reduced problem (RL) or (RR) has a weakly
Theorem 4.3.
or locally weakly stable solution which is also (II
C(2)([a,b])
uL > 0
that
in
(a,b)
and
n) uL(b) < B
Then there exists an
uR(a) < A.
u = uL(t)
or
u = uR(t)
- or (II n)-stable in
the problem (DP2) has a solution
co > 0
or
uR > 0
in
such that for
y = y(t,e)
for
t
of class
[a,b].
in
Assume also
(a,b)
and
0 < c < CO [a,b]
which
satisfies
0 < Y(t,e) - uL(t) < wR(t,e) + r(e) or
0 < Y(t,E) - uR(t) < wL(t,e) + r(e). Here wR and
and
wL
are as given in Theorem 3.2 with
u replaced by
uL
uR, respectively, and
Ir(e)i
ccI/n 1/{n(n_1)}
for the weakly stable case,
ce
for the locally weakly stable case.
Let us consider only the case of
Proof:
uR.
We can prove this case by
defining a(t,e) = uR(t),
6(t,e) = uR(t) + wL(t,e) + where
p = 1
(or
(Cpym-1)1/n,
p = 1/(n-1)), if uR
is weakly (or locally weakly)
stable and proceeding as in the proof of Theorem 4.2.
48
QUASILINEAR SINGULAR PERTURBATION PROBLEMS
IV.
Assume that the reduced problem (RL) or (RR) has a weakly
Theorem 4.4.
u = uL(t)
or locally weakly stable solution C(2)([a,b])
uL < 0
in
or
u = uR(t)
and
(a,b)
Then there exists an
uL(b) > B
uR < 0
such that for
e0 > 0
y = y(t,e)
has a solution
or
for
in
t
in
and
(a,b)
uR(a) > A.
the problem (DP2)
0 < e < e0
[a,b]
of class
Assume also that
which is also (III n)- or (III n)-stable.
which satisfies
-wR(t,e) - r(e) < y(t,e) - uL(t) < 0 or
-wL(t,e) - r(e) < y(t,e) - uR(t) < 0,
wR and wL
where
are as given in Theorem 3.2 with
uR, respectively, and
and
Simply let
Proof:
y -
-y
r
u
uL
replaced by
is as given in Theorem 4.3.
and apply Theorem 4.3.
We next consider the solution
u = u(t)
of the reduced equation (R)
which does not satisfy either boundary condition but which is locally The proofs of the following two
strongly or locally weakly stable.
theorems are only a slight modification of the proofs of Theorems 4.2 and 4.3.
Theorem 4.5.
Let the reduced equation (R) have a locally strongly or
locally weakly stable solution (Iq)- or (Iq)-stable in
also
that for t
in
0 < e < e0
[a,b]
u = u(t) [a,b].
of class
C(2)([a,b])
Then there exists an
the problem (DP2) has a solution
which is such
e0 > 0
for
y = y(t,e)
which satisfies
Iy(t,e) - u(t)I < wL(t,e) + wR(t,e) + r(e).
Here, r
is as given in Theorem 4.2, wR = IB-u(b)Iexp [A(b-t)],
wL = IA-u(a)Iexp [a(t-a)], and
wR
X'= -k/e + 0(1), if
u(t)
is locally strongly stable, while
are as given in Theorem 3.1, if
Theorem 4.6.
u = u(t)
is also (IIn)- or (IIn)-stable in
(a,b), u(a) < A and
[a,b]
wL,
is locally weakly stable.
Let the reduced equation (R) have a locally strongly or
locally weakly stable solution
for
u(t)
0 < e < e0
u(b) < B.
[a,b].
of class
Assume also that
Then there exists an
the problem (DP2) has a solution
which satisfies
C(2)([a,b])
u" > 0
e0 > 0
y = y(t,e)
which in
such that for
t
in
4.2.
Robin Problems:
Boundary Layer Phenomena
0 < y(t,e) - u(t) < wL(t,e) + wL(t,e) +
WL, wR
where
stable, while
are as given in Theorem 4.5, if u wL, wR
49
el/n0
is locally strongly
are as given in Theorem 3.2, if u
is locally
weakly stable.
If the function u
is locally strongly or locally weakly stable and
(III n)- or (III n)-stable, then the result analogous to Theorem 4.6 holds,
provided that
u" < 0
in
(a,b), u(a) > A
and
u(b) > B.
We leave its
precise formulation to the reader.
Robin Problems:
§4.2.
Boundary Layer Phenomena
We now turn to the occurrence of boundary layer phenomena for solutions of the Robin problems ey" = f(t,y)y' + g(t,y) = F(t,y,y'), y(a,e) - Ply'(a,e) = A,
a < t,< b,
y(b,e) = B.
ey" = f(t,y)y' + g(t,y),
(RP3)
a < t < b, (RP4)
y(a,e) - Ply'(a,e) = A,
y(b,e) + P2y'(b,e) = B.
with their associated reduced problems f(t,u)u' + g(t,u) = 0,
a < t < t1 < b, (RL)
u(a) - plu'(a) = A,
f(t,u)u' + g(t,u) = 0,
a < t2 < t < b, (RR)
u(b) = B,
f(t,u)u' + g(t,u) = 0,
a < t2 < t < b,
(RR)
u(b) + p2u'(b) = B, and
f(t,u)u' + g(t,u) = 0,
a < t < b.
(We note that the related problem
with
y(a,e), y(b,e) + p2y'(b,e)
the change of variable
uL
and
ey" = f(t,y)y' + g(t,y), a < t < b,
prescribed, can be handled by making
t . a + b - t
to the transformed problem.)
(R)
and applying the results for (RP3)
We denote the solutions of (RL) and (R) by
u, respectively, while the solutions of (RR) and (RR) will be
50
QUASILINEAR SINGULAR PERTURBATION PROBLEMS
IV.
denoted, without any distinction, by
functions
dR, d
and u
uL, uR
ity apply to these
and
Earlier definitions of stabil-
uR.
except that for the case (RP3), the
are uniformly small in
d1
[a,b-6]
dR, d, di = 6)) while for the case (RP4) the functions and
d2
are uniformly small in
(that is,
dL, dR, d, di,
[a,b].
To provide some insight into the results for such problems, we conThe first two problems are
We leave it to the reader to formulate the statement of the results for a weakly or locally weakly stable solution
u(t)
which is also (III
or (III n)-stable.
The following results follow from the proofs of Theorems 4.7 - 4.9. Corollary 4.2.
Suppose the reduced problem (RL) has a solution
u = uL(t)
satisfying the hypotheses in each of the Theorems 4.7 - 4.9, then the conclusions of each of the theorems hold with out the term
vL(t,a).
uL
replacing
u, but with-
n)-
54
QUASILINEAR SINGULAR PERTURBATION PROBLEMS
IV.
On the other hand, if the reduced problem (RR) has a solution
u =
satisfying the hypotheses in each of the Theorems 4.7 - 4.9, then
uR(t)
uR
the conclusions of each of the theorems hold with
replacing
u, but
without the term wR(t,e). With these results, it is now an easy matter to deal with the problem (RP4), and so we omit the proofs of the following results. Theorem 4.10.
Assume that the reduced equation (R) has a locally strongly
(or weakly) stable solution u = u(t) also (Iq)-stable in 0 < e < e0
of class
Then there exists an
[a,b].
the problem (RP4) has a solution
which is
C(2)([a,b])
e0 > 0
such that for for
y = y(t,c)
in
t
which satisfies
[a,b]
ly(t,e) - u(t)l < vL(t,e) + vR(t,e) + r(e).
Here vL = e(plk)-1IA-u(a)+plu'(a)Iexp[-ke-1(t-a)], e(p2k)-llB-u(b)-p2u'(b)lexp[-ke-1(b-t)],
VR =
if u
is locally strongly stable, and
3.7 if u
is locally weakly stable, and
Theorem 4.11.
also (IIn)-stable in
t
in
0 < e < e0
[a,b]
u = u(t)
[a,b].
u(a) - plu'(a) < A and such that for
r
are as given in Theorem
is as given in Theorem 4.2.
Assume that the reduced equation (R) has a locally strongly
satisfying the hypotheses in each of the Theorems 4.9, 4.10.
Then the
conclusions of each of the theorems hold with
u, but without
uL
replacing
the term vL(t,e). On the other hand, if the reduced problem (RR) has a solution u = uR(t)
satisfying the hypotheses in each of the Theorems 4.9, 4.10,
then the conclusions of each of the theorems hold with but without the term
vR(t,e).
uR
replacing
u,
Interior Layer Phenomena
4,3.
The corresponding result
55
for (III n)-stable cases we leave to the
reader to state.
Interior Layer Phenomena
§4.3.
The remaining part of this chapter will be devoted to a discussion of interior crossing phenomena for solutions of the problems (DP2), (RP3) What we have in mind is the following.
and (RP4).
that the reduced problem (RL) has a solution problem (RR) has a solution in
(a,b)
and the reduced
u = uL(t)
which intersect at a point
with unequal slopes, that is, uL(t0) = uR(to)
and
to
uL' (t0)
Then we have the reduced path
uR(t0).
a < t < to,
U (t) u0 (t) =
(R0)
L
uR(t),
where
u = uR(t)
Suppose, for example,
t0 < t < b,
We ask under what conditions will the problem
u6(t0-) # u0(t0).
(DP2) possess a solution lim
y = y(t,e)
such that
y(t,c) = u0(t)?
(4.4)
E->0+
The answer to this question is not as straightforward as it was in the case of the problem (DP1) considered in the previous chapter. all, if the pair of reduced solutions
uL
and
uR
stable, then such a pair can never attract a sol ution in the sense of (4.4).
uL, U
R
crosses at
First of
are both strongly y(t,c)
of (DP2)
The reason for this is simply that if the pair
t0, then in the small interval
t0-6 < t < t0+6,
the Cauchy problem f(t,u)u' + g(t,u) = 0, u(t0) = o(= uL(t0) = uR(to))
must have two distinct solutions, but this
smoothness of f
and
g
is impossible in view of the
and the fact that
jfI > 0.
Consequently, we
can only expect the relation (4.4) from the crossing of two weakly stable solutions of the reduced equation (R)
then it necessarily follows that
since if
f(t0,o) = 0.
uL(t0) = uR(t0) = o,
With these remarks, we
can now discuss some general results for this crossing phenomenon, first for the Dirichlet problem (DP2) and then for the Robin problems (RP3) and (RP4).
56
QUASILINEAR SINGULAR PERTURBATION PROBLEMS
IV.
To give our first results, let us recall from Theorem 3.9 the function (e/m)1/2Iu'(t0)-u'(t0)Iexp[-(e/m)1/2It-t0I],
if
q = 0,
if
q > 1,
vl(t,e)
2 v[1+q(e(2q+2)!/2m)
where aq+1 = lu'(t0) - u'(t0)I{e(2q+2)!/2m}1/2.
Theorem 4.12.
that for t
in
Assume that the reduced path (R0) is of class
(Iq)-stable in
stable and
[a,b].
Then there exists an
the problem (DP2) has a solution
0 < e < c0
C2, weakly such
e0 > 0 y = y(t,c)
for
which satisfies
[a,b]
cel/(2q+1)
ly(t,e) - u0(t)l < vl(t,e) +
where
vl(t,e)
vl(t,e)
is
with
replaced by
- u'(t-)I
lu'(t+)
1 u'(t0) - uL(t0)I Proof:
The proof is similar to that of Theorem 3.9.
uL(t0) < uA(t0), and define for
a(t,e) = u0(t) -
(e1'm
a < t < b
1)1/(2q+1)
We note that
y > lu"1(2q+1)!.
c > 0
1)1/(2q+1)
e(t,e) = u0(t) + vl(t,e) + (eyra
where
and
We can suppose that
t = t0; indeed, al(to) < a'(t0).
is not differentiable at
a
Nevertheless, by our choice of
y, we
can show as in the proof of Theorem 3.9 that call > F(t,a,a')
that is, a of
vi, a
in
(a,b)
is a lower solution. is differentiable at
eel' < F(t,e,e')
in
{t0},
e, we see that by our choice
As regards
t = t0, and again by our choice of y,
(alb) . {t0}.
Thus the results follow from Theorem 2.2. Theorem 4.13.
Assume that the reduced path (R0) is of class
stable and (IIn)-stable. uL(t0) < uR(t0).
Moreover, assume
Then there exists an
e0 > 0
the problem (DP2) has a solution y = y(t,c) satisfies
uL > 0,
u'R' > 0
C2, weakly and
such that for for
t
in
[a,b]
0 < e < e0
which
4.3.
Interior Layer Phenomena
57
0 < y(t,E) - u0(t) < vl(t,c) + ccl/n,
where
vI(t,e)
is
vI(t,e)
with
in place of
luj(t0) - uL'(t0)I
2
in place of
n-1)
q
and with
lu'(t+) - u'(t0)I.
This result follows by arguing as in the proof of the previous
Proof:
theorem if we define for
a < t < b
and
c > 0
a(t,e) = u0(t),
0(t,e) = u0(t) + vl(t,e) + (c 1)1/n, where y > IullIn!
and Z(n-1)(e(n+l)!/2m)-l/201/2(n-l)It-t0l]-2/(n-1)
vl(t,a) = 2 0[l +
with 0n+1
- e(n+1)!Iuj(t0) - uL(t0)l/2m.
If the weakly stable reduced path
R0
is (IIIn)-stable, then a
result analogous to Theorem 4.13 is valid provided that and
(a,t0) U (t0,b)
u0 < 0
in
We leave its precise formulation
uL'(t0) > uj(t0).
to the reader.
The above results can be generalized to the three-branch reduced path f uL(t), u(t), u0(t) =
a < t < tl,
1 uR(t),
t2 < t < b,
t1 < t < t2,
uL, uR
where the solutions
intersect the middle
reduced equation (R), such that u(t2) = uR(t2)
duced path
and
u0(t)
Theorem 4.14.
u'(t2) # uR(t2).
for
in
t
0 < e < c0
[a,b].
u0(t)
v1(t,e)
is
respectively, by with
t0
and
the problem (DP2) has a solution
C2,
c0 > 0
y = y(t,c)
which satisfies
vl(t,e) t1
and
with
t0
and
1/(2q+1)
Iu'(t0)
Iu'(t1) - uL(tl)I, while
lu'(t+) - u'(t_)l
IuR(t2) - u' (t2) I
is of class
Then there exists an
IY(t,e) - u0(t)I < v1(t,e) + v2(t,e) + ce Here
of the
tl = t2, this becomes the re-
If
Assume that the reduced path
[a,b]
u
above.
weakly stable and (Iq)-stable in such that for
solution
uL(tl) = u(t1), uL(tl) # u'(tl),
- u'(tp)I
v2(t,c)
replaced, respectively, by
replaced, is
vl(t,e)
t2
and
58
IV.
QUASILINEAR SINGULAR PERTURBATION PROBLEMS
The corresponding result for the (II)-stable solution
we
u0(t)
will leave to the reader to formulate, but we wish to state the result for the following reduced path I
ul(t) _ where
,
a < t < t2,
uR(t),
t2 < t < b,
u(t)
u'(t2) # u1(t2).
Here, since the reduced solution
satisfy the boundary condition at
does not
u(t)
t = a, we expect that the solution of
(DP2) which follows the reduced path
will behave nonuniformly at
ul(t)
Similar results hold for the "reflected path"
t = a.
uL(t),
a < t < tl,
u(t)
tI < t < b.
u2 (t) _ Theorem 4.15.
,
Assume that the function
u(t)
weakly) stable and that the reduced
path
stable and (Iq)- or (iq)-stable in
[a,b].
such that for for
in
t
0 < e < e0
[a,b]
is locally strongly (or
ul(t)
is of class
C2, weakly
Then there exists an
the problem (DP2) has a solution
c0 > 0
y = y(t,e)
which satisfies cel/(2q+1)
ly(t,E) - ul(t)I < wL(t,E) + v2(t,E) +
Here, wL
is as given in Theorem 4.1 or Theorem 4.3, according as
is locally strongly or weakly stable, and
v2
u
is as given in Theorem 4.14.
We conclude this chapter with the following theorems for the Robin problems (RP3) and (RP4) whose proofs are similar to those of Theorems 4.12 - 4.15.
Theorem 4.16.
Let the assumptions of Theorems 4.12 - 4.14 hold with the
solution of (RL) replaced by the solution of (RL) for the Robin problem (RP3) or with the solutions of (RL) and (RR) replaced, respectively, by the solutions of (RL) and (RR) for the Robin problem (RP4).
Then the
conclusions of Theorems 4.12 - 4.14 hold with (DP2) replaced by (RP3)
or (RP4) . Theorem 4.17.
Let the assumptions of Theorem 4.15 hold for the Robin
problem (RP3) or with the solution of (RR) replaced by the solution of (RR) for the Robin problem (RP4).
Then the conclusions of Theorem 4.15
hold with (DP2) replaced by (RP3) or (RP4) and with Here
vL
strongly (or locally weakly) stable.
replaced by
wL
is given in Theorem 4.7 (or Theorem 3.5) if
u
is locally
VL.
Notes and Remarks
59
Notes and Remarks 4.1.
The theory developed above applies with little change to the more general equation A(c)
and
ey" - f(t,y,c)y' + g(t,y,e)
and boundary data
B(c), provided that
{f,g} (t,Y,e) - {f,g} (t,Y,O) + 0(1), for
(t,y)
in the appropriate domain, and
{A(c),B(e)} = {A(0),B(0)} + 0(1) 4.2.
for
0 < c < e0.
The Dirichlet problem (DP2) has been studied by many people including Tschen [85], von Mises [64], Oleinik and Zizina [70], Coddington and Levinson [14], Bris [7], Wasow [92], Vasil'eva [87], Erd6lyi [22], Willett [96], O'Malley [72], Cole [55; Chapter 2], Eckhaus [21; Chapter 5], Cook and Eckhaus [16], Chang [9], [10], Dorr, Parter, Shampine [20], Ackerberg and O'Malley [1], Habets [28], Habets and Laloy [31], Kreiss and Parter [56], Matkowsky [63], Howes [39] and Olver [71].
The majority of these papers deal with
solutions of the various reduced problems which are strongly stable and/or (IO)-stable.
The corresponding Robin problems (RP3) and (RP4) have also been We mention only the papers by Bris [7],
The stability conditions given in Definitions 4.1 and 4.2 can be weakened in the following way, as first observed by Coddington and Levinson [14].
Consider first a solution
u = uR(t)
duced problem (RR).
If there is a positive constant
f(t,uR(t)) < -k < 0
in
[a,b]
such that
and
f(a,s)ds > 0
(uR(a)-A) J
of the rek
for all
in
(uR(a),A]
y = y(t,c)
such that
E
uR(a)
or then the problem (DP2) has a solution lim y(t,e) = uR(t) e+0+
for
[A,uR(a)),
a < a+d < t < b,
Similarly, if there is a positive constant f(t,uL(t)) > k > 0
in
[a,b], where
k
u = uL(t)
such that
is a solution of
the reduced problem (RL), then the problem (DP2) has a solution y = y(t,c)
such that
60
IV.
lim
y(t,e) = uL(t)
QUASILINEAR SINGULAR PERTURBATION PROBLEMS
provided that
a < t < b-d < b
for
(n f(b,s)ds < 0
(uL(b)-B) 1
for all
n
(uL(b),B]
in
uL(b)
or 4.4.
[B,uL(b)).
In considering the asymptotic behavior of solutions of (DP2), (RP3)
and (RP4) in the presence of locally strongly or weakly or locally weakly stable solutions of the corresponding reduced problems, we always assumed an additional form of
y-stability.
If such y-
stability is not assumed then the theory becomes incredibly complicated, even in the linear case, that is, f(t,y) = f(t) g0(t)y.
and
g(t,y)
The reader can consult the book of O'Malley [75; Chapter
8] and the article of Olver [71] for discussions of such problems in the linear case.
A corresponding nonlinear theory is nonexist-
ent, in the sense that if our assumptions of y-stability are dropped, then no conclusions regarding the existence and the behavior of solutions of these problems
4.5.
can be drawn.
It is also possible to weaken our assumptions regarding the Robin problems (RP3) and (RP4) in the case that the reduced solution is strongly stable, as was first observed by Bris [7].
Suppose, for
example, that we consider the problem (RP4) and let
u = uL(t)
a strongly stable solution of the reduced problem (RL).
f(a,uL(a)) + Plhy(a,uL(a)) # 0, for
h(t,y) = f(t,y)uL(t) + g(t,y),
the problem (RP4) has a solution y = y(t,e) lim
y(t,e) = uL(t)
Similarly, if
u = uR(t)
is a strongly stable solution of the ref(b,uR(b)) - p2hy(b,uR(b)) # 0,
then the problem (RP4) has a solution
Note that if uL
or
uR
such that
a < t < b.
for
duced problem (RR) such that
lim y(t,e) = uR(t) e*0+
be
Then if
for
y = y(t,e)
such that
a < t < b
is (Iq)-, (IIn)- or (III n)-stable then
these two conditions are automatically satisfied, since
pl
and
are positive and
4.6.
h > 0. P2 y We have not considered shock layer behavior for solutions of (DP2), (RP3) or (RP4) in this chapter.
The reader can consult Howes [39],
[43] for a discussion of these phenomena.
Chapter V
Quadratic Singular Perturbation Problems
45.1.
Introduction
In this chapter we investigate the asymptotic behavior of solutions of boundary value problems for the differential equation ey" = p(t,y) Y'2 + g(t,y),
a < t < b,
(DE)
The novelty here is the presence of the quadratic term in
y'.
The more
general differential equation eY" = P(t,Y)Y'2 + f(t,Y)Y' + g(t,y)
will not be studied, since it can be reduced to the form (DE) in some cases by the familiar device of completing the square.
Our decision to
study the simpler equation (DE) rather than the more general equation stems from a desire to present representative results for this "quadratic" class of problems without having to deal with extra complexities in notation.
45.2.
The Dirichlet Problem:
Boundary Layer Phenomena
We shall first consider the following Dirichlet problem eY" = p(t,Y)Y'2 + g(t,y),
a < t < b, (DP3)
Y(a,e) = A,
y(b,c) = B.
To motivate some of the results to follow, it is useful to pay attention to the results that have been obtained for the model problem (cf. [27])
With these definitions we are now in a position to give results on the types of asymptotic behavior displayed by solutions of the Dirichlet problem (DP3). and
g
pect to
In what follows we tacitly assume that the functions p
are continuous in y
t
and sufficiently differentiable with resNote that in the statement of
in the domains defined above.
each of the theorems below, the constant
c
on the reduced solution under consideration.
is positive, depending only
70
QUADRATIC SINGULAR PERTURBATION PROBLEMS
V.
Theorem 5.1.
Assume that the reduced problem (RL) or (RR) has a strongly
stable solution
u = uL(t)
or u = uR(t)
of class v
also that there exists a positive constant p(t,y) > v > 0
R (uL) if and p(t,y)
+
in
R
(u L)
uL(b) < B
if
p(t,y) < -v < 0
and
uL(b) > B, or (ii) p(t,y) > v > 0 in < -v < 0 in R (uR) if uR(a) > A.
E(Y - IuRI - IPIc2E) > 0, if we choose Y = IuRI + 1, for
0 < c 0
since
by the (I0)-stability of
uL.
uL, and
pyuL2 + gy > m > 0
Now, in the interval
[b-6/2,b], we have
and so the desired inequality follows by setting y > M.
p < 0
remaining interval
pwR2 < c1c
for
set y > M + c1.
[a,b-6/2), there exist
c1 > 0
and
e0 > 0
In the
such that
0 < e < co; and so the desired inequality follows if we Theorem 5.2 now follows from Theorem 2.1.
A similar argument allows us to prove the following theorem. Theorem 5.3.
Assume that the reduced problem (RL) or (RR) has a weakly
stable solution
u = uL(t)
or
also (IIn)- or (IIn)-stable in uR > 0 [b-6,b]
e0 > 0
in
or
(a,b), uL(b) < B
p(t,y) 10 in
such that for
y = y(t,e)
in
[a,b]
u = uR(t) [a,b].
or
uR(a) < A
and
R (uR) f1 [a,a+d].
0 < e < c0
C(2)([a,b])
uL > 0
p(t,y) > 0
in
which is
or R+(uL) f1
Then there exists an
the problem (DP3) has a solution
satisfying
0 < y(t,E) - uL(t) < wR(t,E) + cc1/n or
of class
Assume also that
74
QUADRATIC SINGULAR PERTURBATION PROBLEMS
V.
0 < Y(t,e) - uR(t) < wL(t,e) + cel/n, wL
where the functions u
Theorem 3.2 with
wR
and
are as defined in the conclusion of
replaced by
uR
uL, respectively.
and
The next result follows from Theorem 5.3 with the change of variable
y + -Y Assume that the reduced problem (RL) or (RR) has a weakly
Theorem 5.4.
u = uL(t)
stable solution
or
in
u'R' < 0
R (uL)
fl
solution
or
uR(a) > A, and
p(t,y) < 0
in
R-(uR) n [a,a+d].
or
such that for
c0 > 0
y = y(t,e)
in
[a,b]
0 < e < e0
which
C(2)([a,b])
Assume also that
[a,b].
(a,b), uL(b) > B
[b-6,b]
exists an
of class
u = uR(t)
or
is also (IIIn)- or (IIIn)-stable in
uL < 0
p(t,y) < 0
in
Then there
the problem (DP3) has a
satisfying
-wR(t,e) - cel/n < y(t,e) - uL(t) < 0 or
-wL(t,e) - ccl/n < y(t,c) - uR(t) < 0,
where the functions
wL
Theorem 3.2 with
replaced by
u
and
wR
are as defined in the conclusion of uR
and
uL, respectively.
In the next two theorems, the reduced equation (R) has a solution u = u(t)
which in general satisfies neither of the boundary conditions
but which is locally strongly or weakly stable.
The proofs are similar
to those of Theorems 5.1 and 5.2 and are omitted. Theorem S.S.
Assume that the reduced equation (R) has a locally strongly
(weakly) stable solution
p(t,y) >_v > 0
of class
u = u(t)
(p(t,y) > 0) in
such that
C(2) (ja,b])
if u(a) < A
R+(u) fl {[a,a+6] U [b-6,b))
u(b) < B, p(t,y) < -v < 0 (p(t,y) < 0) in R _(u) fl {[a,a+6] U [b S,b]} if u(a) > A and u(b) > B, p(t,y) > v > 0 (p(t,y) > 0) in R±(u) fl [a,a+d] and p(t,y) < -v < 0 (p(t,y) < 0) in R±(u) fl [b-6,b] if u(a) < A
Assume that the reduced equation (R) has a locally strongly
(weakly) stable solution (IIn)- or (IIn)-stable in
of class
u = u(t) [a,b].
[b-d,b]} for a positive constant 0 < e < c0
(p(t,y) > 0) v.
which is also
C(2)([a,b])
u" > 0
Assume also that
u(a) < A, u(b) < B, p(t,y) > v > 0
that for
are as defined in the
is locally weakly stable.
u
in
in
(a,b),
R+(u) fi {[a,a+d] U
Then there exists an
the problem (DP3) has a solution
e0 > 0
y = y(t,e)
such in
satisfying
[a,b]
cel/nI
0 < y(t,e) - u(t) < wL(t,c) + wR(t,e) +
where
wL, wR
are as defined in Theorem 5.5 (Theorem 3.2) for the
locally strongly (weakly) stable solution.
An analogous result holds when the locally strongly or weakly stable function
u
and
fl {[a,a+d] U [b-d,b]}.
R _(u)
variable
y - -y
p(t,y) < -v < 0
2p(t,y)us(t) = 0
tainly locally weakly stable. is
in
in
and applying Theorem 5.6 to the transformed problem.
u = us = const., then
us
or
It can be proved by making the change of
We remark that if the solution u
if
u" < 0
p(t,y) < 0
is (III n)- or (III n)-stable, provided that
(a,b), u(a) > A, u(b) > B
of (R) is a singular solution in
[a,b]
and so
us
is cer-
Consequently Theorems 5.S and 5.6 apply,
q- or n-stable and satisfies the additional geometric condi-
tions of these theorems.
76
§5.3.
QUADRATIC SINGULAR PERTURBATION PROBLEMS
V.
Robin Problems:
Boundary Layer Phenomena
We turn now to a consideration of similar phenomena for the Robin problems a < t < b,
Ey" = P(t,y)y'2 + g(t,y),
(RP5)
y(a,e) - ply'(a,e) = A,
y(b,c) = B,
and a < t < b,
ey" = P(t,y)y'2 + g(t,y),
(RP6)
y(a,e) - Ply'(a,e) = A, where
and
pl
p2
y(b,e) + P2y'(b,e) = B,
are positive constants.
ey"
(The related problem
p(t,y)y'2 + g(t,y), a < t < b, y(a,e), y(b,e) + p2y'(b,e) can be studied by making the change of variable
t - a+b-t
the theory of (RP5) to the transformed problem.)
prescribed,
and applying
The associated reduced
problems are then p(t,u)u'2 + g(t,u) = 0,
< b,
a < t < t 1
(RL)
u(a) - Plu'(a) = A, p(t,u)u'2 + g(t,u) = 0,
a < t2 < t < b, (RR)
u(b) = B, p(t,u)u'2 + g(t,u) = 0,
a < t2 < t < b, (RR)
u(b) + P2u'(b) = B, and p(t,u)u'2 + g(t,u) = 0,
whose solutions are denoted by
a < t < b,
(R)
uL, uR, and
u, respectively.
The
definitions of stability given at the beginning of this chapter are assumed to apply to the functions
uL, uR
and
In the case of (RP5) the domains
fications.
u with the following modiR+(uR)
and
R (uR)
are
replaced by R(uR) = {(t,y): a < t < b,
(y - uR(t)I < 6),
while in the definition of the domains
R+(u), R _(u), R±(u)
and
R+(u)
the error function is assumed to be uniformly small (that is, bounded Similarly, in the case of (RP6) the error
above by
6) in
function
in each of the domains is assumed to be uniformly small in
[a,b-6].
[a,b].
Boundary Layer Phenomena
Robin Problems:
5.3.
77
The theory for the problems (RP5) and (RP6) is not as straightforward as for their counterparts in Chapter IV, in view of the nonlinear (quadratic) dependence on
y'.
We shall discuss the Robin problem (RP5) first.
results relate to the solution Theorem 5.7.
Assume that the reduced problem (RL) has a strongly (weakly)
+ g(t,Q) - g(t,uR) - evL - EuR > m(vL + Mm I) - kvL + k1vL - EM > 0. Theorem 5.9 now follows from Theorem 2.3.
Such a strong result is not possible if the reduced solution (RR) is not strongly stable, but only weakly stable.
uR of
However, if we im-
pose an additional assumption, we can obtain the following two theorems. Only the first theorem will be proved, since the proof for the second theorem is similar. Theorem 5.10. u = uR(t)
Assume that the reduced problem (RR) has a solution
of class
C(2)([a,b])
which is weakly stable and (Iq)-stable.
Then, if p(t,y)[A - uR(a) + p1uR'(a)] > 0 in
R(uR) f1 [a,a+i], there exists an
the problem (RP5) has a solution
c0 > 0
y = y(t,e)
such that for in
0 < e < co,
satisfying
[a,b]
ly(t,E) - uR(t)l < vL(t,E) + ccl/(2q+l)
where
vL
Proof:
is given in Theorem 3.4 with
u
replaced by
We give the proof only for the case that
such that
uR(a) - p1uj(a) < A. and
p > 0
in
uR R(uR)
uR.
is
for
(10)-stable and a < t < a+d.
The other cases can be proved in a similar manner. Define, for
Then, as in the proof of the previous theorem, these functions can be shown to satisfy the required inequalities.
5.3.
Robin Problems:
Theorem 5.11. u = uR(t)
and
Boundary Layer Phenomena
Assume that the reduced problem (RR) has a solution
of class
which is weakly stable and (II)-stable,
C(2)([a,b])
Then, if
u'R' > 0.
uR(a) - pluR(a) < A in
81
p(t,y) > 0
and
R(uf) fl [a,a+d], there exists an
such that for
co > 0
y = y(t,e)
the problem (RP5) has a solution
in
[a,b]
0 < c < e0,
satisfying
0 < Y(t,e) - uR(t) < vL(t,e) + cel/n where
vL
is given in Theorem 3.5 with
u
replaced by
uR.
A result analogous to Theorem 5.11 can be obtained if the reduced solution
is weakly stable and (III n)-stable, and we leave it to the
uR
reader to provide details.
Similar results can be obtained, if we make similar assumptions with respect to the solution
of the reduced equation (R) which, in
u = u(t)
general, does not satisfy any of the given boundary conditions.
We state
only two representative results, and we omit proofs, since these are combinations of the proofs of Theorems 5.7, 5.8 and 5.9. Assume that the reduced equation (R) has a solution
Theorem 5.12. u = u(t) also in
of class
which is locally strongly stable and
C(2)([a,b])
(Iq)-stable or
R+(u) fl [b-6,b]
(Iq)-stable.
or in
Assume also that
R _(u) fl [b-6,b].
p(t,y)[B-u(b)] > 0
Then, if
kl = k - LP111uR(a) - p1ui(a) - Al > 0, there exists an a solution
Assume that the reduced equation (R) has a solution
Theorem 5.13.
of class
u = u(t)
wR
u.
which is locally weakly stable and also
C(2)([a,b])
(Iq)-stable or (Iq)-stable.
Assume also that
p(t,y)[A - u(a) + plu'(a)] > 0 in
or in
R+(u) fl [a,a+d]
R _(u) fl
[a,a+6], and that
R _(u) fl
[b-d,b].
p(t,y) [B - u(b)] > 0 in
R+(u)
e0 > 0
or in
[b-6,b]
fl
such that for
y = y(t,e)
in
satisfying
[a,b]
-ccl/(2q+1) < Y(t,E)
(a)
-v
(b)
- u(t) < VL + WR + cel/(2q+1)
u(a) - plu'(a) < A
if
L
-
and
-
and
ccl/(2q+l) < y(t,c)
_WR
u(b) < B;
cel/(2q+1) < y(t,e) - u(t) < WR + ccl/(2q+1)
if u(a) - plu' (a) > A (c)
Then there exists an
0 < e < e0, the problem (RP5) has a solution
u(b) < B;
- u(t) < vL + cel/(2q+l),
-
if -v
(d)
u(a) - plu'(a) < A
L
if Here, vL
- w
R
-
and
u(b) > B; and
cc1/(2q+1) < y(t,c) - u(t) < cel/(2q+1),
u(a) - plu'(a) > A
and
is given in Theorem 3.4 and
u(b) > B.
wR
is given in Theorem 3.1.
We expect analogous results for the Robin problem (RP6) as well. We state only the following result, leaving it to the reader to formulate others.
Theorem 5.14. tion
u = u(t)
(I0)-stable.
Assume that the reduced problem (RL) (or (RR)) has a soluof class Let
L =
C(2)([a,b])
max lp(t,y)j.
which is strongly stable and also
Then if
R(u)
k2 = k - Lp21IuL(b) + p2uL(b) - BI < 0, (or if
kl = k - Lp111uR(a) - pluR(a) - Al > 0), there exists an
such that for in
[a,b]
0 < e < E0, the problem (RP6) has a solution
satisfying
E0 > 0
y = y(t,e)
5.4.
Interior Layer Phenomena
83
Iy(t,e) - uL(t)I 0, if uL(to) < w < uR(t0)
+ g(to,Q)
(*)
< 0, if where
Assume that the reduced path
a = uL(t0) = uR(to).
u''(to) < w < uL(to),
Then there exists an
0 < e < e0, the problem (DP3) has a solution
co > 0
y = y(t,e)
such that for in
[a,b]
satisfying y(t,e) = u0(t) + 0(e).
The following corollary holds if
Ip(t,y)I > v > 0
in
R(uL,uR),
since the inequality (*) is automatically satisfied.
Assume that the reduced path
Corollary 5.15.
stable in
[a,b]
in
Ip(t,y)I > v
u = u0(t)
and that there is a constant R(uL,uR).
v > 0
is strongly
such that
Then the conclusion of Theorem 5.15 is
valid.
If the reduced path
u0
is only weakly stable in
[a,b], it is
still possible to prove results analogous to Theorem 5.15 provided that u0
is also (Iq)- or (IIn)-stable.
We have the following two theorems,
which can be proved in the same manner as Theorems 3.9 - 3.10. Theorem 5.16. and
Assume that the reduced path
(Iq)-stable in
0 < e < e0
[a,b].
u = u0(t)
Then there exists an
the problem (DP3) has a solution
is weakly stable
c0 > 0
y = y(t,e)
such that for in
[a,b]
satisfying y(t,E) = u0(t) + 0(vl(t,e)) + 0(el/(2q+l)), where
by
vI
is as defined in Theorem 3.9 with
IuR(to) - uL(to)I
Iu'(t+)-u'(t0)I
replaced
interior Layer Phenomena
5.4.
Assume that the reduced path
Theorem 5.17.
and (IIn)-stable in in
uR > 0
and
0 < a 0
in
(a,tL),
Then there exists an
uL(to) < uR(t0).
the problem (DP3) has a solution
e0
eo > 0
y = y(t,e)
cel/nI
0 < y(t,e) - u0(t) < vI(t,e) +
by
uL
satisfying
[a,b]
where
is weakly stable
u = u0(t)
Assume also that
[a,b].
(tR,b)
such that for in
85
is as defined in Theorem 3.10 with
v1
- u'(ta)I
Iu'(to)
replaced
IuR(to) - uL(to)I If the function
[a,b], then an analogous
is (III n)-stable in
u0
result holds, provided that
uL < 0
in
(a,tL), uR < 0
in
and
(tR,b)
uL(t0) > uR(t0) . uL
We suppose next that the functions
t2
(tR,b), respectively, with
in
and
intersect a third
uR
of the reduced equation (R) at points
u
solution
in
t1
and
(a,tL)
The next two theorems
tL < tR.
relate to the reduced path
fuL(t), u = ul(t) =
u(t)
,
uR(t),
Theorem 5.18.
a < t < tl, tI < t < t2, t2 < t < b.
Assume that the reduced problems (R
C(2)([tl,t2])
and
(Iq)-stable in
stable and that for
and
0 < e < e0
[a,b].
uL(tl) = u(tl),
at points
u'(t2) # uR(t2)
Assume also that the reduced path
(a,b).
(R) and (R ) have R C(2)([a,ti]),
),
of Lclass
C(2)([t2,b]), respectively, such that
uL(t1) # u'(tl), u(t2) = uR(t2) in
u = uR(t)
and
u = uL(t), u = u(t)
solutions
tl < t2 is strongly
u = ui(t)
Then there exists an
the problem (DP3) has a solution
e0 > 0
y = y(t,e)
such in
satisfying
[a,b]
Iy(t,E) - ul(t)I < vl(t,E) + V2(t,E) + CE1/(24+1)
where
and
v1
Theorem 5.19.
v2
Assume that the reduced path
and (IIn)-stable in u" > 0 uR'(t2).
in
are as defined in Theorem 4.21.
[a,b].
(tl,t2), uR > 0
Then there exists an
problem (DP3) has a solution
Assume also that in
and
(t2,b)
co > 0
u = ul(t)
uL > 0
in
(a,ti),
uL(ti) < u'(tl), u'(t2) 0
there exists an y = y(t,E)
(tl,t2)
and
Assume also that
[a,b].
u" > 0
in
(t2,b).
Then
such that the problem (DP3) has a solution
e0 > 0
whenever
in
is of class
u = ul(t)
C 1)( [a,b]), weakly stable and (IIn)-stable in
0 < e < e0.
In addition, for
in
t
[a,b]
we
have that 0 < Y(t,e) - ul(t) < cel/n
We note that if the
C(1)-path
u1
is (III)-stable
suit corresponding to Theorem 5.21 holds, if
addition, if uI(tl) and
u'(t2) # uR(t2)
u'(tl)
and
ul < 0
u'(t2) = uR(t2)
in
or
Let us next give some results for the reduced path
u(t),
a < t < t2,
uR(t),
t2 < t < b.
u2 (t) These results can be applied to the path
{uL(t). a < t < tl, u(t),
In
uL(tl) = u'(tl)
then the obvious results are valid and may be ob-
tained directly from Theorems 5.18 and 5.19.
u3 (t)
then the re(a,b).
tl < t < b,
5.4.
Interior Layer Phenomena
87
by making the change of variable
t -
a + b - t.
Since the function
is not required to satisfy the boundary condition at pate that solutions of (DP3) described by
u
t = a, we antici-
will exhibit boundary
u2
layer behavior there, as well as interior layer behavior at
t = t2.
The
results which follow are a combination of our previous results on boundary layers and interior layers, and so their proofs are omitted. Assume that the reduced problems (R) and (RR) have solu-
are as defined in the conclusion of Theorem 5.22. u2
If the reduced path
that is, if
is weakly stable and of class
C(1)([a,b]),
u'(t2) = uR(t2), then the results corresponding to Theorems
5.22 - 5.24 are valid mutatis mutandis.
We note that if
u'(t2) = uR(t2),
v2(t,e) = 0.
then
Lastly we state our results on the interior crossing phenomena for solutions of the Robin problem (RP6).
The results relating to the prob-
lem (RP5) are left to the reader to formulate. Theorem 5.25.
Assume that the reduced problems (R ) and (R ) have soluR L and u = uR(t) of class C(2)([a,tL]) and C(2)([tR,b]),
u = uL(t)
tions
tL > tR, uL(t0) = uR(to)
respectively, such that uR(t0)
at a point
u = u0(t)
in
to
(= e)
and
uL (to) #
Assume also that the reduced path
(tR,tL).
( UL(t),
a < t < to,
Sl uR(t),
t0 < t < b,
is strongly stable and (Iq)-stable in
[a,b], and that
r > 0, for
uL(t0) < w < uR(t0),
< 0, for
uR(t0) < w < uL(t0).
P(t0,(Y)w2 + g(t0,a)
(*) (1`
Then there exists an
co >0 such that for
(RP6) has a solution
y = y(t,c)
in
[a,b]
0 < e < e0
the problem
satisfying
y(t,e) = u0(t) + 0(el/(2q+1))
We note that if
jp(t,y)T > 0
in
R(uL,uR), then the inequality
(*) is automatically satisfied.
Theorems 5.16 - 5.21 hold for the problem (RP6) mutatis mutandis, and so we proceed directly to a consideration of the analogs of Theorems 5.22 - 5.24 for the reduced path
Interior Layer Phenomena
5.4.
J
u = u2 (t) _
89
a < t < t2,
u(t),
UR(t), t2 < t < b.
Assume that the reduced problems (R) and (RR) have solu-
Theorem 5.26. u = u(t)
tions
and
u = uR(t)
C(2)([t2,b]), respectively.
(Iq)-stable in
strongly stable and
C(2)([a,t2])
of class
and class u = u2(t)
Assume also that the path
is
u(a) - plu'(a) = A
[a,b], and that
or
(u(a) - plu'(a) - A)[p(a,u(a)){pll(A-u(a))}2 + g(a,u(a))] > 0. Then there exists an has a solution
such that for
£0 > 0 in
y = y(t,£)
0 < c < £0
the problem (RP6)
satisfying
[a,b]
ly(t,£) - U2(t)l < vL(t,e) + v2(t,c) + c£1/(2Q+1), where
vL
placed by
u, and
and (IIn)-stable in
and
in in
u" R> 0
0 < £ < £0
re-
is as defined in the conclusion of Theorem 5.22.
v2
Assume that the reduced path
Theorem 5.27.
p(t,y) > 0
uR
is as defined in the conclusion of Theorem 5.9 with
uR(t2), and weakly stable, then the results corresponding to Theorems 5.26 and 5.28 hold mutatis mutandis with
v2 = 0.
Notes and Remarks 5.1.
The theory developed above for the problems (DP 3)' (RP.) and (RP6)
can accommodate, with minor modification, the more general problem in which
p(t,y), g(t,y), A
and
p(t,y,c) = p(t,y,0) + 0(1),
A(e) = A(0) + 0(1) 5.2.
and
B
are functions of
a
satisfying
g(t,y,e) = g(t,y,0) + o(l),
B(c) = B(0) + 0(1).
Surprisingly little work has been done on the Dirichlet problem (DP3).
Motivated by a brief discussion in Section 5 of the paper
by Dorr, Parter and Shampine [20], Howes has developed a reasonably coherent boundary layer theory for (DP3) in [39].
He has also dis-
cussed a corresponding interior layer theory in [39) and [38] which includes not only the interior crossing phenomena treated above but also shock layer phenomena.
The classic interior crossing
result of Haber and Levinson [27] has been extended in several directions by Vasil'eva [87], O'Malley [74] and Howes [43].
The
Robin problems (RP5) and (RP6) have been considered by Vasil'eva [87], Macki (62), Searl [84] and Howes [40].
Chapter VI
Superquadratic Singular Perturbation Problems
56.1.
Introduction
In previous chapters we have presented fairly comprehensive results for boundary value problems involving the ey" = f(t,Y,Y'),
differential equation
a < t < b,
subject to the fundamental restriction: f(t,y,z)
= O(lzj2)
as
IzI - oo.
It is natural for us to ask if similar results can be extended to these
f
boundary value problems when f(t,y,z)
= O(Izln)
as
is subjected to the restriction:
(zI - oo
for
n > 2.
A partial and somewhat negative answer was given many years ago by Vishik and Liusternik [90] for the Dirichlet boundary value problem Ey" = f(t,y,y'),
Y(a,e) = A,
a < t < b,
(6.1) (6.2)
Y(b,E) = B.
They showed that if f = p(t,Y)Y'n + fl(t,Y,Y'), where
n > 2, Ip(t,y)I > v > 0 f1(t,Y,z) = O(Izln-1)
as
and
Izl
..
then every solution of (6.1), (6.2), if it exists (cf. (E14) below), satisfies
91
92
y'(a,e) = 0(1)
VI.
SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS
and
y'(b,e) = 0(1)
as E- 0+.
[Note that these authors actually considered the Cauchy problem (6.1), for 0 < n < 2, together with the initial conditions
y(a,c) = A, y'(a,e) =
This Cauchy problem is equivalent to the above Dirichlet
C/ey, C,y > 0.
problem, as the reader can see by solving the simple example ey" _ -vy',
y(l,e) = B, where
y(0,e) = A,
v > 0,
A short calculation shows that here
y = 1
and
A < B.
C - v(B - A).
However,
n = 2, they showed that this equivalence is obtained only if k/e , k > 0. Also compare this with our discussion at the y'(a,e) ee
when
beginning of Chapter I.]
Thus, in contrast to the solutions for similar problems studied in
previous chapters, Vishik and Liusternik's problem does not exhibit a boundary layer characteristic! is the following. f(t,u,u') = 0
u(a) # A
and
Let
u = u(t)
Another way of formulating their result be a solution of the reduced equation
which does not satisfy either boundary condition, that is, u(b) # B, then there is no solution
y = y(t,e)
of (6.1),
(6.2) satisfying lim+ y(t,e) = u(t) e+0
for
a < t < b.
This implies that for arbitrary choices of A
and
B, the Dirichlet prob-
lem (6.1), (6.2) has no solution for all sufficiently small values of
Therefore, the Dirichlet problem (6.1), (6.2), where n > 2 f = 0(jy'jn)
as
C.
and
ly'l + W is not, in general, a well-posed problem,
since we are interested in existence of solutions for small values of
E.
These remarks are best illustrated by the classic counterexample due to Coddington and Levinson [14] (see also [23], [94]): Ey" = -Y'
-
Y(O,e) = A,
y'3 S f(Y'),
0 < t < 1, (E14)
Y(l,e) = B.
By quadratures we obtain the general solution (x+c1)/e
y(t,e) = ±e arc sin (e where
cl, c2
)
+ c2,
are arbitrary constants.
To choose these constants to
satisfy the boundary conditions, we run into a difficulty. ficiently small values of
ly(t,E) - Al = 0(c),
c, in fact, if
For all suf-
0 < e < 21A - BJ/a, we have
6.2.
A Dirichlet Problem
and so
y(l,e)
cannot be equal to
solution, unless solution
93
B = A.
B, unless
B = A.
Thus there is no
latter case, we obtain the constant
In this
y(t,E) a A = B.
For this counterexample, note that f(Y') = O(Iy'13), as
I Y') + 00,
and that the only real solution of the reduced equation u(t) = constant.
This reduced solution
both boundary conditions, unless (E14) as
f(u') = 0
u(t) a constant
is
cannot satisfy
A = B. and so there is no solution to
c + 0', unless A = B, which is in agreement with the result
of Vishik and Liusternik.
The above results seem to indicate that there is very little that one can do with Dirichlet boundary value problems.
However, we hasten to
point out that, fortunately, the Robin problem for this class of differential equations turns out to be well-posed and solvable.
That this is
so should not be surprising, in view of our discussion of it in Chapter V, and also in view of the result of Vishik and Liusternik, which implies that the solution
y(t,e)
for the Robin boundary value problem should
satisfy
Y'(a,E) = 0(1) = pll(y(a,E) - A) and
Y'(b,e) = 0(1) = p21(Y(b,e) - B). and
This will be the case if y(a,e)
y(b,e)
are
0(1)
as
e - 0+.
Thus, solutions of such problems have uniformly bounded derivatives at the endpoints.
§6.2.
A Dirichlet Problem To study these superquadratic boundary value problems in detail, we
first consider the following class of Dirichlet problems ey" = h(t,y)f(t,y,y') a F(t,y,y'),
a < t < b (DP4)
y(a,c) = A,
y(b,E) = B.
Here the function the function in a domain
f
h(t,y)
is of the same type as in Chapter III, while
is continuous and satisfies
V0(u)
and for all
fined in Chapter III, where
z
in 1R.
u = u(t)
f(t,y,z) > p > 0 The domain
V0(u)
for
(t,y)
is as de-
is a solution of the reduced equation
94
VI.
h(t,u) = 0,
SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS
a < t < b.
(R4)
The stability properties of
stable, then
are as given in Definitions 3.1, 3.3
u(t)
and 3.3, with m replaced by
For example, if
mu-1.
hy(t,y) > mu-1 > 0
for
(t,y)
in
u(t)
V0(u).
is (I0)-
We also assume
that
f(t,y,z) = 0(Izln)
We note that for
lzi - - for
as
n > 2.
n = 2, the results of Chapter III for the Dirichlet prob-
lem apply directly to the problem (DP4), provided that the stability con-
m
stant
mu-1.
is replaced by
The following two theorems give the basic results for the Dirichlet problem (DP4). Theorem 6.1.
Assume that the reduced equation (R4) has an (Iq)- (or (IIn)-)
stable solution u(b) = B for
u = u(t)
u" > 0
and
of class
in
C(2)([a,b])
such that
Then there exists an
(a,b).
the problem (DP4) has a solution
0 < e < e0
u(a) = A,
co > 0
y = y(t,e)
such that in
[a,b)
satisfying u(t) < y(t,c) < u(t) + ce1/p.
where on
p = 2q + 1
m, loll
Proof:
and
n), and
(or
c
is a known positive constant depending
p.
Define for
t
in
[a,b]
and
c > 0
a(t,e) = u(t), a(t,e) = u(t) + r(e), where
r(e) = (eym
1)1/p,
for
y > p! M and M > lu"I.
a < a, a(a,e) < A < a(a,e), a(b,c) < B < 0(b,e), and (since
u" > 0).
It only remains to show that
Clearly,
ca" > F(t,a,a')
ce" < F(t,a,s').
We have
F(t,a,S') - ea" = h(t,a)f(t,a,a') - call p-1
aih(t,u)rl/j! + ayh(t,)rP/P!]f(t,a,s') -
[h(t,u) +
cull
j=1
> (m)J-Ieym 1/P!)u - em > 0,
since y > p! M. (t,g)
belongs to
= f(t,e) = u(t) + e(a - u(t)), 0 < e < 1, and
Here V0(u)
Consequently, a
and
for all sufficiently small a
c, say
0 < c < co.
satisfy all of the required inequalities.
The proof will follow from Theorem 2.1 if we can show that whenever
6.3.
Robin Problems:
Boundary Layer Phenomena
and a < y < $
ey" = F(t,y,y') then
ly'(t,e)I < N
fies
u'(a) < y'(t,c) < u'(b).
and a(t,e) < y(t,e)
a'(b,e) = u'(b).
This is because
imply that
y(b,c) = u(b) = a(b,c)
and
J c [a,b],
Indeed, the solution
J x
on
on
95
satis-
y'(a,s) > a'(a,e) = u'(a), while
a(t,c) < y(t,c)
imply that
u'(a) < y'(t,e) < u'(b)
Therefore
y(t,c)
y(a,c) = u(a) = a(a,c)
y'(b,c) < y" > 0
since
for
a < y < 5. Theorem 6.2 is the "concave" version of Theorem 6.1. Theorem 6.2.
Assume that the reduced equation (R4) has an
stable solution and
u(b) = B for
0 < c < s0
u = u(t)
u" < 0
in
of class (a,b).
C(2)([a,b])
Then there exists an
the problem (DP4) has a solution
(Iq)- or (III n)-
such that
u(a) = A,
such that
e0 > 0
y = y(t,e)
in
[a,b]
satisfying u(t) - ccI/p < y(t,e) < u(t), where the constants Simply let
Proof:
c
and
y + -y
p
are as defined in Theorem 6.1.
and apply Theorem 6.1 to the transformed prob-
lem.
We remark that since it is assumed that domain
form
VO(u)
in which the functions
h
and
u(a) =.A f
V0(u) = {(t,y): a < t < b, ly - u(t)l < S}, where
arbitrarily small (but fixed) positive constant.
and
u(b) = B, the
are defined is of the d > 0
is an
Consequently, for such
problems (Iq)-, (IIn)- and (IIIn)-stability are essentially equivalent to their "tiled" counterparts.
§6.3.
Robin Problems:
Boundary Layer Phenomena
We turn now to the following classes of Robin problems cy" = h(t,y)f(t,y,y'),
y(a,e) - ply'(a,c) = A,
a < t < b, y(b,c) = B,
and
ey" = h(t,y)f(t,y,y'),
a < t < b, (RP8)
y(a,s) - ply'(a,c) = A, where
pl
and
p2
y(b,c) + P2y'(b,c) = B,
are positive constants.
The analogs of Theorems 6.1
and 6.2 for these problems follow easily from our discussion of the prob-
96
VI.
SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS
lems (RP1) and (RP2) in Chapter III, and so we omit the proofs of the next two theorems.
Assume that the reduced equation (R4) has an
Theorem 6.3.
plu'(a) < A, u(b) = B such that for
e0 > 0
y = y(t,e)
in
of class
u = u(t)
stable solution
[a,b]
and
u" > 0
0 < e < e0
such that
C(2) ([a,b])
in
(Iq)- or (IIn)-
u(a) -
Then there exists an
(a,b).
the problem (RP7) has a solution
satisfying
u(t) < y(t,E) < u(t) + VL(t,e) + cel/P
Here
is as defined in Theorem 6.1, and
c
Theorems 3.4 and 3.5 for
p = 2q+1
and
is as defined in
vL(t,e)
p = n, respectively.
The following result refers to the analogous problem (RP8). Assume that the reduced equation (R4) has an (Iq)- or (II
verse inequalities, then results analogous to Theorems 6.3 and 6.4 hold true.
Tiis can be seen by using the change of variable
y - -y
and
applying Theorems 6.3 and 6.4, respectively.
§6.4.
Interior Layer Phenomena It is now an easy matter to consider the occurrence of interior layer
behavior for the problems (DP4), (RP7) and (RP8).
The situation described
in the following theorems arises most frequently when two solutions of the reduced equation (R4) intersect at a point
t0
slopes, as we have discussed earlier in Chapter III.
in
(a,b)
with unequal
We omit the straight-
forward proofs.
Theorem 6.5.
Assume that the reduced equation (R4) has an (Iq)- or (IIn)-
stable solution
u = u(t)
of class
C(2)([a,b]), except at the point
t0
Interior Layer Phenomena
6.4.
in
(a,b)
where
u'(t0) < u'(t0) and
u(a) = A, u(b) = B an
e0 > 0
u" > 0
in
ju"(tt)I < -.
and in
Assume also that Then there exists
(a,t0) U (t0,b).
0 < e < £0
such that for
y = y(t,e)
97
the problem (DP4) has a solution
satisfying
[a,b]
u(t) < y(t,c) < u(t) + vl(t,e) + ceI/p, where
c
3.10 for
is a positive constant and p = 2q+1
is defined in Theorems 3.9 and
vI
p = n, respectively.
and
It is clear that a result analogous to Theorem 6.5 can be obtained if the solution
u
is (Iq)- (or (III n)-) stable, and
u'(tp) > u'(t+)
and
u" < 0
in
(a,t0) U (t0,b).
We leave its exact formulation to the reader. In the same manner we can prove the next two results which deal with 'interior crossing' phenomena for the Robin problems (RP7) and (RPB), when the reduced solution
Theorem 6.6.
(a,b)
where
is (Iq)- (or (III n)-) stable and
concave.
there exists an
and
and
in
ju"(t±)I < -.
u" > 0
such that for
e0 > 0
y = y(t,c)
C(2)([a,b]), except at the point
of class
u'(t-) < u'(t4.)
u(a) - plu'(a) < A, u(b) = B
a solution
u
Assume that the reduced equation (R4) has an (Iq)- or (IIn)-
stable solution u = u(t) in
Similar
is (Iq)- (or (IIn)-) stable and convex.
u
results can be obtained when
in
(a,t0) U (t0,b).
0 < £ < £0
t0
Assume also that Then
the problem (RP7) has
satisfying
[a,b]
u(t) < y(t,£) < UM + vL(t,e) + v1(t,£) + ceI/p where
c
and
vL
are as given in Theorem 6.3 and
vI
is as given in
Theorem 6.5. Theorem 6.7.
Assume that the reduced equation (R4) has an (Iq)- or (IIn)-
stable solution in
(a,b)
u = u(t)
where
C(2)([a,b]), except at the point
of class
u'(t0) < u'(t+)
and
u(a) - plu'(a) < A, u(b) + p2u'(b) < B Then there exists an
We now return to the general equation given at the beginning of this chapter, namely, ey" = f(t,y,y'),
a < t < b.
In the sequel it is assumed that f(t,y,z) = 0(Izln)
IzI -- for n> 2.
as
Let us first give two definitions of stability for the solution of the reduced equation f(t,u,u') = 0.
(R)
They are obvious extensions of earlier ones. A solution
Definition 6.1. to be (I
q)_1
u = u(t)
of the reduced equation (R) is said
(IIn)- or (III n)-stable in
[a,b]
if the respective inequal-
ities in Definitions 3.1-3.3 hold for the function in
h(t,y) = f(t,y,u'(t))
V0(u) = {(t,y): a < t < b, Iy - u(t)I < 6}. A solution
Definition 6.2.
u = u(t)
of (R) is said to be stable in
if there is a positive constant
[a,b]
k
such
Ify,(t,y,y')l > k > 0
that ly'
in
V1(u) = {(t,y,y'): a < t < b, Iy - u(t)I < S,
- u'(t)l < d}.
With these stability properties we can now discuss the Dirichlet
problem ey" = f(t,y,y'),
a < t < b, (DP5)
y(a,E) = A,
y(b,E) = B,
and we obtain the same results as Theorems 6.1 and 6.2. Theorem 6.8.
Assume that the reduced equation (R) has an (Iq)- or (II
stable solution u(b) = B
u = u(t)
u" > 0
and
0 < e < e0
for
in
of class
C(2)([a,b])
such that
Then there exits an
[a,b].
the problem (DP5) has a solution
e0 > 0
y = y(t,e)
n)-
u(a) = A,
such that in
[a,b]
satisfying u(t) < y(t,e) < u(t) + CE: 1/p,
where
p = 2q + 1
on
lu"I
m,
Proof:
and
(or
n)
and
c
is a known positive constant depending
p.
We proceed as in the proof of Theorem 6.1 by defining the same
bounding pair
6.5.
A General Dirichlet Problem
a(t,e) = u(t),
where
r(c) _ (ey m
99
B(t,e) = u(t) + r(e), 1)1/p
for y > p! Iu"I.
Then it is clear that
a < B, a(a,e) = A < B(a,e), a(b,e) = B < B(b,e), call = Cu" > f(t,a,a') = 0, and
c6" < f(t,B,B').
This last inequality follows by virtue of our
stability assumption and our choice of y, that is, p-1
f(t,6,6') - e6" = f(t,u,u') +
2'.'f(t,u,u')B1/j!
E
j=l
3
+ 8Yf(t,&,u')$P/p! - cull
m(eym 1)/P! - elu"l > 0.
Finally we can show by arguing as in the proof of Theorem 6.1 that the function
f
satisfies a generalized Nagumo condition with respect to
a
B, and so the conclusion of Theorem 6.8 follows from Theorem 2.1.
and
The next result is the concave version of Theorem 6.8. Theorem 6.9.
Assume that the reduced equation (R) has an
stable solution u(b) = B for
and
u = u(t)
u" < 0
in
of class
such that
C(2) ([a,b])
Then there exists an
[a,b].
the problem (DP5) has a solution
0 < e < e0
(Iq)- or (III n)-
u(a) = A,
such that
e0 > 0
y = y(t,e)
in
[a,b]
satisfying u(t) - ce1/p < y(t,s) < u(t), where the constants
c
and
are as defined in Theorem 6.8.
p
A stronger result than the above two results can be obtained, if the reduced solution Theorem 6.10. u = u(t)
u(t)
Assume that the reduced equation (R) has a stable solution
of class
satisfying
C(2)([a,b])
Then there exists an has a solution
is stable in the sense of Definition 6.2.
such that for
e0 > 0
y = y(t,c)
in
[a,b]
and
u(a) = A 0 < e < e0
u(b) = B.
the problem (DP5)
satisfying
(y(t,e) - u(t)j < cc,
where Proof:
c
is a known positive constant depending on We suppose first that
f
I
< -k < 0.
f
and
u.
Clearly, there exists an
y R. > 0
such that
Ify(t,y,y')I < £
in
V1(u).
Setting
z = y - u(t)
in the proof of Theorem 4.1, we are first led to the problem
as
SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS
VI.
.100
ez" = f(t,u+z, u' + z') - cu"
= fy[.]z + fy["]z' - eu", z(b,e) = 0,
z(a,e) = 0, where
denote the appropriate intermediate points, and
and
then to the problem ez" + kz' + iz = -cu" z(b,c) = 0.
z(a,c) = 0, If
0 < c < k2/4R, one of the two negative roots of the corresponding
auxiliary equation
cA2 + kA + R = 0
is
A = -R/k + 0(e).
Then the
function
r(t,e) = eycI(exp[-A(b-t)] - 1) is a solution of cr" + kr' + tr = -cy. It has the following properties: 0 < r < cc
r(b) = 0, for some
and
-cc < r' < 0,
c > 0.
We define the bounding pair a(t,c) = u(t) - r(t,c), s(t,e) = u(t) + r(t,e),
and we need only verify that inequality for
s
call > f(t,a,a'), since the differential
follows by symmetry and the other required inequali-
ties clearly hold true.
By Taylor's Theorem, however, we obtain
ca" - f(t,a,a') = cu" - cr" +
-c u" J
- er" - kr' - Rr
c(Y - Iu") > 0 by choosing y >
I u" I.
Finally, it is not difficult to see that
for any solution of
cy" = f(t,y,y')
the theorem follows from Theorem 2.1.
y'(t,c) - u'(t) = 0(c)
which satisfies
fy, > k > 0, then we would use the bounding functions
a(t,e) = u(t) - r(t,e), s(t,e) = u(t) + r(t,c),
a < y < $, and so
We remark that in the proof, if
A General Robin Problem
6.6.
101
where P(t,e) = ey&-1(exp[A(a - t)]
56.6.
- 1).
Boundary and Interior Layer Phenomena
A General Robin Problem:
We now turn our attention to the nonlinear Robin problem a < t < b,
ey" = f(t,y,y'),
(RP9)
y(a,E) - Ply'(a,E) = A, It turns out that if
pl > 0
y(b,E) + P2y'(b,e) = B. and
p2 > 0, and if appropriate stability
assumptions hold, then these problems will have solutions, irrespective
f with respect to
of the growth of
y'.
To formulate these stability
assumptions, let us suppose that the reduced problem a < t < b,
f(t,u,u') = 0,
(RR)
u(b) + p2u'(b) = B,
has a smooth solution solution of (RP9).
uR
u = uR(t)
which we will use to approximate the
Since, in general, uR(a) - p1u.(a) # A, we require The first requirement is, of course,
to have two stability properties.
that
uR
is stable in the sense of Definition 6.2, that is,
fy(t,y,y') < -k < 0
(6.3)
in the region O1(uR) _ {(t,y,y'): a < t < b, Iy'
Iy - uR(t)I < 6
- uR(t)I < 6).
The second requirement is new and its motivation can be seen from the stability results for the following class of initial value problems eY" = f(t,y,y'),
a < t < b, (IVP)
y(a,c) = uR(a),
Note that here
y'(a,e) = p11(uR(a) - A)
y(a,e) - p1y'(a,e) = A.
It is known (cf. [61], [87])
that the solution of (IVP) is uniformly close to the reduced solution uR
such that either
for all
uR(a) _
or
(uR(a) - Qf(a,uR(a),A) > 0, for all
A
in
(uj(a),Q
or
[E,uR(a)).
Therefore, the second require-
102
VI.
ment is that if
SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS
uR(a) - p1uj(a) # A, then (6.4)
(uR(a) - P1uR(a) - A)f(a,uR(a),A) < 0, A in
for all
(uj(a), p1l(uR(a) - A)]
or in
[p11(uR(a) - A), uR(a)).
This inequality provides us with the required "boundary layer stability" of the function
uR, and for nonlinear functions
f, it serves to define
the admissible boundary layer jumplA - uR(a) + pluR(a)I. Finally, we must ensure that at
uR
approximates the solution of (RP9)
t = b, and so we assume that
P2fy(b,uR(b),uR(b)) - fy(b,UR(b),uA(b)) 0 0. Theorem 6.11. u = uR(t)
(6.5)
Assume that the reduced problem (RR) has a solution
of class
has a solution
satisfying the relations (6.3) - (6.5).
C(2)([a,b])
Then there exists an
such that for
e0 > 0
y = y(t,e)
in
[a,b]
0 < e < e0
the problem (RP9)
satisfying
Y(t,e) = uR(t) + 0(c.(p1k)-lexp[-k(t-a)/e]) + 0(e), k = IA - uR(a) + p1uR'(a)l.
where Proof:
The bounding functions are defined in the usual manner and using
the standard techniques, it is not difficult to verify that the required inequalities are indeed satisfied.
must also verify that if y = y(t,c)
In order to apply Theorem 2.3, we is a solution of
which lies between the lower and upper solutions, then formly bounded.
This is true, however, since
ey" = f(t,y,y') y'(t,e)
is uni-
y'(t,e) = uj(t) +
0(Rpl1 exp[-k(t-a)/e]) + 0(c); see [40] for complete details.
We remark that if, instead, the reduced equation smooth solution
u = uL(t)
which satisfies
f = 0
has a
uL(a) - pluL(a) = A, then a
result analogous to Theorem 6.11 can be obtained mutatis mutandis. Lastly we formulate a result for the Robin problem (RP9) which displays angular interior layer behavior which is similar to that discussed by Haber and Levinson [27] for the Dirichlet problem (cf. Chapter V). However, we must proceed with care as the
following example shows.
The problem is
0 m > 0 in the region
D1(u).
(6.9)
With these assumptions, we can formulate the follow-
ing result.
Theorem 6.12. tions
Assume that the reduced problems (R ) and (R ) have soluL R of class C(2)([a,t0]), C(2)([t0,b]), res-
u = uL(t), u = uR(t)
pectively, which satisfy the relations (6.6) - (6.9).
Then there exists
104
an
SUPERQUADRATIC SINGULAR PERTURBATION PROBLEMS
VI.
0 < E < EO
such that for
e0 > 0
in
y = y(t,e)
[a,b]
the problem (RP9) has a solution
satisfying vLI(2k)-lexp[-klt-tol/e])
y(t,E) = u(t) + 0(EIVR
+ 0(E).
The proof proceeds in the now familiar manner by defining the
Proof:
appropriate bounding functions.
solution y = y(t,e)
of
It is also not difficult to see that any which lies between the bound-
Ey" = f(t,y,y')
ing functions is such that Y'(t,c) = u'(t) + 0(IVR VLI/2 exp[-kjt-t0I/E]) + 0(E), and so we can apply Theorem 2.3 to obtain the required result.
A Comment
§6.7.
We conclude this chapter with the remark that under certain circumstances (cf.
[44]) solutions of the Dirichiet problem (DP5) exist and In particular, if
exhibit boundary and shock layer behavior. 0(IY'In-1)
as
f(t,y,Y') = p(t,Y)Y'n +
and if path
vanishes at a value of
p (t,y)
in
t
in
Iy'I
-
-.
for all
[a,b]
y
or along a
[a,b] xIR1, then the result of Vishik and Liusternik pro-
hibiting boundary layer behavior does not apply.
As a simple illustra-
tion, consider the problem Eyit = -ty'3 = f(t,Y'),
y(0,E) = 0,
0 < t < 1, (E16)
y(l,e) = B > 0.
Its solution, which can be found by quadratures, satisfies
lim y(t,E) =B for 0 0
which satisfies
lim+ y(t,c) = 0
where
for
in
t
[a+d,b-S],
Indeed, the reader should have no difficulty showing
0 < 6 < b-a.
that each component of the solution estimate in
(7.1)
y(t,c)
satisfies the more precise
[a,b]:
Yi(t,c) = O(IAilexp[-(m i/c)1/2(t-a)])
+ O(IBileXp[-(mi/c)I'2(b-t)]), for
i = 1,...,N.
Generalizing to the system (S1), we may anticipate
that if there is a positive constant
matrix of H
evaluated at
YTJOY > mIIYII2 for all
y
in
m
such that
JO(t), the Jacobian
y = 0, satisfies the inequality (7.2)
[a,b],
in IRN, then the problem (S1) will have a solution which satis-
fies the limiting relation (7.1), provided ently small.
IIAII
and
IIBII
are suffici-
This restriction (7.2) is precisely the condition that the
coefficient matrix of the corresponding linearized system sould be positive definite.
To consolidate the above heuristic ideas, let us first define the following region in gtN+1
V = {(t,y): a < t < b, 11Y11 < d(t)}. Here in
d(t)
is a smooth positive function such that
[a,a + 6/2], d(t) ° IIBII + 6
[a + 6, b - d], where
6 > 0.
in
[b - 6/2, b]
d(t) = IIAII + 6 and
d(t)
=_ 6
the reduced solution of (R1) may be regarded as the vector analog of Definition 3.1.
in
The following definition of stability for
108
VII.
Definition 7.1.
The zero solution
said to be norm-stable
u E 0
SINGULARLY PERTURBED SYSTEMS
of the reduced system (R1) is
[a,b]) if there is a positive constant m
(in
such that
YTJ(t,y)y > m1Iy1I2 where to
for
(t,y)
in
(7.3)
D,
is the Jacobian matrix of H with respect
J(t,y) _ (8H/8y)(t,y)
Y.
The condition (7.3) is obviously stronger than the condition (7.2).
We will see that if u ° 0 small
e > 0
is norm-stable, then for all sufficiently
there exists a solution of the semilinear problem (S1) sat-
isfying the limiting relation (7.1).
This is the content of the first
theorem (cf. [54], [41]). Theorem 7.1.
Assume that the zero function is a norm-stable solution of
The result is a consequence of Theorem 2.4 if we can construct
a scalar function P = P(t,y,e) = IIyII - y(t,E)
with
p
satisfying (2.5), or equivalently with
Ey" < YTH(t,y)/IIyII
whenever
y
satisfying
y = IIyII
(Refer to the discussions immediately following Theorem 2.4). Define
y = y(t,E) = IIAIIexp[-(ma-1)1/2(t-a)] + IIBIIexp[-(ma-1)1/2(b-t)].
Then y
satisfies
IIyII < y, is in
Ey" = my.
Note that the set
V for all suffficiently small
(t,y), with
E, say
From the Mean Value Theorem and (7.3) it follows that for
yTH(t,y)/I IrI I where have
(t,g)
=
a < t < b,
0 < e < E0. (t,y)
in
D,
yTJ(t, )y/I Iyj I _ ml Iyl I,
is the intermediate point, and so, whenever y = IIyII
we
The Semilinear Dirichlet Problem
7.2.
109
YTH(t,Y)/IIYII - Ey" > gym, - ey" = 0.
We conclude from Theorem 2.4 that for solution
y = y(t,e)
of class
0 < e < c0, the problem (SI) has a such that
C(2)([a,b])
p(t,y(t,e),e) < 0,
that is, IIy(t,E)II < Y(t,E) It is possible to prove a more general result than Theorem 7.1, in which a limiting relation of the form (7.4) is still valid (cf. [41]). We simply replace the inequality (7.3) by the condition that there exist a smooth, real-valued function
h = h(t,z)
such that, for
(t,y)
in
V,
(YT/IIiiI)H(t,r) > h(t,IIrlI), and
h
satisfies:
h(t,0) = 0; 8 (t,0) > m > 0,
1E h(a,s)ds > 0 0
for
for
a < t < b;
0 < C < IIAII,
-
and fn
h(b,s)ds > 0
for
0 < n < IIBIJ.
0
(See Remark 3.3 at the end of Chapter III and Example 8.21 in Chapter VIII.)
Thus far we have concentrated on the bounds for the norm of the solution of (S1), and in so doing, we have ignored the behavior of individual components as
H
side
[68], [69]).
(cf.
e -
0+.
Let us now seek conditions on the righthand
which will allow the study of each component of the solution We first define the
N
regions in IR:
Ui = {yi: Iyil m > 0 J (7.8)
yTH(t,y)/IIYII > h(t,IIyII), for
in the region
(t,y)
b - [a,b] x {y
is a small positive constant.
in 1RN: IIyII < d}, where
Now we may recall from the results of
Chapter III that under the condition (7.7), the Robin problem for h(t,z) in
has a nonnegative solution
z(t,e)
also be close to zero, if we can show that result of the inequality (7.8).
ez" _
which is asymptotically zero
Consequently, the norm of a solution
[a,b].
d
of (S2) will
y(t,e)
IIy(t,c)II < z(t,e), as a
The precise statement is contained in
the following theorem. Theorem 7.3. in
[a,b]
Assume that the reduced system (R1) has the solution
and that there exists a scalar function
for
(t,z)
in
[a,b] x {z: Izi < 6}, satisfying (7.7) and (7.8).
there exists an a solution
and continuously differentiable with respect to
t, z
with respect to
u = 0
h = h(t,z), continuous
e0 > 0
such that for
0 < e 0, define the scalar function
z = z(t,c)
For
t
in
p(t,y) = IIYII - z(t,c),
is the nonnegative solution of the scalar problem in
by the Cauchy-Schwarz inequality. Q, it follows that
ez" < (yT/IIYII)H(t,y)
whenever
Similarly,
qp(b) + p'(b) < 0.
Finally
z = IIYII; indeed,
(yT/IIYII)H(t,Y) - ez"
> h(t,IIYII) - ez" h(t,z) - ez" = 0, by our choice of the function
solution
z, provided
a
is sufficiently small, say
Thus Theorem 2.5 tells us that the problem (S2) has a smooth
0 < e < e0.
y = y(t,c)
satisfying
p(t,y(t,e)) < 0
[a,b], that is,
in
IIY(t,e)II < z(t,e) < (em 1)1/2I IAIlexp[_(me-1)1/2(t_a)] +
(CM1)1/2IIBIIeXp[-(me-1)1/2(b-t)],
by virtue of Theorem 3.4.
We note that if the reduced system H(t,u) = 0 zero solution
has a smooth, non-
u = u(t), and if the inequalities (7.3) and (7.7), (7.8)
are modified accordingly, then results analogous to Theorems 7.1 and 7.3 are valid.
These modified conditions allow us to derive an estimate for
Ily(t,e) - u(t)II
in
0+.
[a,b]
as
a -
It is also possible to obtain
componentwise bounds on solutions of the Robin problem (S2), in much the same way we obtained bounds in Theorem 7.2 on solutions of the Dirichlet problem (Sl).
Indeed, for the Robin problem (S2) the assumptions (7.5)
and (7.6) reduce to the simpler assumptions that for
t
in
and
[a,b]
i = 1,...,N Hi(t,Yl,...,Yi_1,0,y.
(7.5a)
11...,yN) = 0
and
(8Hi/ayi)(t,Y1,...,yN) > mi > 0,
for
Iyil < 6.
(7.6a)
That is to say, we need only verify (7.6a) along the zero solution.
With
these ideas, we leave it to the interested reader to formulate for (S2) a result which is analogous to Theorem 7.2.
114
VII.
SINGULARLY PERTURBED SYSTEMS
The Quasilinear Dirichlet Problem
97.4.
We turn finally to an examination of the existence and the asymptotic behavior of solutions of the quasilinear vector problem ey" = F(t,y)y' + g(t,y), y(a,c) = A, Here
tinuous in
y(b,c) = B.
is a continuous
F
a < t < b,
(NXN)-matrix-valued function and
g
is a con-
N-vector-valued function, and each is continuously differentiable
y, on
[a,b] x1RN.
Depending on the properties of
F
and
tions of (S3) can exhibit a variety of asymptotic behavior as
g, soluc-* 0*;
indeed, as we have already noted in Chapter IV, the scalar form of (S3) is already fairly complicated.
The principal difficulty in studying the
system (S3) arises from the coupling of the first-order derivatives in the righthand side.
It is perhaps not surprising then that we must treat
this problem under some rather restrictive conditions on
F.
We will
study (S3) in the same manner as the semilinear problem (S 1), by first
considering norm-bound estimates on its solutions, followed by componentIn order to apply
wise estimates, in the spirit of O'Donnell's work [68]. O'Donnell's techniques we must assume that
F
is a diagonal matrix, that
is, we assume (S3) is a weakly coupled system in which the derivative of the i-th component appears only in the i-th equation.
Norm-bound results
can also be obtained for systems which are not necessarily weakly coupled; however, the estimates on the norm are
usually much cruder than the cor-
responding estimates on the individual components. Motivated by the scalar theory of Chapter IV, let us begin by considering solutions of the two reduced problems in
(a,b)
F(t,u)u' + g(t,u) = 0,
u(a) = A
(RL)
F(t,u)u' + g(t,u) = 0,
u(b) = B,
(RR)
and
which are stable in the sense of Definitions 7.3, 7.4 respectively.
We
first define the regions
D(uL) _ {(t,y): a < t < b, IIy-uL(t)II < dL(t)} and
D(uR) = {(t,y): a < t < b, 11y-uR(t)II < dR(t)}. Here
dL
is a smooth positive function such that
dL(t) = IIB-uL(b)II + 6
The Quasilinear Dirichlet Problem
7.4.
for
in
t
[b-6/2,b]
dL(t) F 6
and
a smooth positive function such that in
and
[a,a+6/2]
A solution
Definition 7.3.
for
dR(t) E 6
115
for
t
in
[a,b-S], while
dR(t) E IIA-uR(a)II + S t
u = uL(t)
in
in
(t,y)
dR
is
t
[a+S,b].
of the reduced problem (RL) is
said to be norm-stable if there exists a positive constant for
for
k
such that
V(uL)
zTF(t,y)w > kzTw, z, w
for all
in IR
N
A solution
Definition 7.4.
u = uR(t)
of the reduced problem (RR) is
said to be norm-stable if there exists a positive constant for
in
(t,y)
k
such that
V(uR)
zTF(t,y)w < -kzTw,
in e.
z, w
for all
These definitions are obvious extensions of Definitions 4.1 and 4.2 for the scalar analog of (S3).
They imply that the matrix
tive definite (negative definite) along layer at
t = b (t = a).
uL
F
is posi-
(uR) and within the boundary
This is a rather strong restriction on
F; how-
ever, this can be slightly weakened by means of certain integral condi[47] and Example 8.24 in the next chapter).
tions (cf.
We can now state
a basic result for the quasilinear problem (S3). Theorem 7.4. tion
Assume that the reduced problem (RL) has a norm-stable soluof class
u = uL(t)
such that for
0 < E < c0
C(2)([a,b]).
Then there exists an
the problem (S3) has a solution
e0 > 0
y = y(t,c)
satisfying
IIY(t.E) - h(t)II < IIB - uL(b)Ilep[-k1E-1(b-t)] + KE, where
K
Proof:
is a known positive constant
0 < k1 < k.
In order to simplify the proof, let us introduce the new depen-
dent variable
v = y-uL(t), in terms of which the problem (S3) becomes
the problem
EV" = f(t,V,V',E), v(a,E) = 0, where
and
v(b,c) = B - uL(b),
116
SINGULARLY PERTURBED SYSTEMS
VII.
f(t,v,v',e) = F(t,v + uL(t))v'
+ F(t,v + uL(t))uL(t) + g(t,v + uL(t)) - euL(t). The antitipcated application of Theorem 2.4 prompts us to define, for 0 < e < k2/(4L), the function
then Theorem 2.4 is applicable, if we can show that (vT/llvll)f(t,v,v',c)
- ey" > 0
(a,b)).
(in
Expanding by the Mean Value Theorem gives us
(vT/I lvI I)f - ey" (vT/IIvii)[f(t,0,0,0) Y
=
+ F(t,v+uL(t))v'
+ (af/av)(t,r1,0,0)v - euL(t)] - ey"
(vT/Ilvll)F(t,v + uL(t))v'
+ (vT/Ilvll)(3f/av)(t,n,o,o)v - eL - ey", since
f(t,0,0,0) - 0.
point and
L =_
max
Here
(t,n,0,0)
IIuL(t)Il.
Now if
is the appropriate intermediate a
is sufficiently small, say
[a,b]
0 < e < CO, then the point
(t,v + uL(t))
and so the norm-stability of uL ity
belongs to the region
V(uL),
allows us to continue with the inequal-
7.4.
The Quasilinear Dirichlet Problem
(vT/I
117
lvIl)f - ey" > kvTv'/I IvI I - 9I Ivl I - eL - EY" -ka1IIB - uL(b)Ilexp[Xl(b-t)] -ka2cvL-lexp[X2(a-t)] -RIIB - uL(b)Ilexp[al(b-t)] -9.evR-1exp[a2(a-t)] + ev - eL
- ex IIB - uL(b)IIeXp[Xl(b-t)] -
ex2cvR-1exp[J' 2(a-t)]
= 0,
if we set
v = L, since
cX
+ kai + R = 0, i = 1,2.
Therefore it follows
from Theorem 2.4 that the problem (Si), and hence the original problem (S3) have, respectively, C(2)-solutions that
v = v(t,c), y = y(t,e), such
p(t,v(t,e)) < 0, that is,
IIv(t,e)II = IIY(t,e) - h(t)II < IIB - uL(b)IIexp[-klc-1(b-t)] + Kc in [a,b], 0 < k1 < k
for
K = L£-1(exp[X2(a-b)] - 1).
and
The companion result for a boundary layer at
t = a
follows easily
from Theorem 7.4 and Definition 7.4 via the change of variable
t - a+b-t.
We leave its precise formulation to the reader. For classes of problems such as (S3), it is often advantageous to seek componentwise-bounds, rather than norm-bounds, on the solutions.
In
order to accomplish this, let us assume in what follows that the matrix F
is diagonal, say
tions
fi.
cY
F(t,y) = diag{f 1(t,y),...,fN(t,y)}
for smooth func-
Then the system (S3) can be written in component form as
= fi(t,Y)y! + gi(t,Y),
a < t < b, (S4)
Yi(a,c) = Ai,
Y1(b,c) = Bi.
Since the righthand side of the i-th equation depends only on
y
!
and
i
does not depend on
y!
(j # i), we say that the quasilinear system is
weakly coupled.
Let us now look for solutions of (S4) which exhibit boundary layer behavior at t = b
t = a; analogous results for boundary layer behavior at
then follow in the usual manner.
With our first assumption that
118
VII.
the reduced problem (RR) has a solution
SINGULARLY PERTURBED SYSTEMS
u = u(t)
of class
C(2) ([a,b]),
we define the regions i = 1,...,N,
V. = {yi: lyi - ui(t)l < di(t)},
is a smooth positive function such that
where each
d i
ui(a)l + 6
for
with
and
[a,a+6/2]
in
t
a small positive constant.
6
di(t) - 6
for
di(t) = JAi in
t
[a+6,b],
The second assumption is that
u
additionally satisfies the reduced differential equation in the following strong sense, namely, for
i = 1,...,N
fi(t,yui)ui + gi(t,yui) = 0, for all j # i.
(7.9)
(t,yui) - (t'yl' ...Ayi-1'ui'yi+1'" ''N )
with
in
yj
Dj,
A solution of the reduced equations (7.9) will be called a strong
reduced solution, to distinguish it from the reduced solution of the reduced system (RR).
As was the case with the semilinear problem (S1) (cf.
(7.5)), the second assumption is precisely the condition which allows us to decouple the system, and thereby apply the scalar theory of Chapter IV to the problem (S4).
Lastly we require
to be stable in the follow-
u
ing sense.
A strong reduced solution
Definition 7.5.
u = u(t)
lem (RR) is said to be componentwise-stable (in
[a,b]) if there are
such that
ki
positive constants
of the reduced prob-
fi(t,y) < -ki < 0, N for all
(t,y)
in the region V _ [a,b] x
V
11
i=l
With this notion of stability, we have the following result (cf. [68]). Theorem 7.5.
Assume that the reduced problem (RR) has a componentwise-
stable strong solution exists an solution
E0 > 0
u = u(t)
such ;hat for of class
y = y(t,E)
Proof:
0 < ki < ki
and each
Then there
C(2)([a,b]).
the problem (S4) has a satisfying, for
i = 1,...,N,
KiE, Ki
is a known positive constant.
It is enough to consider just the i-th component.
for definiteness, that tions
0 < e < c0 C(2)([a,b])
lyi(t,c) - ui(t)I < IAi -
where
of class
We assume,
ui(a) > Ai, and so we define the bounding func-
The Quasilinear Dirichlet Problem
7.4.
119
ai(t,E) = ui(t) - wi(t,e) - Wi(t,e) and Si(t,E) = ui(t) + Wi(t,E), where
wi(t,E) = (ui(a)-Ai)exp[Ai(t-a)]
and
Wi(t,c) = eviRi1(exp[ui(t-b)]-1),
in order to apply the "componentwise" version of Theorem 2.4. and
ai - -kic-1
Eat + kiA + Ri
ui -
-Rikil
0 <e < k2(4Ri)-1), Ri
(for
Here
are the negative roots of the quadratic is a positive constant
such that ((afi/Byi)(t,y)ul + (Bgi/Byi)(t,y)I < Ri, for
(t,y)
V, and
in
max
vi
Iu (t)I.
Clearly
ai < Si, ai(a,c) fi(t,Yai)ai + gi(t,yai),
The vector
yvi
aj < y. < Sj
is equal to
cs
< fi(t,ysi)S1 + gi(t,y81).
(y1, ....yi-1'v1,yi+1'" .,yN), where
for appropriate bounding functions
a., S., j # i, that is,
yj = uj(t) + 0(IAj-uj(a)Iexp[-kjc-1(t-a)] + 0(c), and so for
c
(a,b):
sufficiently small, say
0 < c < c0.
yj
satisfies the required inequality, as the verification for analogously.
is in
We only verify that Si
V.
ai
proceeds
Differentiating and expanding via the Mean Value Theorem,
we have ca
- fi(t,yai)ai - gi(t,yai)
= CO - eaiwi
-
cp1(Wi + Evl)
- fi(t,Yui)ui - gi(t,yui)
+ [(afi/ayi)(t,nai)ui + (agi/ayi)(t,nai)](wi + Wi) + fi(t,yai)[aiwi + ui(Wi + eviRil)], nai = (yl' ..., yi-1,ui + 0(ai-ui)'yi+1" ... N)' 0 < 0 < 1, is the appropriate intermediate point. Since, by assumption, fi(t'yui)ui +
where
gi(t'yui) ' 0 ity
and
fi(t,yai) < -ki < 0, we can continue with the inequal-
120
SINGULARLY PERTURBED SYSTEMS
VII.
eai - fi(t,yai)ai - gi(t,yi)
> - evi - ex2wl - eui(W1 + v
ii
- tiwi - Ii(Wi + evitil) + evi - kiaiwi
- ku(W. +
evltil)
= 0,
eai + kiai + ti = 0
owing to the fact that
and
We conclude from Theorem 2.4 that for has a solution
of class
y = y(t,e)
yi(t,e) < di(t,e)
eui + kiwi + fi = 0.
0 < e < e0
C(2)([a,b])
the problem (S4)
satisfying
ai(t,e) m > 0
h(t,IIyII)) or
(for
h
(8Hi/ayi) > mi > 0
such that for
(yT/IIyII)H(t,y) >
i = 1,...,N.
Since these
are "scalar" conditions, we can easily apply the theory of Chapter III on higher-order stability conditions (cf. Definitions 3.1-3.6) to (S1) and (S2). regard.
The papers [41] and [46] are relevant in this
122
VII.
7.4.
The conditions (7.5) and (7.9), which guarantee that a reduced solu-
SINGULARLY PERTURBED SYSTEMS
tion is a "strong" solution, deserve a brief comment.
It turns out
that these conditions are not invariant, under even a linear change of variables.
In other words, it may be possible to transform a
system like (SI) or (S4), not originally having any reduced solution satisfying (7.5) or (7.9), respectively, into a new system for which these conditions obtain. 7.5.
Theorem 7.1 is due originally to Kelley [54] (cf. also [41]), Theorems 7.2, 7.5 and 7.6 are due to O'Donnell [68], [69], Theorem 7.3 is due to Howes [46] and Theorem 7.4 is due originally to Chang [12], who used a "diagonalization" method of approach.
Earlier work
on related problems includes the papers of Levin and Levinson [61], Levin [59], [60], Harris [32], Hoppensteadt [37], Chang and Coppel [13], Howes and O'Malley [47], as well as the monograph of Vasil'eva and Butuzov [88].
Additional references may be found in the mono-
graphs of Wasow [93] and O'Malley [75], and in O'Malley's long survey article [73].
Chapter VIII
Examples and Applications
Part I - SCALAR PROBLEMS Examples of Semilinear Problems and Applications
§8.1.
Example 8.1.
Consider the Dirichlet problem
ey" = (y - u(t))2q+1, y(-l,c) = A, where
-1 < t < 1,
y(l,e) = B,
is a nonnegative integer.
q
If the function
u(t), defined for
-1 < t < 1, is twice continuously differentiable or has a bounded second derivative, then by Theorem 3.1, for sufficiently small chlet problem has a solution lim+ y(t,e) = u(t)
y = y(t,e)
e > 0, the Diri-
which satisfies
in
(8.1)
e->0
where
0 < d < 1.
Moreover, the behavior of the solution
boundary layers at
t = -1
and/or
t = 1
u(-l) # A
(if
y(t,e)
in the
and/or
u(1) # B) can be described by means of the layer functions given in the conclusion of Theorem 3.1. If we choose
u(t) _ Itl = max{-t,t}, then the reduced solution is
not differentiable at
t = 0.
In this situation the reduced solution
is best regarded as the union of the stable path and the stable path
u+(t) = t
in
[0,1].
3.9 to deduce the existence of a solution fies the limiting relation (8.1). y
is also obtained from this theorem.
123
u
[-1,0]
We can then apply Theorem y = y(t,e)
t = 0
which also satis-
of the reduced paths
Note that, as
larger, the thickness of the angular layer at pondingly.
in
The precise behavior of the solution
in a neighborhood of the crossing point
u_, u+
u _(t) = -t
t = 0
q
becomes
increases corres-
124
VIII.
Example 8.2.
Let us consider next the related problem
ey" = (y
- UM)
y(-l,E) = A,
where in
2n,
-1 0
provided
0 < JEI < f,
0
we find that if
as
CAI < f and
IBI < T2, then the problem has a solution
e - 0+ such that lim+ y(t,e) = 0 E+
in
[6,1-d],
by virtue of O'Malley's result [76]. The problem also has solutions exhibiting what is termed spike layer behavior, in that the solutions are asymptotically zero except at regularly spaced points.
In a neighborhood of such a point the solution
8.1.
Examples of Semilinear Problems and Applications
has a spike of finite height which does not vanish as 8.1, 8.2.
u2 = 0
125
a - 0+; cf. Figures
This follows, again from a result of O'Malley [76], because
is a maximum point of the potential energy functional
'1'(Y) = -fA (s-s3)ds = (y4-A4)/4 - (y2-A2)/2
and T(J) = Y'(0) > 0
JAI < f), with
T.
not a maximum point of n > 2
plies that for each integer four solutions
y = y(t,e)
1im+ y(t,a) = 0
e
in
as
e -
(if
O'Malley's result im-
the problem (with
CAI, IBI < 72) has
satisfying
0+
[6,14],
y y
I
T2 B
A
J t
1/2
0
1/2
Figure 8.1
Spiked Solutions of Example 8.3 for n = 2.
1
126
VIII.
n
A
A
v .. 1/3 1/2 2/3
EXAMPLES AND APPLICATIONS
t
U 1/3 1/2 2/3
1
1
fl
A
"UU 1/3 1/2 2/3
t
uuu 1/3 1/2 2/3
1
Figure 8.2
Spiked Solutions of Example 8.3 for n - 3.
t 1
Examples of Semilinear Problems and Applications
8.1.
127
with the exception that
1+ y(ti,a) = T for ti = i/n The four solutions for the cases
(1 < i < n-1).
n = 2
and
are pictured in
n = 3
Figures 8.1 and 8.2.
We note finally that since 4`(O) = `Y(-T), with -r not a maximum point of
y = y(t,e)
four solutions
n > 2
J181 Al, < I) also has for each integer
'Y, the problem (if
lim+ y(t,e) = 0
as
a -
0+
satisfying
in
a+0
with the exception that
r.
1+ Y(te) _
We consider next the related problem
Example 8.4.
0 < t < 1,
ey" = Y3 - y = g (y) , y(O,E) = A,
Since
g = -h
Y(l,e) = B.
we see now that the reduced solutions
are stable, while
u2 = 0
ul = 1
and
u3 = -1
Let us look at the function
is unstable.
ul
The integral conditions require that (cf. Remark 3.3)
first.
(s3 - s)ds > 0
for
E
between
1
and
A
or
B,
1
and a short calculation shows this inequality holds provided B > -1.
Consequently, if A,B > -1
y = yl(t,e)
as
1
and
then the problem has a solution
such that
e + 0+
li0 m+
A > -1
in
By symmetry we note that an analogous result holds for the reduced solution
u3 = -1.
Namely, if the boundary values A and
A, B < 1, then the problem has another solution 1iQ+ y3(t,e) = -1
y1, y3
B
satisfy such that
in
In particular, we note that if A = B = 0 three solutions:
y = y3(t,e)
and
y2
0.
then the problem has at least
VIII.
128
EXAMPLES AND APPLICATIONS
Finally let us hasten to point out that the problem has solutions which display discontinuous interior layer (shock layer) behavior (cf. Remark 3.4).
As an illustration, suppose that
A < -1
O'Malley [76] has shown that the problem has a solution
and
B > 1.
y = y(t,e)
Then as
satisfying
e + 0+
1-1
in
[6, Z - d],
1
in
[2 + d, 1-d],
lim+ Y(t,e) _
that is, y
transfers from
u3
to
which shrinks to zero as
t = 1/2
in a neighborhood of the point
u1
a - 0+.
He has also shown that when
A = B = 0, for example, the problem has for each nonnegative integer two solutions with limiting values and
-1.
which switch
1
n
n-times between
1
Thus, because in this example there are two stable reduced
solutions separated by an unstable one, we see that there is a countably infinite number of solutions. Application 8.1.
The following boundary value problem arises as a model
problem in the theory of nonpremixed combustion (cf. [97])
-1 < t < 1,
ey" = Y2 - t2 S h(t,y), Y(-',E) = Y(1,E) = 1.
Here
(assumed to be very small) is a ratio of diffusive effects to
a
the speed of reaction, and t = 0
is a distance coordinate, chosen so that
t
is the location of the flame, where the fuel and the oxidizer meet
each other and react.
The functions
y - t
and
y + t
represent the
mass fractions of fuel and oxidizer, respectively. In the limit of infinite reaction rate solutions
path
uI(t) = t
u(t) = Itl
and
u2(t) _ -t.
e = 0, we obtain the reduced
From these we form the stable
(known in combustion theory as the Burke-Schumann ap-
proximation [97]), that is, in
[-1,1] 2
2y (t,u(t)) > 0
2 > 0.
and ay
Theorem 3.10 then tells us that for sufficiently small ary value problem has a solution
y = y(t,e)
in
[-1,1]
e > 0
the bound-
satisfying
Y(t,E) = Itl + 0((e1/3/a)(1 + altl/e1/3)-2),
where a at
t = 0
is a known positive constant. is of order
c
1/3
Thus the thickness of the flame
Examples of Semilinear Problems and Applications
8.1.
129
The article of Williams [97] also discusses the same boundary value problem for more general differential equations of the form
ey = (y2
-
(8.2)
n > 1,
t2)n,
and m > n > 1.
syn = (Y+t)n(Y-t)m,
(8.3)
Using the theory of Chapter III, the reader should have no difficulty in seeing that the thickness of the flame at t = 0 in the case of model e1/(2n+1) (8.2) is of order and of order el/(m+n+l) in the case of model (8.3).
We turn now to a consideration of some related Robin problems. Example 8.5.
The first Robin problem is
eY" _ (y - u(t))2q+1,
-1 0, and the theory
tells us that the solution of the problem satisfies lim+ y(t,e) = y
in
[a,b-d].
e+0
There remains the case when
F(y+) = F(y_).
The existence of bound-
ary layers is now precluded, and so we look for another type of limiting behavior as tion for all a(t,e) 2 y-
c + 0+.
e > 0 and
(Note that the boundary value problem has a soluby virtue of Theorem 2.1, which is unique, since
B(t,s) 2 y+
are lower and upper solutions, respectiv-
Because the reduced solutions are constants, the only type of
ely.)
behavior possible is that involving a shock layer connecting the states y-
and
y+
at some point
t0
in
(a,b).
In order to locate this
transition point, let us begin by noting that the solution satisfies y'(t,e) = const. exp[F(y(t,e))], and so
y'(t,e) > 0
since by assumption
(8.8)
y_ < y+.
the original differential equation in the form e(ln y')' = f(y).
Thus we can rewrite
8.2.
Examples of Quasilinear Problems and Applications
133
which in turn allows us to write the two equations Jt
C in y'(t0,e) - e in y'(a,e) = J 0 f(y(s,e))ds
(8.9)
a
and
e in y'(b,e) -
in y'(t0,e)
f(y(s,e))ds.
(8.10)
J t
But
y'(a,C) = y'(b,e)
by virtue of (8.8), since
F(y_) = F(y+), and so
adding (8.9) and (8.10) gives 1t
0 f(y(s,e))ds +
0 =
f(y(s,e))ds.
(8.11)
ft
a
0
Finally, since y_,
a < t < t0-d,
y+,
t0+d < t < b,
lim+ y(t,e)
(that is, there is a shock layer at
t0), if we take the limit as
a - 0+
of both sides of (8.11), we obtain from the Dominated Convergence Theorem the limiting relation
t
0=
! 0 f(y_)ds + a
j
b
t0
f(y+)ds = f(y-) (t0 - a) + f(y+) (b - t0).
It follows that the shock layer is located at
to = [f(y_)a - f(y+)b]/[f(y_) - f(y+)]. For example, if connecting at
y_
f(y) = -y, then in order for there to be a shock layer to
y+, we must have
y+ = -y_ > 0; the layer is located
t0 = (a+b)/2.
The relation
F(y_) = F(y+), known as the Rankine-Hugoniot shock
condition (cf. [17; Chapter 3], [55; Chapter 4]), arises in modelling compressible flows and chemically reacting flows, as shown in the next application.
Application 8.3.
This problem concerns the description of the one-
dimensional, steady-state flow pattern arising from the injection of a gas at supersonic velocity into a duct of uniform or diverging crosssectional area when a back pressure is applied.
Complications such as
the effect of viscous stresses on the duct walls is gas is assumed to be perfect and polytropic.
neglected, and the The time-independent laws
of conservation of mass, momentum and energy can be expressed in the
134
EXAMPLES AND APPLICATIONS
VIII.
following dimensionless form by referring all quantities to appropriate lengths, physical constants and upstream conditions (cf. [18]): d/dx(pyA) = 0,
(8.12) 2
y dx
(Yp)-1
+
(8.13)
dz (PT) = uP-1 a 2 dx
and
y dz + (Y-1)T[dx + y dz (ln A)] - Y(Y-1)UP-1(dx) wwpp-1
=
Here
r
8.14)
dx 2
is the dimensionless distance measured from the entrance of the
x
is the dimensionless velocity of the gas relative to the velo-
duct, y
city of sound, p between
is the density, y 5/3, T
and
1
is the adiabatic index with a value
is the dimensionless temperature, u
efficient of viscosity, and
Pr
is the co-
is the Prandtl number, taken equal to
Finally A = A(x) (A(0) = 1) is the dimensionless cross-sectional
3/4.
area of the duct relative to the area of the duct entrance.
Crocco [18] we have omitted terms of the form udA/dx
Following
and uPT1 dA/dx
By first neglecting the second-order
in (8.13) and (8.14), respectively.
stress terms in these equations we obtain easily two equations for isentropic flow (cf. [18]) Ay[1 - (Y21
y2)1/(Y-1) = const.
and
y
= 1 -
Y21) Y2
Upon substituting the expression
T(x) = 1 -
y-1
y2(x)
into (8.13) and
using (8.12) we obtain, after a straightforward calculation, an equation for the velocity
y
of the form
uY(POc0)-lAY d4 _ [(Y21) y-Y
1)ff
- d[ln A(1 -
Y21
y2)].
dx The quantities
p0. c0
are respectively an upstream reference density
and the upstream velocity of sound.
sionless term viscosity
u
py(p0c0)-1 = e
Let us now assume that the dimen-
is small, that is, the coefficient of
is small for fixed values of
y, p0
the singularly perturbed quasilinear problem in
and (0,1)
c0.
Then we have
8.2.
Examples of Quasilinear Problems and Applications
E Ay
2 da
=
L(Y21)
Y-Y I]d
dx
[In A(1 -
135
y-1) Y2)]2
(SL) Y(O,E) = Y_,
Y_ > Y+ > 0.
Y(l,e) = Y+>
(This is basically the same boundary value problem used by Pearson [77] in his numerical experiments with
A(x) = 1 + x2.)
The original physi-
cal problem can be restated now in terms of (SL) as follows: supersonic velocity
given a
at the entrance of the duct (x = 0), determine
y_
what subsonic velocity
at the end of the duct (x = 1) produces a
y+
supersonic-subsonic transition in the interior of the duct and also the location of this transition.
Let us consider first the case of a uniform duct, that is, A(x) - 1. The problem (SL) reduces to the simple form
0<x 0
is continuous and vanishes only at
-y-2
the func-
which can be regarded as a dimensionless critical velocity [17; 3].
Consequently, f(y) > 0
for
y > yc
and
f(y) < 0
for
(2/Y+l)1/2,
yc =
Chapter
0 < y < yc.
The Rankine-Hugoniot shock condition is
(+1 2
-1 - (Y+ 1) -1 Y_ + Y_ Y+ + Y+ 2
that is, 2
Y+Y- = Yc =
2 (Y+1)
,
which is known as Prandtl's relation (cf. [17; Chapter 3]). an initial supersonic velocity velocity
y+ = (Y+1) y_1
lim+ y(x,e) _ e+0
such that (SLO) has a solution satisfying
Y_,
0 < x < x0-d,
y+,
x0+d < x < 1.
Here
f (Y+) x0
f(Y+)-f(Y_)
Thus, given
y- > yc, there is a unique subsonic
Y_
Y_ + Y+
136
VIII.
EXAMPLES AND APPLICATIONS
is the location of the shock layer representing a supersonic-subsonic This formula for
transition.
allows us to conclude that if the
x0
supersonic inlet velocity is very large, then the major portion of the x0
flow is supersonic since
The shock layer sits
is close to unity.
close to the end of the duct.
We turn finally to a consideration of (SL) when the cross-sectional area of the duct increases in the downstream direction, that is, dA/dx > 0 for
()
0 < x < 1.
( +l
equation and
c = 0
u-u-1)du
=
we first obtain solutions of the reduced
d[ln A(1
-
Y-1
u2)]
satisfying
u1(0) = s
implicitly as
u2(1) = y+ ul(x)(1 -
Setting
(y-1
= y-(1 _
ui(x))1/(Y-1)
Y21
y2)1/(Y-1)/A(x)
and u2(x))I/(Y-1)
Y21
u2(x)(1 -
A(0) = 1.)
(Recall that
respectively.
(or
y-2
Not unexpectedly, these are the
in
[x2,1]
with
+l
y+y-I, has a solution
x0
y > yc = ((Y+l))1
ul(x) > yc
in
F(u1(x0)) = F(u2(x0)), for in the interval
, that is, Prandtl's relation must hold at
A
(x2,x1).
short calculation shows that this condition is equivalent to yc =
and
[0,x1]
Then our theory applies pro-
x1 > x2.
vided that the Rankine-Hugoniot equation F(y) =
Since 2
is positive (or negative) for
0 < y < yc), we must require that
0 < u2(x) < yc
y2)1/(Y-1)/A(x),
+
beginning of our discussion.
isentropic relations obtained at the f(y) _ (Y2I)
(i-i 2
= A(l)y+(l -
2
x0.
uI(x0)u2(x0)
We conclude
(Y+l)
that under these assumptions, the problem (SL) has a solution satisfying F
lim+ y(x,e) Ct
ul(x),
0 < x < x0-6,
u2(x)'
x0+d < x < 1.
The solution describes a supersonic-subsonic transition at the nonconstant states
u1(x)
and
between
x0
u2(x).
In order to illustrate this result, let us consider the case A(x) = 1 + x2, y- = 0.9129, y+ = 0.375 with this data and with the term
and y = 7/5.
The problem (SL),
added to the righthand side,
-ey dx was treated numerically by Pearson [77] whose results afford a means of
comparison with ours.
For y = 7/5
is slightly less than
y-, and so
and with value
yc
A(x) = 1 + x2 at a point
the critical velocity y_ > yc > y+.
the reduced solution x2
in
u2
y
=
(_(Y2 (Y+l))
1/2
c
With this choice of y+ assumes the critical
(0,1/2), that is, f(u2(x2)) = 0.
As
8.2.
Examples of Quasilinear Problems and Applications
u1, it is easy to see that
regards
137
ul(x) > yc; whence, f(u1(x)) > 0
in
Finally, one can show that the Rankine-Hugoniot equation
[0,1].
has a unique solution
F(ui(x0)) = F(u2(x0))
approximately
x0
in
(x2,l)
This compares well with the value
0.6.
which is
x0 = 0.634
ob-
tained numerically by Pearson for the slightly modified version of (SL) with
of order
c
Example 8.9.
10-8.
The reasoning employed in Example 8.8 extends to more gen-
eral problems of the form
a < t < b,
ey" = f(y)y' + g(y), y(a,e) = y-,
y(b,c) = y+,
provided that the boundary values are solutions of the reduced equation, For instance, we know from Example 8.8 that
that is, g(y-) = g(y+) = 0.
a = y- = -1, b = y+ = 1, f(y) = -y
for
solution connecting
y_
However, if
e + 0+.
to
y+
g = 0, this problem has a
and
across a shock layer at
t = 0, as
g(y) = 1 - y2, then this same result also holds.
Asymptotically the g-term has no effect on the behavior of solutions, provided that the f-term has the properties given above. Example 8.10.
Consider now the problem
0 < t < 1,
ey"=-Yny' +Y, y(O,e) = A,
where
y(l,e) = B,
When
is a positive real number.
n
n = 1, it is the classic
Lagerstrom-Cole model problem about which much has been written (cf. [55; Chapter 2], [20], [39]).
boundary layer at Bn)1/n
t = 0.
then the function
uR(t) = (n(t-1) +
is a strongly stable solution of the corresponding righthand
reduced problem.
Suppose now that
n
First, if n
two cases to consider. ent of
Let us look first for a solution with a
If Bn > n
is a natural number.
is even, then
There are
-yn, the coeffici-
y', is nonpositive throughout the layer, and so Theorem 4.1 implies
that the problem has a solution satisfying lim+ y(t,e) = uR(t) e+0
in
[d,l],
Second, if n
for all values of
A.
of A
is important.
and
uR(O)
is odd, then the relative position
If A > uR(O) = (Bn - n) 1/n
0 < A < uR(O), then the coefficient of
(8.15)
d > 0,
y'
or if
is again nonpositive in the
layer, and so the limit (8.15) also holds for such values of
A.
However,
138
VIII.
EXAMPLES AND APPLICATIONS
if A < 0, then we must apply the integral condition of Coddington and Levinson [14]
(cf. Remark 4.3) which allows values of A
n
(uR(O)
A 1
then
uR(tR) = °, and it is not
However, for the intermediate case
clear how to proceed.
uR(tR) = 1, and we can say a few words about this case. solution is clearly
n = 1,
The zero reduced
(I0)-stable (cf. Definition 4.4), and so the reduced
path
un(t) =
J0,
0 < t
0, then Theorem 4.12
is weakly stable (cf. Definition 4.5).
tells us that for sufficiently small
e > 0, the problem has a solution
satisfying in
Y(t,e) = u0(t)
[S,1].
lm+ i (Of course, if A = 0 Next, if
then this limit is assumed at
-1 < A < 0
duced solution
and
-A < 1-B
uL(t) = A + t
t = 0
as well.)
then the weakly stable lefthand re-
intersects the zero solution at
tR, and we have a situation described by Theorem 4.14.
tL = -A 1-B, there are no angular crossings
8.2.
Examples of Quasilinear Problems and Applications
since
tL > tR.
solution as
uL
In this case, one can show (cf. [20], [39]) that the
c -+ 0
t0 = 1/2(1-B-A)
in
connects
uL
(tR,tL).
Finally, for
and
is strongly (weakly) stable in
with boundary layers at
uR
across a shock layer at A < -1
(A = -1) the function
[0,1], and so there are solutions
or with shock layers at
t = 1
uR, depending on the relative sizes of A
and
139
and
B.
joining
to
uL
See [20] or [39]
for all of the details.
The last phenomenon we discuss is the existence of boundary layers relative to the (I0)-stable zero function, when
If n
n
is a natural number.
B = 0, then Theorem 4.5 tells us that the problem
is even and
has a solution satisfying in
lim+ y(t,e) = 0
for all values of
[6,1]
A, with
However, if n
if A = 0.
S = 0
then by Theorem 4.5 we have for all values of A > 0
is odd,
B < 0, a solu-
and
tion satisfying lim+ y(t,e) = 0
in
E+0
For such values of
A, B
and/or
n, either
uL
or
uR
does not exist
or is unstable. Application 8.4.
For our final application of this section we consider a
catalytic reaction in a one-dimensional fixed-bed reactor packed with catalyst in the presence of axial diffusion.
for the dimensionless concentration
Ey"=y' +g(Y),
x
The boundary value problem
is then (cf. [80; Chapter 4])
0<x< 1,
Y(O,e) - p1Y'(O,e) = A, where
y
Y(l,E) = p2Y'(l,E) = B,
is the dimensionless axial coordinate, a
is the reciprocal of
the Peclet number (the diffusional analog of the Reynolds number), and g
is the reaction rate term, of the form
reaction.
The positive terms
pl = pl(E)
g(y) = yn
and
for an n-th order
p2 = p2(e)
are mass
transport coefficients of the type considered in Application 8.2. axial diffusion is weak then
a
If the
is small, and the Robin problem is sin-
gularly perturbed.
We consider only two cases.
First, if p2(e) = 0(E)
then Corollary
4.2 tells us that the problem has a solution with a boundary layer at x = 1, that is,
140
EXAMPLES AND APPLICATIONS
VIII.
lim+ y(x,e) = uL(x)
in
[0,1-6],
g+0 where
uL, the solution of the reduced problem
plu'(0) = A, is assumed to exist in
u' + g(u) = 0, u(0) -
Second, if p2 = 0(1)
[0,1].
or
larger, then Corollary 4.3 tells us that there is a solution which satisfies this limiting relation in all of g(y) = yn, for
n
[0,1].
Here we are taking
a natural number.
Depending on the nature of the function
g, O'Malley [75; Chapter 7]
has shown that this problem may have more than one solution.
Multipli-
city of solutions, which are steady states of the associated time-dependent partial differential equation, is often an important consideration in reactor design, since it implies that the reactor may be capable of operating in more than one stable regime. Example 8.11.
We conclude this section with a problem about which we
know very little, namely
-1 < t < 1,
cY" = ty, + Y 3 - Y,
= -1,
Y(-1,c) -
Clearly
uL
-1
and
uR a 1
Y(l,c) + Y'(l,e) = 1.
are solutions of the corresponding left-
and righthand reduced problems, respectively.
Since these functions are
(I0)-stable, we can apply Theorem 4.10 to deduce the existence of two solutions 1im
yl, y2
of the problem as
satisfying in
c - 0+
[-1,1]
Y1(t,e) = -1
and lim+ y2(t,e) = 1.
There are however other solutions of the reduced equation -- the zero function which is unstable and the one-parameter family of functions
2 2 1/2 ,
uc(t) = ct/(1 + c t ) 8.3.
The functions
uc
where
c
is a constant, as shown in Figure
are all capable of supporting boundary layers
at the endpoints, since the coefficient of
near
t = -1
origin.
y'
is negative (positive)
(t = 1), but they lose stability upon passing through the
We strongly suspect, though, that the
uc's do participate in
describing the asymptotic behavior of other solutions of the problem. It might be instructive to collect some numerical data in this direction.
8.3.
Examples of Quadratic Problems and Applications
141
..................... c < 0
-1
Figure 8.3
Solutions of the Reduced Equation of Example 8.11
58.3.
Examples of Quadratic Problems and Applications
Example 8.12.
Consider the problem
EY" = -Y12 - yy', Y(O,a) = A,
0 < t < 1,
y(l,c) = B,
treated by Dorr, Parter and Shampine [20; 55] in the case
A < B.
Our
first step is to examine the reduced equation
0 = u'(u'+u), and we find
that
uR(t) = Bel-t
uL(t) = A, uL(t) = Ae-t, uR(t) s B
and
solutions of the corresponding reduced problems. equation has no singular solutions. see that Ae
t
,
_ -2y' - y, and so
Setting
are four
Note that this reduced
f(t,y,y') _ -y'2 - yy', we
fy,(t,uL,ul) = y,[uL] _ -A, fy,[uL]
yf
fy,[uR]
-B
and
fY[uR] = Bel-t.
Thus the stability of these
142
EXAMPLES AND APPLICATIONS
VIII.
reduced solutions depends critically on the values of A
and
We
B.
distinguish the following cases. A,B > 0.
Case 1.
Ae_t
A, B, uL(t) =
Clearly for such
and
necessary for boundary layer behavior, that is, uL(1) = Ae For these values of
> B
and
A, B, it follows from Theorem 5.1
y = y(t,e)
that the problem has a solution
_E B
We then check the sign restrictions
are the stable reduced solutions.
uR(0) = B > A.
uR(t)
as
such that in
e + 0+
[0,1]
B - (B-A)exp[-Bt/e] < y(t,e) < B.
if
B > A > 0,
and
Ae-t
-
(Ae-1-B)exp[-Ae-1(1-t)/c] < y(t,e) < Ae-t + ey, if
where
y
is a known positive constant.
Ae-1 > B > 0,
We next check for the occurrence
of an angular crossing involving these reduced solutions. consider values of
ties fail simultaneously; namely, Ae-1 < B see that uL (t0)
uL(t0) = uR(t0) = B
for
and
t0 = ln(A/B)
B < A.
for w
in
It is easy to
(0,1), with
in
_ -B < uR(t0) = 0, and since the inequality
.u(w+B) > 0
That is, we
for which the above boundary layer inequali-
A, B > 0
f(t0,uL(t0),w)
(-B,0), the theorem of Haber and Levinson [27]
We conclude that if
(cf. Corollary 5.20) applies.
0 < Ae-1 < B < A,
problem has a solution satisfying
lim+ y(t,E) = e+0 Case 2.
Ae-t,
0 < t < t0,
B,
t
In this case, the functions
A > 0, B < 0.
uR(t) = Bel-t
< t < 1. o-
are the stable reduced solutions.
see that only the boundary layer inequality
uL(t) = Ae-t
Since
uL(l) > B
obtains, and so
by Theorem 5.1 the problem has a solution satisfying
lim+ y(t,e) = Act in
[0,14].
E+O
In this case there are no interior crossings. Case 3.
This is the reflection of Case 2 and so the
A < 0, B > 0.
solution satisfies lim+ y(t,E) = B
e0
in
[6,1].
and
B < 0 < A we
8.3.
Examples of Quadratic Problems and Applications
A, B < 0.
Case 4.
reduced solutions
143
This is also a reflection, of Case 1, in which the and
uL(t) - A
uR(t) = Bel-t
are stable.
We see
easily that the problem has solutions with the following asymptotic behavior:
lim+ y(t,c) = A
in
if A > B,
[0,1-6],
e+0
lim+ y(t,E) = Bei-t e+0
in
if
[6,1],
Be > A,
and finally 0 < t < to,
A,
lim+ y(t,e) e+0 for
Be
(0,1).
Two cases remain:
first case, uL - A
R
since
uL < 0.
in this case.
A < 0, B = 0
(i)
and (ii) A = 0, B < 0.
is strongly stable; however, since
and f[0] = fy[0] - 0. U
Be < A < B,
t0 < t < 1,
,
in
t0 = ln(Be/A)
if
-
-
1-t
But
Thus
uL(1) = A < B
and
In the
B = 0, uR = uR = 0 does not intersect
uL
does not determine the asymptotic behavior
uL
Let us examine
uR - 0
more closely.
Clearly,
- 0
is
an upper solution of the problem, and a short calculation shows that
,.-1 -r+ a'%
.
1/2,-1
,
-
=
(A-' - (1-T(e))/e
is a lower solution for
and fying
B = 0
e+0+
- 7(1-T(e))e 1 /2,
of order
y(t,e) = 0
in
1-T(e) < t < 1,
el/2, provided that y > 0
We conclude from Theorem 2.2 that if A < 0
the problem has, for sufficiently small
a(t,s) < y(t,e) < 0 lim
1/2 )-1
T(e) > 0
is appropriately chosen.
1/2
in
e, a solution satis-
[0,1], that is,
[6,1].
Arguing in a similar manner, we see that in case (ii) the problem has a solution satisfying lim+ y(t,e) = 0
in
[0,1-6].
e+0
These two cases deserve to be given special care, since appropriate partial derivatives are neither strictly positive nor negative.
144
EXAMPLES AND APPLICATIONS
VIII.
The problem
Example 8.13.
0 0
A, B
which satisfies in
a unique [0,1]
min{A,B,O} < y(t,e) < max{A,B,O}.
The differential equation is the one discussed in Example 8.13, and so we know that the reduced equation has the solutions their envelope
u(t) = 1/4(t+c) 2,
us = 0, and paths formed from finite combinations of
these curves, for example,
u(t)
0
1/2 -1/2(1+4A) or if B 0. F, and
is the only real zero of of Definition 6.1.
Since
u
The function
u(t) = t2
is clearly (Iq)-stable in the sense
u(O) - 0, u(l) = 1
and
u" __ 2, the Dirichlet
problem (cf. Example 8.1). cy" = F(t,y,y'), y(O,e) = 0,
0 < t < 1,
y(l,c) = 1,
has by virtue of Theorem 6.1 a solution fying in
y = y(t,e)
as
s
+
0+
satis-
[0,1]
t2 < y(t,£) < t2 + (2£)1/(2q+l).
u'(0) = 0
On the other hand, since
Theorem 6.3 tells us that the Robin
problem (cf. Example 8.5) ey" = F(t,y,y'), -y'(O,e) = A,
has a solution as
0 < t < 1,
y(l,e) = 1, a - 0+
satisfying in
[0,1] (2£)1/(2q+l)
t2 < y(t,e) < t2 + v(t,e) +
for any value of A > 0.
Here
v(t,e) = 0
if A = 0, and if A > 0,
then
v(t,e) = A£1/2 exp[-t/e1/2]
(q = 0)
and v(t,e) = AT-1ge1/(2q±2)(1 + Tt/£1/(2q±2))-i/q
(q > 1),
for
T = T(q) =
[A2gg2q+2/(q+1)]1/(29+2)
Application 8.6.
The superquadratic theory is often useful in solving
problems involving the form
the curvature of surfaces, since nonlinearities of (1 + y'2)3/2 - 'y'I3, as ly'l - -, arise naturally. As an
illustration, we note (cf. [3; Chapter 1] or [78]) that the elevation of the free surface of a liquid meeting a plane, vertical rigid wall (at
t = 0) is described by the problem
y
0 < t < L,
ey" = Y(i + y'2)3/2
let < w/2, y(L,e) = 0(e),
y'(0,e) = tan e, for p
EXAMPLES AND APPLICATIONS
VIII.
156
and
e2 = T/(pg)
Here
an arbitrarily large positive constant.
L
is the density of the liquid, g
is the gravitational constant, T
the coefficient of surface tension, and
is
is the contact angle, the
e
angle the surface makes with the wall, measured from the horizontal axis, y = 0.
If
is small then
T
is small, and the problem for the eleva-
c
tion is singularly perturbed.
Suppose first that the liquid is water, and therefore, that Since
0 < e < 1/2.
u E 0
is an (10)-stable reduced solution and
-u'(0) = 0 < tan 0, Theorem 6.3 tells us that the elevation y in
satisfies
[0,L)
0 < y(t,e) < (tan e)e1/2exp[-t/el/2] + 0(e). However, if the liquid is mercury, then the contact angle is negative (-1/2 < e < 0),
pression"
y
we now conclude that the "de-
-u'(0) = 0 > tan a
Since
satisfies in
[0,L]
0(e) + (tan e)el/2exp[-t/el/2] < y(t,e) < 0. Example 8.17.
Consider now the problem
eyV1 = -Y' -
y'3, 0 < t < 1,
pY(O,e) - y'(0,e) = A,
p > 0.
Y(l,e) = B,
(We saw earlier that the Dirichlet problem
f(u') = -u' - u'3
fu(0) = -1
we make the corresponding reduced solution satisfy
that is, u = uR(t) - B. y(t,e) E B
The reduced equation
as its only real solution, and since
u' = 0
has
ey" = -y' - y'3, y(0,e) = A,
if A # B.)
c - 0
y(l,e) = B, has no solution as
Suppose first that
is the solution.
for all values of
A
between
0
and
If A = 0
p = 0.
However, if A # 0
then
-A, A 0 0.
u(1) = B, then
Af(x) = -AA(1+a2) > 0
Consequently Theorem
all values of A the solution satisfies in
6.11 tells us that for
[0,1]
y(t,e) = B +
Suppose finally that
p > 0.
tion, while if A # pB
If A = pB
then
y(t,e) E B
is the solu-
then
(p5 - A)f(A) = -A(PB - A)(1 + A2) < 0,
for all values of
A
between
0
and
6.11 we see that for all values of A
pB - A, X j 0. and
B
Again from Theorem
8.4.
Examples of Superquadratic Problems and An Application
Y(t,e) = B + 0(elpB-Alexp[-t/E]) Example 8.18.
0 < t < 1,
Y(O,E) - y'(O,e) = A,
The reduced equation and
we make
u1
Y(l,e) + Y'(l,e) = B.
f(u') = u' - u'3 = 0
u3 = 0, which satisfy and
u1(t) = t + B -2
u2
u3(1) +
satisfy
and
However, if A < B - 3
then
y(t,c) = t + B - 2
and
1
u3
is the solution.
(B - 3 - A)A(1-A2) < 0
B - 2 - A, A # 1.
implies that the solution satisfies in
satisfy
If A = B - 3
first.
ul
(ul(0) - 1 - A)f(A)
between
A
Consider
Theorem 6.11
[0,1]
y(t,s) = u1(t) + 0(Z e(B-3-A)exp[-2t/E]).
Finally, if A > B - 3 and
B - 2 - A, A
then
6.11 it follows that for
(8.23)
(B - 3 - A)A(1 - A2) > 0
1, provided that
Thus
j = 1,2, that is,
for
B
ui = 1,
fu(0) = 1.
and
u2(t) = -t + B + 2, and we make
u1(0) - u1(0) = A, and so
for all values of
now has three solutions
fu,(±1) = -2
u3(0) - u3(0) = A, that is, u3 = A. then
[0,1].
Consider next the related problem
Ey" = Y' - y'3,
u2 = -1
in
157
B - 2 - A > 0.
for
A
between
1
Again from Theorem
the solution satisfies in
B - 3 < A < B - 2
[0,1]
y(t,c) = ul(t) + 0((c/k)IB-3-AIexp[-kt/e]), for a positive constant
k < 2.
The asymptotic behavior described by of that described by
is clearly a reflection
u2
B + 3 < A
In particular, if
ul.
then the relation (8.23) [(8.24)] obtains with the term
IB - 3 - Al
replaced by
IB + 3 - Al
Consider next the reduced solution y(t,e) E A
B - A < 1.
B - A > -1.
u3
is the solution, while if A < B
(A-B)A(1 - A2) < 0
for
A
between
Similarly, if A > B Consequently for
that the solution satisfies in
0
and
k < 1.
u1
[B + 2 < A < B + 3]
replaced by
u2
and
inside, the Landau symbols.
A.
then
If
A = B
then
(A-B)f(A) =
B - A, A # 0, provided
then this inequality obtains provided
B - 1 < A < B + 1 [0,1]
y(t,e) = A + 0((e/k)IB - Alexp[-kt/e]), for a positive constant
(8.24)
Theorem 6.11 tells us
EXAMPLES AND APPLICATIONS
VIII.
158
Summarizing to this point, we have estimates of the solution for all
values of A and
B - 2 < A < B -
B, except those satisfying
B + 1 < A < B + 2, for which Theorem 6.11 is inapplicable.
1
and
Thus we are
led to consider the angular paths
fA,
0 < t < t0,
u5(t) t+B-2,
(A,
-
u4(t) =
0 < t < t0,
llt
t0 < t < 1,
-t+B+2,
t0 < t < 1.
It follows directly that
t0 = A - B + 2
is in
(0,1)
if and only if
B - 2 < A < B - 1, while
t0 = B - A + 2
is in
(0,1)
if and only if
B + 1 < A < B + 2. that
Consider just
(aR - aL)f(A) = A(1-A2) > 0
implies that for
B - 2 < A < B -
u4.
For
for
0 < A < 1, and so Theorem 6.12
aL = 0
oR = 1
and
we see
1
Y(t,s) = u4(t) + 0(Z(c/k)exp[-kIt - t0I/e]) in
[0,1], for a positive constant
B + 1 < A < B + 2
with
u4, t0
Finally, if A = B - 2
k < 1.
[A = B + 2]
[-t + B + 2], while if A = B - 1
A similar result holds for
replaced by
or
u5, t0.
then
lim+ y(t,e) = t + B - 2
A = B + 1, then
lim+ y(t,e) = 0.
None of these limits is a surprise; the convergence is, of course, uniform in
[0,1].
Example 8.19.
cyli = y
The solutions of the nonautonomous problem
- tY' - Y'3 a f(t,Y,Y'), -1 0 for
-1/3 < A < 2/3, and so we conclude that
I
lim+ y(t,e) _
us(t), -1/3 < t < 1.
uR(t),
Example 8.20.
We close this section with the problem -1 < t < 1,
ey" = y + ty' + yn = f(t,y,y'), y(-l,e) - y'(-l,c) = A, for
n m> 3
y(l,e) + y'(l,e) = B,
The function
an integer.
u = 0
is clearly (10)-stable in
the sense of Definition 6.1, and it is also locally strongly since
fyJ0] = t.
Suppose first that
n
is odd.
y'-stable
In order to proceed
we consider the two inequalities (u(-l) - u'(-1) - A)f(-1, u(-l),A) < 0, for
A
between
and
0
-A, A # 0, and
(u(1) + u'(1) - B)f(l,u(1),X) < 0, for for
A
between
and
0
B, A
At < 1, A # 0, since
values of
B # 0.
n
0.
The first inequality clearly obtains
is odd, while the second obtains for all
If A = B = 0
then
y(t,e) __ 0, and so we
by arguing as in the proof of Theorem 6.11 that if n then for all values of lim+ y(t,E) = 0 c+0
B
is odd and
JAI < 1,
the solution satisfies
in [-1,1].
On the other hand, if n
deduce
(8.26)
is even then these inequalities obtain for all
values of A > -1, A # 0, and
B > -1, B # 0, respectively.
have the limiting relation (8.26).
We again
Examples of Semilinear Systems and An Application
B.S.
161
Part II - VECTOR PROBLEMS
Examples of Semilinear Systems and An Application
58.5.
Example 8.21.
Let us illustrate the norm-bound theory of Chapter VII by
u = 0, and it is clearly stable, in the sense of Definition 7.1, since the 1-3yi - 1
matrix
is positive definite, for
J(0,0)
J(y)
1 - 3y2
1
, the
2
Jacobian matrix. y2 (1-3y 2)
Finally, the quadratic form yTJ(y)y = yi(1-3yi) +
is positive definite, only for vectors
lyll < l/, and A2
and
B1, B2
11jg.
ly21 1
hi,Yl(O,y2)
and
h2,Y2(Y1,0) = 1 + yl > 1.
Finally we observe that
hl,yl(Y1.
2) = (i 2y1)(1+y2) > 0
and h2,y2(Yl,Y2)
_ (1-2Y2)(i+yi) > 0,
Therefore, for boundary values such that
for all values of yl,y2 < 1/2. Ai < 1/2
and
(i = 1,2), Theorem 7.2 states that the problem
B. < 1/2
has a solution
y = y(t,e)
lim+ y(t,e) _ (0 0)T
as
satisfying
a - 0+
(8.27)
in
We note that this bound on the boundary values can be improved by using the less restrictive integral conditions that 711
O
h1(s,A2 or B2)ds > 0
for all values of cf.
[69].
ni
012 h2(A1 or B1,s)ds > 0,
and
between
0
and
A.
For boundary values such that
or
Bi, ni # 0 (i = 1,2);
Ai,Bi < 3/2, i = 1,2, we de-
duce therefore the existence of a solution satisfying the limiting relation (8.27).
Application 8.7.
The scalar theory of Application 8.2 concerned itself,
of course, with the behavior of a single reactant undergoing mal catalytic reaction.
an isother-
Suppose however that we have a system of N
reactants, each component of which undergoes such a reaction, influenced by and influencing the other
N-i
components.
Then by arguing as in
Application 8.2 (cf. also [2; Chapter 5]), we see that the steady-state behavior of the concentrations can be governed by a boundary value problem of the form
8.5.
Examples of Semilinear Systems and An Application
0 < x < 1,
ey" = h(y),
PY(O,e) - Y'(O,e) = A,
T
Here an
Qy(l,e) + Y'(l,e) = B,
is the vector of normalized concentrations, h
Y = (yl " 'YN)
N-vector-valued function of y
tics, and
N x N
is
which represents the nonlinear kine-
is the normalized distance.
x
163
P, Q
are positive semidefinite
matrices which contain the various transfer coefficients between
the bulk flow and the solid phase, and
e2
is the reciprocal of the Thiele
modulus, assumed to be the same for each reaction. Suppose now that evaluated along
h(0) = 0
and that the Jacobian matrix
J
of h
is positive definite, in the sense that there is a
0
positive constant mI
for which
(8.28)
yT Jy > ml 2 I IYI I2 , y
for all
in 1RN.
YT h(Y)/I IYI I _ for
It follows from the Mean Value Theorem that m2
i IYI I .
sufficiently small.
IIYII
the problem has a solution
I IY(x,c) I for
Consequently Theorem 7.3 tells us that
y = y(x,e)
as
+ a - 0*
satisfying in
[0,1]
(,T/m) I JAI Iexp[-mx/,T] + (Fe/m) I IBI I exp[-m(l-x)/,r]
from Aris's discussion [2; Chapter 5] of the pseudo-steady-state hypothesis in an enzyme reaction.
reduced equation matrix
J
of h
Clearly
u = 0
is the only solution of the
h = (h1 h2)T = (0 0)T, and the corresponding Jacobian evaluated along
0
is
-(k-a) -1 1
Since
k
k
1
is positive, a necessary and sufficient condition for
positive definite is that
J
to be
164
EXAMPLES AND APPLICATIONS
VIII.
(k-A+1)2 < 4k,
in which case inequality (8.28) obtains with m1 = {k+l - [(k-1) 2 + (1_A+1)2]1/2}/2. (Here we have used the results [26; Chapter 8] that a real symmetric
