Hua C., Wong R. Differential Equations and Asymptotic Theory in Mathematical Physics
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232
and
M(z)
=
{E2(2)Ai2(x)
+
E-2(x)Bi2(z)}1/2, (3.11)
where E-'(x)
=
l/E(x) and
c
denotes the negative root
of
the equation
Ai(z)
=
Bi(z)
with smallest absolute value. Numerical calculation has shown that
c
=
-0.36605, correct to five decimal places. The modulus and the weight have
the well known asymptotic behavior
M(~)
7F-1/2121-1/4,
2
-+
fm,
(3.12)
and
E(X)
2112
exp
(:23/2),
2
+
+m.
(3.13)
The error terms E~(A,z) and
E~(A,z)
in (3.7) satisfy
and
where
p
=
SUP
{~FIX~~/~M~(X)}
=
1.04.
*
. .
(3.16)
(-,=m)
The above method can be extended to the more general equation
YlW
=
A2.(A,
Z)Y(Z),
(3.17)
where
.(A,
x)
has an asymptotic expansion
(3.18)
as
1x1
-+
00,
and
aO(2)
has a zero
20
in the interval
(u1,u2).
Applying the
Eanger transformation (3.2)
-
(3.3) with
u(x)
replaced by
uo(z),
equation
(3.17) becomes
.1(2)
+-+...
.2(.)
A2
.(A,
2)
N
uo(2)
+
-
A
(3.19)
d2w
--
-
{a
+
wo
+
$(A,
C)>w,
dC2
(3.20)
233
and
with
(3.21)
(3.22)
and
$s(()
=
aS+2(()/h(()
for
s
2
1.
An asymptotic series solution to
(3.19)
is given by
O0
As(<)
w(()
-Ai
(
X2l3<
+
")c,
x1,3
(3.23)
s=o
where
6,
=
(a(<)
is defined by
(3.24)
4.
Simple Pole
Returning to equation
(2.18),
namely
Y"(4
=
{X2a(z)
+
b(z)}y(z),
(4.1)
we now assume that
u(z)
has a simple pole (say) at
zo
and
(z
-
z~)~b(z)
is analytic.
For
simplicity, we
also
assume that
u(z)
has the same sign as
z
-
zo.
In this case, Olver
[7]
has introduced the transformation
and
which transforms
(4.1)
into the new equation
234
where
(4.5)
and
2(x)
=
(dC/dx)2
=
4Ca(x). If
b(x)
has a simple or double pole at
50,
then
d(<)
has the same kind of singularity at
C
=
0.
Denote the value
of
C24(C)
at
C
=
0
by
i(u2
-
I),
and write (4.4) in the form
with
+(<)
=
cG(c)
-
i(u2
-
l)<-'.
Note that
+(<)
is analytic at
C
=
0.
In terms of the original variable,
i(u2
-
1)
is the value
of
(x
-
x~)~b(x)
at
x
=
20,
and
1
-
4u2
b(x)
+
4a(x)a11(2)
-
5a'2(x)
+(O
=
-
+
-
64a3
(x)
(4.7)
16c 4a(x)
The differential equation (4.6) has a regular singularity at
C
=
0,
and
we suppose that the range of
C
is a real interval
(a1,aZ)
which contains
C
=
0
and may be unbounded. We consider separately the intervals
[0,
a2)
and
(al,O].
If
the term
+(C)/C
is neglected, then (4.6) becomes
d2w
X2
u2
-
1
=
{
2
+
c.}w.
Two linearly independent solutions of (4.8) are
C1/21u(Xc1/2)
and
C1/2Kv(X<1/2).
It
can be demonstrated, as in the previous sections, that if
C-1/2+(C)
is absolutely integrable on
[0,
a2),
then equation (4.6) has two
linearly independent solutions
wl(X,
C)
and w2(X,
C)
such that
Wl(X,C)
=
C1/21u(XC'/2)[1
+
O(A-l)],
(4.9)
wz(X,C)
=
C1/2Kv(XC1/2)[1
+
O(X-l)],
(4.10)
as
1x1
4
00,
where the 0-terms hold uniformly with respect to
C
E
[0,
a2).
When
C
is negative,
two linearly independent solutions of the re-
duced equation (4.8) are
~~11~2Ju(X~~~1~2)
and
lc/1/2Yv(Alc11/2).
Hence,
if
C-1/2+(C)
is
absolutely integrable on
(a1,0],
equation (4.6) has two
so-
lutions
Wl(X,C)
=
IC11/2Ju(~IC11/2)[~
+
o(X-l)l,
(4.11)
235
as
1x1
-+
00,
which hold uniformly for
C
E
(a1,0]
5.
Examples
As an illustration of the results obtained in Secs. 2-4, here we give three
concrete examples. The first one demonstrates that the Liouville-Green
(WKB)
approximation is indeed a powerful tool
for
approximating solu-
tions
of
linear second-order differential equations. The second one provides
a
uniform asymptotic approximation for the polynomials orthogonal with
respect to the weight exp(-x4) on the real line. The final example deals
with the Jacobi function
pe")(t),
t
>
0.
Example 1. In an interesting paper
[9]
on a very abstract topic (namely,
Hardy fields),
M.
Rosenlicht considered solutions of the equation
y//(x)
=
x"y(x) (5.1)
as
x
+
00.
In view of the rapid growth of the coefficient function x" as
x
-+
00,
it is really not easy to guess the large x-behavior
of
the solution
y(x). Let us try the results in Sec. 2, and choose
X
=
l,a(x)
=
x"
and
b(x)
=
0
in (2.18). The control function F(x) in (2.12) is given by
Clearly,
Ux,m(F)
+
0
as
x
-+
00.
Hence, (2.20) gives the recessive solution
SX"/2dX),
A dominant solution is provided by (2.27), namely,
y3(x)
-
5-44
exp
(/
xx/2dx),
x
-+
03.
x
+
03.
(5.3)
An asymptotic expansion of the integral xXI2dx can be obtained by inte-
gration by parts.
Example
2.
In
[6],
Nevai has studied the asymptotic behavior of the
orthogonal polynomials
pn(x)
=
TnXn
+
Tn-lxn-l
f..
.
,
Tn
>
0,
(5.4)
associated with the weight function exp(-x4) on the real line
R.
These
polynomials satisfy the recurrence relation
XPn(X)
=
an+lPn+l(x)
+
anPn-l(x),
n
=
0,1,.
. . ,
(5.5)
236
with
po(z)
=
70
>
0
and
pl(z)
=
yoz/al.
The coefficients
a,
are deter-
mined successively from the equation
(5.6)
n
4a:(ai+1+
a:
+
a,-1)'
2
n
=
1,2,-..
,
where
a;
=
0
and
a:
=
I'(i)/I'(i).
A
two-term asymptotic expansion for
a,
has been given by Lew and Quarles
[4].
They showed that
If we let
&(.)
=
a:+1
+a:
+
x2,
(5.8)
then Shohat
[lo]
and Nevai
[5]
independently showed that the function
satisfies the differential equation
z"
+
f(n,
z)z
=
0,
(5.10)
where
f(nlz)
=
4~:[4$,(z)$,-1(z)
+
1
-
4a:z2
-
4z4
-
2z2$n(z)-1]
(5.11)
-
4z6
-
4~~$,(2)-~
-
3z2$,(z)-1
+
62'
+
q5,(~)-~.
If we make the change of variable
4n
3'
x
=
x1/4w
with
A=-
(5.12)
then equation
(5.10)
becomes
d2z
Ul(W)
+-+...
@(W)
-
dw2
=
hz[uo(w)
+
A2
(5.13)
where
ao(w)
=
(4w6
-
3w2
-
1)
=
(2w2
+
1)'(w2
-
1)1
al(w)
=
-(1+ 2w2)
and
20w4
-
64w2
+
17
a2(w)
=
-
9(1+2w2)2
'
(5.14)
Since
ao(w)
vanishes at
w
=
fl,
we have exactly the extended form of the
turning point problem mentioned in
(3.17)
-
(3.18)'
and the result
(3.23)
can be applied. The details of this study is given in
[l].
237
Example
3. Let
a,P
and
p
be real numbers with
p
>
0
and
(Y
#
-1,
-2,.
. . .
The Jacobi function is defined by
1
1
2
-(a+,B+l++p);a+l; -sinh2
t
(5.15)
for
t
>
0,
where
2Fl(a,
b;c;z)
is the Gaussian hypergeometric function.
This function is related to the Jacobi polynomial
The last formula furnishes the extension of the polynomial
PP’p’(x)
to
arbitrary values of the degree
n.
From (5.15) and (5.16), it is evident that
and. for this reason,
,pf”’(t)
is called the Jacobi function.
satisfies
It
is known that
,pjL””’(t)
is the unique, even, Cm-function on
R
which
v’’(t)
+
[(2a
+
1)
cotht
+
(2p
+
1)
tanht]v’(t)
(5.18)
+
[p2
+
(a
+
p
+
1)2]v(t>
=
0
and
v(0)
=
1;
see
[2,
p.21.
If
we set
u(t)
=
(sinh
t)a++
(cosh
t)Pf+,pp9p)(t),
(5.19)
then it is easily verified that
u(t)
=
0.
L-p2
p2
+
4
-
-
4
sinh2
t
cash'
t
(5.20)
When
a
>
-4,
we also have
u(0)
=
0.
ourselves to the case
a
2
0
and introduce the new variables
To apply the asymptotic theory of Olver discussed in Sec.
4,
we restrict
(-<)+
=
t,
C<O;
W(C)
=
(-C)&(t).
(5.21)
The transformed equation is given by
where
(5.23)
238
Note that
$(C)
is analytic at
C
=
0.
For
negative
C,
equations
(4.11)
and
(4.12)
give two asymptotic solu-
tions to
(5.22),
one involving the Bessel function
Ja(pG)
and the other
involving
Y,(pe).
To
identify the function
(-<)au(t)
in
(5.21)
with one
of these two solutions
or
a linear combination of them, we note that from
(5.19)
and
(5.21)
we have
C
--$
0-
(5.24)
Since
Ja(z)
N
(z/2)"/r(a
+
1)
for
z
near zero, it follows from
(4.11)
and
(5.21)
that
(5.25)
for details, see
[ll].
References
1.
Bo
Rui and R. Wong,
A
uniform asymptotic formula for orthogonal polyno-
mials associated with
exp(-x4),
J.
Approx. Theory
98
(1999), 146-166.
2.
T.
H.
Koornwinder,
Jacobi functions and analysis on noncompact semi-
simple Lie group,
in "Special Functions: Group Theoretical Aspects and
Applications", R. A. Askey,
T.
H. Koornwinder, and W. Schempp,
eds.,
D.
Reidel, Dordrecht, Holland,
1984, 1-85.
3.
R. E. Langer,
The asymptotic solutions
of
ordinary linear differential equa-
tions of the second order, with special reference to a turning point,
Trans.
Amer. Math. SOC.
67
(1949), 461-490.
4.
J.
S.
Lew and
D.
A. Quarles,
Jr.,
Nonnegative solutions of a nonlinear recur-
rence,
J.
Approx. Theory
38
(1983), 357-379.
5.
P.
Nevai,
Orthogonal polynomials associated with
exp(-x4), in "Second Ed-
monton Conference
on
Approximation Theory", Can. Math. SOC. Conf. Proc.
3
(1983), 263-285.
6.
P.
Nevai,
Asymptotics for orthogonal polynomials associated with
exp(-x4),
SIAM
J.
Math. Anal.
15
(1984), 1177-1187.
7.
F.
W.
J.
Olver,
The asymptotic solution of linear differential equations of the
second order
in
a domain containing one transition point,
Philos. Trans. Roy.
SOC. London Ser. A249
(1957), 65-97.
8.
F.
W.
J.
Olver,
Asymptotic and Special Functions,
Academic Press, New
York,
1974.
Reprinted by A.
K.
Peters, Wellesley,
1997.
9.
M. Rosenlicht,
Hardy fields,
J.
Math. Anal. Appl.
93
(1983), 297-311.
10.
J.
Shohat,
A
differential equation for orthogonal polynomials,
Duke Math.
J.
5
(1939), 401-417.
11.
R. Wong and &.-Q. Wang,
On the asymptotics of the Jacobi function and its
zeros,
SIAM
J.
Math. Anal.
23
(1992), 1637-1649.
239
LECTURE
IV
BOUNDARY LAYER
THEORY
The paper will certainly prove
to
be one
of
the most extraordinary papers
of
this
century, and probably
of
many centuries
Sydney Goldstein
1.
Introduction
At the third International Congress
of
Mathematicians (1904) in Hei-
delberg, Ludwig Prandtl [16] presented
a
short paper entitled On fluid
motion with small frictions. In 1972, at the Symposium on the Future
of
Applied Mathematics, George Carrier
[3]
made the remark that “This
success is probably most surprising to rigor-oriented mathematicians (or
applied mathematicians) when they realize that there still exists no theo-
rem which speaks to the validity or the accuracy
of
Prandtl’s treatment
of
his boundary-layer problem; but seventy years
of
observational experience
leave little doubt
of
its validity and its value”. With comments like these,
one can readily appreciate the important position
of
Prandtl’s principle
of
boundary-layer theory in the minds
of
applied mathematicians.
To
illustrate Prandtl’s idea,
K.O.
Friedrichs used the simple example
d2u
du
dx2
dx
E-
+
-
=
a+2b~
(u(0)
=
0,
u(1)
=
1,
in his 1942 lecture notes [5]. Friedrichs’ analysis can be easily extended to
the more general singularly perturbed two-point boundary-value problem
EY”(X)
+
a(x)y’(x)
+
b(x)y(z)
=
0,
(1.2a)
Y(-1)
=
A,
y(1)
=
B.
(1.2b)
It is now well known that if u(x) is positive, then the asymptotic solution
which holds uniformly in the interval
[-1,1]
is given by
240
This formula is provided in at least eight standard texts; see, e.g.,
[2,
p.4251,
[6, pp.53
&
581,
[7,
p.591, [9, p.681,
[lo,
p.4211,
[ll,
p.2891, [13, p.941 and
[17,
p.1091. Despite what Carrier had said above, the lack of rigor in Prandtl's
boundary-layer theory does raise some concern from mathematicians who
believe that arguments based on purely heuristic reasoning may lead to
incorrect results.
A
derivation of equation (1.3) is given in Sec.
2,
where we also point out
that care must be taken in the use of (1.3) when exponentially small terms
are involved. In Sec.
3,
we consider a case in which the coefficient function
a(.) in (1.2a) has a zero. More precisely, we discuss the case when
a(.)
N
(YZ
and b(x)
P,
asx --+
0,
(1.4)
where
a
#
0
and
P
are constants.
Our discussion will be divided into
three subcases; namely, (i)
a
>
0
and
P/a
#
1,2,...; (ii)
a
<
0
and
@/a
#
0,
-1,
-2,.
. .
;
(iii)
a
>
0
and
P/(Y
=
1,2,.
. .
,
or
a
<
0
and
P/(Y
=
0,
-1,
-2,.
. . .
In case (i), as we shall see, the solution has an internal-layer
behavior. The so-called Ackerberg-O'Malley reasonance refers to case (iii)
.
Sec. 4 is devoted to the nonlinear equation
&UN
+
u2
=
1,
-1
<
x
<
1,
(1.5)
(1.6)
with boundary conditions
u(-1)
=
u(1)
=
0.
2.
Derivation
of
(1.3)
Since
E
is small, it is natural to set
E
=
0
in (1.2a)
so
that we obtain
the reduced equation
a(x)Y'(x)
+
b(x)Y(x)
=
0.
(2.1)
The general solution of this equation is
Y(x)
=
Kexp(l'*dt), a(t)
K
being an arbitrary constant. The boundary condition at
x
=
1
immedi-
ately suggests that
K
=
B
and
In general, Y(x) can not satisfy the boundary condition at
x
=
-1. Hence,
the approximate solution Y(x) is valid only in an interval near
z
=
1. This
interval is known
as
the outer region, and
Y(x)
is
called the outer solution.
241
In the interval near
x
=
-1,
which is known as the
boundary-layer
(or
inner) region,
we make the change of variable
E
=
and define
Y(Z)
=
Y(EE
-
1)
=
%(E).
B”(E)
+
U(EE
-
1)g’(E)
+
Eb(EE
-
1)g(E)
=
0.
YiL
(0
+
4-1)Y;L(E)
=
0.
Clearly,
g(E)
satisfies the new equation
Setting
E
=
0
gives another simplified equation
(2.3)
This equation can again be solved explicitly, and the general solution is
given by
yEL
(0
=
~1+
~ze-+l)E,
where
C1
and
C2
are arbitrary constants. We call
yEL
(0
the
boundary-layer
(or
inner)
solution. The boundary condition at
z
=
-1
gives
C1
+Cz
=A.
Hence,
y
EL
(E)
=
A
+
Cz(e-a(-l)E
-
1).
(2.4)
The first step in Prandtl’s matching principle is to set the two limits
lim
Y(z)
=
Bexp(S1
wdt)
x--1
-1
4t)
and
lim
yBL
(E)
=
A
-
C2
E-m
equal. Thus,
In (2.5) and (2.6), we have made use
of
the assumption that
u(z)
is positive
in
[-I,
11.
The second step in Prandtl’s matching principle is to define the
uniform approximate solution by
Yunif(z)
:=
Y(z)
+
yBL(E)
-
common part;
that is,