Hua C., Wong R. Differential Equations and Asymptotic Theory in Mathematical Physics
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Differential Equations
81
Asymptotic Theory
in Mathematical Physics
SERIES
IN
ANALYSIS
Series Editor: Professor Roderick Wong
City University
of
Hong Kong, Hong Kong, China
Published
Vol.
1
Wavelet Analysis
edited by
Ding-Xuan
Zhou
Vol.
2
Differential Equations and Asymptotic Theory
in
Mathematical Physics
edited by
Hua
Chen and Roderick
S.C.
Wong,
Differential Equations
&
Asymptotic
Theory
in Mathematical Physics
Wuhan University, Hubei, China
20
-
29
October
2003
Editors
Chen
Hua
Wuhan University, China
Roderick
Wong
City University
of
Hong Kong, Hong Kong
K
@World
Scientific
NEW JERSEY
.
LONDON
.
SINGAPORE
'
BEIJING
'
SHANGHAI
.
HONG
KOh
Vol. 2
Published by
World Scientific Publishing Co. Pte. Ltd.
5
Toh Tuck Link, Singapore 596224
USA
ofice:
27 Warren Street, Suite 401-402, Hackensack,
NJ
07601
UK
office:
57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A
catalogue record
for
this book is available from the British Library.
DIFFERENTIAL EQUATIONS AND ASYMPTOTIC THEORY
IN
MATHEMATICAL PHYSICS
Copyright
0
2004 by World Scientific Publishing Co. Pte. Ltd.
All
rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means,
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or
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ISBN
981-256-055-6
Printed
in
Singapore
by
World Scientific Printers
(5)
Pte Ltd
PREFACE
This book consists of lecture notes for four mini courses delivered at
the Conference on Differential Equations and Asymptotic Theory in Math-
ematical Physics held in Wuhan University, China in October,
2003.
Each
course contains five one-hour lectures. These notes give the readers a very
good idea about recent developments in the area of asymptotic analysis,
singular perturbations, orthogonal polynomials, and application of Gevrey
asymptotic expansion to holomorphic dynamical systems. It also includes
invited papers presented at the conference. The topics include the asymp-
totic behaviour for the formal solutions of the singular partial differential
equations in complex domains and the applications of the partial differ-
ential equations in compound crystals, pricing of real options, hydraulic
engineering, diblock copolymer, etc.
The conference has more than one hundred participants. We take this
opportunity to thank
all
of
them, and in particular, the authors of the
articles included in this book.
The Editors
Hua CHEN
Roderick S.C. WONG
V
This page intentionally left blankThis page intentionally left blank
CONTENTS
Preface
..................................
v
PART
I
MINI-COURSES
Ismail, Mourad E.
H.
Lectures on Orthogonal Polynomials
.................
1
Ramis, Jean-Pierre
Gevrey Asymptotics and Applications to Holomorphic
Ordinary Differential Equations
....................
44
Ward, Michael J.
Spikes for Singularly Perturbed Reaction-Diffusion
Systems and Carrier's Problem
...................
100
Wong, Roderick
S.
C.
Five Lectures on Asymptotic Theory
................
189
PART
I1
INVITED
PAPERS
Bohun, C.
S.,
F'rigaard,
I.,
Huang,
H.X.
and Liang
S.
Q.
A Perturbation Model for the Growth of Type
111-V
Compound Crystals
.........................
263
Chen,
H.
and
Yu,
C.
Asymptotic Behaviour of the Trace for Schrodinger
Operator on Irregular Domains
...................
280
Jiang,
L.
S.
and Ren,
X.
M.
Limitations and Modifications of Black-Scholes Model
......
295
Li,
T.T.
(Li,
Daqian)
Exact Boundary Controllability of Unsteady Flows in a
Network of Open Canals.
......................
310
vii
...
Vlll
Miyake, M. and Ichinobe,
K.
Hierarchy of Partial Differential Equations and Fundamental
Solutions Associated with Summable Formal Solutions of
a
Partial Differential Equation of non Kowalevski Type
. . . . . . .
330
Tahara,
H.
On the Singularities of Solutions of Nonlinear Partial
Differential Equations in the Complex Domain,
I1
.
. . . . .
.
. .
343
Tan,
Y.
J.
and Chen,
X.
X.
Identifying Corrosion Boundary by Perturbation Method
.
. .
.
.
355
Wei,
J.
C.
Existence and Stability
of
Lamellar and Wriggled Lamellar
Solutions in the Diblock Copolymer Problem
.
. .
.
. . .
.
. . .
365
LECTURES ON ORTHOGONAL POLYNOMIALS
MOURAD
E.
H.
ISMAIL*
Department
of
Mathematics
University
of
Central Florida
Orlando,
FL
USA
32816
E-mail: ismail@math.ucf. edu
These lecture notes contain
a
brief introduction to orthogonal polynomials and
mention some applications which are not readily available in books on special func-
tions and orthogonal polynomials. Since these lectures were prepared for
a
con-
ference on differential equations and asymptotical theory in mathematical physics,
we naturally emphasized differential equations satisfied by orthogonal polynomials
and attempted
to
explain the role asymptotics play in the theory of orthogonal
polynomials. Due to space and time limitations we have not treated orthogonal
polynomials on the unit circle, nor have we touched the techniques
of
obtaining
asymptotics of large zeros of special functions or largest and smallest zeros
of
orthogonal polynomials. The article by Rod Wong
65
in these proceedings com-
plements this article and provides details of asymptotic techniques.
1.
Construction
of
Orthogonal Polynomials
Given a probability measure
p
supported on a subset
of
R
and having
finite moments,
Jw
zndp(z),
we wish to construct
a
sequence
of
polynomials
{p,(z)}
such that
(i)
pn(z)
has precise degree
n
for
all
n,
p,(z)
=
y,zn+ lower order
terms,
7,
>
0,
(ii)
J,P?n(z)Pn(z)dP(z)
=
&Tl,n.
The Gram-Schmidt procedure shows that
{p,(z)}
exists and is unique.
Let
*Research supported by
NSF
grant DMS
99-70865.
1