B o u n d a r y Value Problems for P a r t i a l Differential Equations and Applications i n Electrodynamics
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BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS IN ELECTRODYNAMICS IN. E . TovMAsyAN Stale Engineering
EdiTEd
University of Armenia
by
L . Z . GEvoRkyAN State Engineering
University of Armenia
C . V . ZAltARyAN State Engineering
Univeristy of Armenia
World Scientific Vb
Singapore • NewJersey
• London • Hong Kong
Published by World S c i e n c e Publishing Co. Pie. Lid. P O Box 128. Fairer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street. River Edge, NJ 07661 UK office; 73 Lynton Mead, Tolteridge, London N20 SDH
BOUNDARY V A L U E PROBLEMS FOR P A R T I A L D I F F E R E N T I A L EQUATIONS A N D APPLICATIONS I N E L E C T R O D Y N A M I C S Copyright © 1994 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book or pans thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN 981-02-1351-4
Printed in Singapore by Utopia Press.
PREFACE
The
book
is
differential and
developed
equations,
technology.
practical
this
and
to
calculus,
I , the
differential
as
growth of
resolution
in
are
is
of
indicated. of
Cauchy
detailed
differential
convenient
new
for
problems
of
functional
I I I , the
From
the
that of
charges
i f
absent,
of of
and
sufficient
role
in
chapter
is
not
problem
this
a
well-posed allowed
f o r general
resolution
of
are
method is
in
to
general,
formulae
systems
of
and
partial
of
of
boundary
analytic such
analytic
value
functions
equations
are
functions,
problems
in
the
obtained. formulae
of
asymptotic
medium
the
of the
s o l u t i o n s of
Maxwell's
of propagation o f e l e c t r o m a g n e t i c energy
electromagnetic
are
systems
functions
and
representations
obtained the
necessary is
i n the class of
of
asymptotic
i s t h e law
of
half-space.
investigations
which
general
class
obtained
which
methods
ellipse,
for
the
A
problems
integral
b o u n d e d by
chapter
the
on t h e
Efficient
and
in
this
I I , singular equations
considered.
propagation
the for
methods
problems
main
analysis of
equations
chapter
particular,
oriented to of
Fourier transformation
the
problem
problem,
c o r r e c t boundary value
obtained.
as
value
The
resolve
In
number
functions,
well
the
the
mentioned
partial physics
theory
great
investigated.
solvability
investigation
equations
a
half-space
However
domains,
developed
problem
chapters.
boundary
equations
polynomial
are
the
for
applications in
technologies.
book c o n s i s t s o f 6
condition
value
constantly applied.
chapter
In
boundary
various wide
i t contains
in different
analysis are
In
the has
book t h e t h e o r y o f a n a l y t i c
operational
The
addition
applications,
application In
In
for which
contains
energy energy
i t
follows,
in
charges,
the
velocity
of
diminishes is
are
formulae
and
propagated
tends with
to
the
zero. I f speed
of
light. In
chapter
cylindrical The
problem
field of
IV,
the
parallel of
potentials conductors
determination
and
are of
the
c a p a c i t i e s of
these
quantities
i s reduced t o t h e system of equations
analytic
functions.
This
two
insulated
determined.
system
V
is
with
of
electromagnetic
the s h i f t
resolved
by
i n the
class
successive
approximations decreasing In are
and these
geometric
New
analytic
and e f f i c i e n t machinery
efficiently
Diriclet
differential
equations
Chapter
concerning system
permits,
and
t o t h e exact
of resolutions
f o r
Poincare
solution
of qualitative
equations
with
example
problems
theorem.
o f algebraic equations
with
t o
mentioned.
resolve,
f o r elliptic
I . The main
a n d some Here
variable
The r e s u l t s
I at the investigation I , I I , VI
equations
are
more
systems
we
of
consists of
topological
coefficients
of this
chapter
the theoretical
chapters
I I I , I V , and V
chapters
I and I I t o t h e e f f i c i e n t
o f system o f
investigate
o f boundary problems
compose
part
properties of solutions
a parameters
Zuidentaerg-Tarsky's
Chapters
as
i n t h e d o m a i n s b o u n d e d by e l l i p s e .
generalized functions.
chapter
methods
V I , i s a n addendum t o c h a p t e r
investigation
differential
of
tend
c h a p t e r V, n u m e r o u s b o u n d a r y v a l u e p r o b l e m s f o r e l l i p t i c investigated
the
approximations
progression.
i n the class
are applied i n
i n the half-space.
part
are the application
problem also the
of
t h e book
of the results
resolution
of concrete
and
of the
application
problems. The
main advantages o f t h i s
1.
In
this
differential 2.
The
book,
equations
efficient
The
obtained
It
i s especially
opens
are
f o r more
general
o f wide
class
boundary
value
(Ch. V ) ;
applied
to
the
investigation
of
(Ch. I l l and I V ) .
important
o f Maxwell's
u p a new
of solution
results
problems
(Ch. I , V I ) ;
e q u a t i o n a r e shown
electromagnetic fields
solution
value
are considered
methods
problems f o r e l l i p t i c 3.
book a r e t h e f o l l o w i n g :
t h e boundary
t o mention
equation
approach
that
t h e asymptotic
(Ch. I l l ) o b t a i n e d
f o r the investigation
formulae
i n t h e book,
which
o f the electromagnetic
fields. The
methods
solution
developed
o f t h e problems
in this arising
book
may
be
i n different
applied domains
also
f o rt h e
o f science
and
technology. This
book
i s based m o s t l y
considerable part This
book
specialists radio,
can
be
including
electro
on t h e i n v e s t i g a t i o n s
o f i t i s published here
and
used
by
professionals
mathematics, heat
mechanics,
engineering,
engineering.
VI
of t h e author
for the first
and
i n
t h e wide
physics, many
and a
time. range
as w e l l
other
of
as i n
realms
of
L.Z.
Gevorgyan,
contribution text
i s read
supply would
the like
G.V.
i n the by
ft.A.
author t o thank
Zakaryan,
and
preparation of Andryan,
with
A.O.
precious
a l l these
V.N.
the
Babayan,
remarks
persons.
VII
Tovmasyan
layout of
and
and
this S.A.
have
a
book.
The
great full
Hairapetyan
improvements.
The
who
author
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CONTENTS
PREFACE
1
V
BOUNDARY
VALUE
PROBLEM
FOR
GENERAL
SYSTEMS
OF
DIFFERENTIAL
EQUATIONS I N THE HALF-SPACE 1.1.
Introduction
1.2.
Fourier Transformation
1.3.
Cauchy P r o b l e m
1.4.
P r o o f o f Theorems 1.1,
1.5.
Examples
1 1 of
Functions
Belonging
t o the
7
Class M
2
f o r Equation
(1.1]
9
1.2 a n d 1.3
17 26
THE SYSTEM OF SINGULAR INTEGRAL EQUATIONS I N THE CLASS OF ANALYTIC FUNCTIONS Reduction
o f Equation
Equation
i nthe Class
2.2.
Analysis o f t h e Equation
(2.1)
32
2.3.
Analysis o f t h eEquation
(2.2)
35
2.4.
E f f i c i e n t Method
2.5.
Reduction to
2.6.
o f Holder
30
f o r Solving t h eEquation
(2.1)
37
o f Sinqular Integral
Equations
(2.1) and (2.2)
Representation
Simply
Analytic
(2.1) t o t h e S i n g u l a r I n t e g r a l
o f Some C l a s s e s
Equations
Integral in
3
Z9
2.1.
Connected
41
o f Functions
Domains
through
Which
are Analytic
Functions
Which a r e
i nt h eUnit Circle
45
ASYMPTOTIC FORMULAS FOR SOLUTION OF MAXWELL'S EQUATIONS AND THE LAWS OF PROPAGATION OF ELECTROMAGNETIC ENERGY AT GREAT DISTANCES 3.1.
Cauchy Problem
3.2.
Formula
49 f o r Maxwell's
Equations
f o r S o l v i n g Cauchy P r o b l e m
System
f o r Maxwell's
System 3.3.
Harmonic O s c i l l a t i o n s
3.4.
Harmonic O s c i l l a t i o n s
49 Equations 53
o f E l e c t r o m a g n e t i c Waves
along t h ex-axis
56 o f E l e c t r o m a g n e t i c Waves
a l o n g t h e axes x , y and z
60
IX
3.5.
Cauchy P r o b l e m
f o r M a x w e l l ' s E q u a t i o n s System
G e n e r a l Boundary
with
more 6
Data
3.6.
On t h e B e h a v i o r o f Some I n t e g r a l s w i t h
3.7.
A s y m p t o t i c Formulas
3.8.
Law o f P r o p a g a t i o n o f E l e c t r o m a g n e t i c E n e r g y
f o r Solutions
Parameter
65
o f Cauchy's Problem f o r
M a x w e l l ' s E q u a t i o n s System
68 a t a Great 7
Distance 3.9. 3.10.
4
Cauchy P r o b l e m
f o r Maxwell's
Resume o f C h a p t e r
E q u a t i o n s System
a t 0, oW,
(1.9)
P ( < r ) y ( 0 ) = b, Hr(b,,
where
,D. )
y ( t ) = ( y , ( t ) ,... , y The of
solution
n
is
(t))
of
the
given
the
problem the
(1.9),(1.10)
r
we
i
Definition condition, uniquely in
A(o-)=E (or ia
constants
1.4. i f
the
The
and
i s looked
f o r i n the
class
d e p e n d i n g on
t^O,
(1-11)
a.
basic problem
problem
(1.1),
(1.9),
(1.7)
(1.10)
satisfies at
every
Lopatinsky's fixed
ceR
shown
that
a
is
solvable.
the ,
are
introduce the
vector
found.
estimate:
|y(t) |=C(l+t) ,
Now
constant
i s t h e s o l u t i o n t o be
functions satisfying
where C and
(1.10)
above
mentioned
papers
d e t d ( o - ) * 0 a t VtreR") a n d
[1-10], the
4
i t has
equation
been
(1.1)
is reqular,
i f then
Lopatinsky's (1.1),
condition
(1.7).
det^(ir)^0.
Now
The
provides
we
extend
exact
unique
this
solvability
result
formulation
of
to
the
of
include
obtained
the
problem
the
case
of
is
as
results
follows. Theorem is
1.1. For unique
necessary Theorem
1.2.
Lopatinsky's
I f
the
condition
n=2 h a s i n f i n i t e
may
Definition
1.5.
at
i t
satisfy (1.7) a t
(1.1),
(1.7) does n o t
dependinq
on m a t r i c e s
independent
solutions of
(1.1),
Thus,
( 1 . 7 ) has a
how t h i s
finite
n u m b e r c a n be f o u n d
o f non-homogeneous
problem
i n t r o d u c e a new
problem
0" «R",
not
(1.1),
f r o m one t o i n f i n i t y .
the solvability
e n d we
The
point
s o l v a b l e a t a-a
be a r b i t r a r y ,
(1.7)
solutions.
then,
when t h e p r o b l e m
about
case ? For t h i s
condition
(cf.(7))
does
problem
independent
independent s o l u t i o n s ,
what c a n be s a i d this
(1.7)
t h e homogeneous
i f n = l and t h e p r o b l e m
arises:
number o f l i n e a r l y
in
(1.1),
(1.1),
Lopatinsky's condition.
a n d P (IT) , t h e n u m b e r o f l i n e a r l y
n = l a question
and
problem
number o f l i n e a r l y
homogeneous p r o b l e m
at
o f t h e problem
i tsatisfies
Lopatinsky's condition
.4(c), B((T), the
that
then
The e x a m p l e s show t h a t satisfy
solvability
and s u f f i c i e n t
(1.1),
(1.7) s a t i s f i e s
i f t h e problem
(1.9),
Lopatinsky's
(1.10)
is
uniquely
. 0
The f o l l o w i n g Theorem a
1.3.
finite
number
Lopatinsky's points In
theorem
answers t h e above mentioned
of
linearly
condition
independent
i s violated
(1.1),
solutions
i s always
o f t h e problem
Theorems
l . l ,
1.2,
(1.1),
and
1.3
only i f number o f
solvable.
c h a p t e r , t h e f o r m u l a f o r t h e number o f l i n e a r l y
solutions
( 1 . 7 ) has
i f and
i n no more t h a n a f i n i t e
and t h e non-homogeneous p r o b l e m
this
question.
L e t n = l . Then t h e homogeneous p r o b l e m
independent
(1.7) i s obtained f o r n = l . are proved
i n section
1.4
of
this
chapter. We
formulate
Lopatinsky's conditions
i n terms
o f m a t r i c e s A(a)
, B{o~)
and P ( c ) . Let
T
point
i n the characteristic
i n R
including PeA(ff)s0 the 11
these
a n d F(
condition:
(1.1)
(1.1)
Fourier the
both
is
rate
a
with
r j ( x , 0 ) = g ( x ) , XEK",
where
i f
sense:
Cauchy P r o b l e m
Let
of
that
formula:
( V ( X ) V ( C ) , i p ( x ) ) = (» ( t ) , ^ ( x ) p ( x ) ) , peS, where
2
I - *)
ipeS
(1.23)
r r"
The
with
s
( o , |cr|+». So i n ( 1 . 4 7 ) a
(1.51)
i t follows
13
that
the
elements
b
jk
(X,cr)
m) o f t h e m a t r i x
(j=l
{A (a) A+B(ir) ) ,
p
|h (A, -)l^Ur k
where the
C is
| c | s r , |X|sp Q
([14],p.16)
0
a
( A ( a ) X+B(cr) ) ' i n
o n r . So t h e m a t r i x
can
g
theestimate:
a t l*l=ff ,lAl*P ,
0
c o n s t a n t depending
o
domain
satisfy
be
represented
by
Cauchy's
integral
a),
(at fixed
fi 1
(J(.r)X+B(cr))~ =S(X,cr) + ^
where
^(o-)*',
ICl^T,,.
£
I*I P -
(1.52)
0
6=m-l-p,
f(*.«^)= - 2S1
j i- ' ")^* ")^ ^!' 1
0
0
1
( 1
-
5 3 )
l£l-P„ j
Kj(
J
w h e r e U(x,t)
i s a m-dimensional
According solutions
to of
functionals
lemma
the
1.1,
vector
the
homogeneous
J V ( t ) are
problem
linearly
function
vector
o f c l a s s M. J U(x,t)
functions
(1.1),
independent,
(1.7), equation
and
are
the
since
the
(1.99)
implies
1 that
the
independent.
vector-functions Theorem
1.2
U(x,t)
(j-1,2,...)
i s proved.
23
are
also
linearly
P r o o f of theorem at
infinite
1.3.
L e t n = l and L o p a t i n s k y ' s c o n d i t i o n be
numbers o f p o i n t s f ( J - 1 , 2 , . . . ) .
rank
We
shall
prove
that
.
homogeneous
latter
means:
j=l,2
< m,
fit,)
The
violated
(1.100)
(1.1),
problem
(1.7)
i n the class
h a s i n f i n i t e number o f l i n e a r l y i n d e p e n d e n t s o l u t i o n s . ) l l L e t a=(oc ,...,cc ) b e a n o n t r i v i a l s o l u t i o n o f a l g e b r a i c
K
system:
i
J (S -W(:|,))« B
) P(?j)« •
0.
(1.101)
0-
( 1 . 102)
J
o f v e c t o r a a r e complex numbers.
The e l e m e n t s While proving
J U(x,t) are
nontrivial
(1.103) under the
J = u(F ,t) expl-ixe ), i
i t follows
that
these
n = l and
i£
the class
M
problem
(1.69),
(1.69),
(1.70),
b
system
y
. As
we
(1.70) (1.72)
virtue
solvable
number
of
have
be
shall
shown
when p r o v i n g
i n t h e same
M*
with
independent
(1.70),
(1.72)
solutions.
at finite
number
(1.69),
t h e theorem
i s equivalent
So in
of
(1.70)
1.1,
t o the
the
problem
class.
i s violated,
(1.72)
(1.7)
(l.l),
independent
violated
From
independent.
solve t h e problem
i n the class
(1.103)
( c f (1.92)).
problem
6.6
at finite
and
6.7
r e s p e c t t o V(0)
solutions.
number
(chapter
and t h e c o r r e s p o n d i n g homogeneous
linearly
t h e system
we
o f theorems
(1.70),
always
of
number o f l i n e a r l y
functions:
.
(1.7)
are l i n e a r l y
t h e homogeneous
. First,
As L o p a t i n s k y c o n d i t i o n •--
(1.1),
solutions
Lopatinsky's condition
F
the vector
j-1,2,...
i
(1.100)
the condition
points
the
a
s o l u t i o n s of t h e problem
c l a s s M h a s an i n f i n i t e
Let
in
1.1 we h a v e shown t h a t
theorem
Hence
of points 6,
6.6)
section
i n the class
S'
system
finite
has
the general
a
is
solution
i s d e t e r m i n e d by t h e f o r m u l a : N 0
^
H0)=L+) k =1
24
k C^L,
(1.104)
where
L i s a
particular
solution
o f non-homogeneous
problem
[1.70),
k [1.72)
and
L
homogeneous complex Thus, (1.69),
( f t = l , . . . ,N)
problem
t h e problem
then
(1.69),
i n t h e class
Since t h e r i g h t
1.3),
(1.72),
independent
and
C
j f
solutions
C ,.•.,C
a
t
e
of the
arbitrary
constants.
(1.104)
(1.72),
are linearly
(1.70),
side
(1.70)
i s reduced
of the condition
according
t o theorem
Cauchy's p r o b l e m
t o Cauchy's
problem
M*.
(1.69),
1.5
(1.104) s a t i s f i e s
and t h e f o r m u l a
the equation
(1.66)
(section
(1.104) has t h e s o l u t i o n : N
7(t)-w({r,t) where (o(c,t} Let
i s the matrix
( i = F ( f (x) ) ,
where
of
(1.105)
(1.65).
f ( x ) sM .
I n Chapter
6,
section
6.6,
i t
i s
]
shown the
(theorem
system
6.S)
(1.70),
that
i n this
L=F(
where
case
the solution L
(j=0,...,W)
of
(1.72) can be r e p r e s e n t e d a s :
g j
(x)),
j=0,...,K,
(1.106)
g^fxJeM^.
Hence, t h e v e c t o r
f u n c t i o n u(cr,t)Z. ( j = 0 , . . . , N ) can be e x p r e s s e d a s :
] ) <ji(o-,t)L=F 'U(x,t))
,
x
j=0,
. . . ,K,
(1.107)
J
w h e r e U ( x , C ) a r e some m - d i m e n s i o n a l Thus,
the general
solution
vector
(1.105)
functions
o f t h e problem
o f the class (1.69),
M.
(1.104)
is:
7(t)
By v i r t u e
= F I u<x,t)
+\
o f lemma 1 . 1 , t h e v e c t o r
cu'x.t)
(1.108)
function: N
U(x,t)
= Btx t)*T
C U{X,t).
t
k
25
( 1 . 109)
is
the solution Thus,
of
points
(1.7)
F ,F , . . . t h e
complex
prove
linearly
now
(1.1),
(1.7).
Lopatinsky's condition
i s d e t e r m i n e d by
arbitrary We
of t h e problem
i f n = l and
the
general formula
is violated
solution
of
at finite
the
( 1 . 1 0 9 ) , where
number
problem
(1.1),
(ft=l,...,N)
are
numbers.
that
the vector
J U(x,t)
functions
independent, obviously,
N)
(j=0
from formula
(1.107)
we
are
have:
J J u ( o - , 0 ) L = F (U(JC,0)) .
(1.110) J 1
The
latter
and
the
equalities
u ( i r , 0) = u ( c )
and
k>(o~) L=L(
j = l , . . . ,N)
imply: J ) L • F^([7{x,0)),
j=l,
(1.111)
J
Since then
the vector
the
formula
(j=l,...,W) (1.104)
finite of
(1.111)
are also
1.6.
implies
linearly
in particular
Theorem
result
I f n=l
and
number o f p o i n t s ,
At
(1.72)
t h e end
independent class
s'.
that
are
the
linearly
vector
i n d e p e n d e n t . The
independent, ) U{x,t)
functions
latter
and
the formula
in Lopatinsky's condition
t h e number o f
t h e homogeneous p r o b l e m
number o f t h e l i n e a r l y (1.70),
L(j-1,...,N)
functionals
(1.1),
linearly
is
violated
independent
(1.7) i n t h e c l a s s M
at
a
solutions
i s equal t o the
i n d e p e n d e n t s o l u t i o n s o f t h e homogeneous
system
i n t h e c l a s s S' .
of Chapter solutions
H e r e we
6,
section
6.6,
we
find
o f t h e homogeneous s y s t e m
mean l i n e a r
independence
t h e number o f (1.70),
(1.72)
over t h e f i e l d s
of
linearly i n the complex
numbers. 1.5.
Examples
Example
1. L e t us
c o n s i d e r t h e system au — 1
at
dV + a — 'ex 3
of equations:
+ b U =0,jf<ER',t>0, 1
26
1
(1.112)
a v
au b U
2
2
Sx where to
U={0
a ,b
,b sf!
can
easily
J
2
2
1
solution
are given
be
found,
which
belongs
numbers.
that
t h e system
of equations
i s r e g u l a r i f and
i f : bjSO , b > 0 ,
a a ib
;
Let
t h e c o n d i t i o n (1.114)
of
the
we
must
impose one
Me
take
i t as:
system
(1.112),
be
boundary
«
and
(
function
from
Applying
i s uniquely
problem
(1.112),
solution. (1.115)
I f 0^=0,
has
We
0^*0,
w h e r e C>0 class
H.
We
take
and
lR
f(x)
is a
and
1.6
we
given
(1.7)
(1.113).
is a
given
obtain:
(1.112),
(1.115)
(1.113),
a ro ;
the
i n the class H
an
+
elliptic
one
^£
-
a
1
2 C
U
i s always
solvable
independent
(1.112),(1.113), solutions.
in half-plane:
xex',t>0,
= 0,
i n the
(1.116)
2 x
t o be
condition i n the
3
(1.115)
non-homogeneous
linearly
independent
equation
i s the solution
t h e boundary
f (x)
2
£
consider
fi(x,t)
to
(1.115)
a =0 t h e homogeneous p r o b l e m
4 * where
1.3
number o f l i n e a r l y
at
regularity
(1.112),
a +a^«0, a n d
homogeneous p r o b l e m has
S
of
according
t h e system
constants,
the problem
(1.113),
infinite
E x a m p l e 2.
Then t h e o r d e r
Therefore,
s o l v a b l e . I f 11^-0,0^*0,
the corresponding
(1.114)
o
1.1,
class M
.
M-
I f a^O
1.1.
i
0 ) = f (x) ,XER' ,
2
real
the class
theorems
Statement
i s one.
+ aU (x,
are given
;
c o n d i t i o n on
i
where
]
satisfied.
(1.113)
a U (x,0)
and
to
3
2
verify
(1.113)
flx
i s the
;
1
One
2
( x , t) , t ? ( x , T.))
(
t h e c l a s s M,
only
2
+ a — '=0,Jt6R', t > 0 ,
found,
which
form:
" ^ ° ' + pf7(X,0) = f (x) ,
function
from
belongs t o the
the
class
«R*, M,
a
(1.117, and
8
are
real
constants. The
equation
(1.116)
may
be
rewritten
27
as
t h e system
(1.1) u s i n g
the
following notation: Bjt t)-V(X,t)
,
r
The e q u a t i o n Applying
a
U (x,t)=
U
(1.116) i s r e g u l a r and i t s o r d e r
theorems
l.l,
1.3
and
1.6
t
^ '
2
)
•
of regularity
t o t h e problem
(1.116),
i s 1. (1.117}
we o b t a i n : Statement class
M
1.2.
The
i s always
solutions
of
non-homogeneous p r o b l e m
solvable
the
corresponding
(1.116),
(1.117)
N of linearly
a n d t h e number
homogeneous
problem
i n the
independent
i s defined
as
follows: N=0 a t a*Q,
B*C
B
0
(1.119)
i
t o be f o u n d ,
which
belongs t o
i n t h e form:
x
< ')
vector-function
e q u a t i o n (2.9) , t h e n cj(z)=0.
n
with
are
simple
analytic
, . . . ,ej (z) ) s a t i s f i e s Let u(z)
the
theorems.
t h e number
non-homogeneous
I f
of
numbers.
function f ( z ) satisfies
2,1,
analytic
accordinq
solutions
o f t h e homogeneous e q u a t i o n (2.9) a r e f i n i t e
Theorem only
independent
dependence
o f complex
statements r e s u l t
homogeneous e q u a t i o n solutions
linearly
(2.9).
assumed o v e r t h e f i e l d The
are
g
homogeneous e q u a t i o n
complex
of
lemma 2 . 1 .
variable
z
since
i n domain
K(z,t)
D' ,
then
([13],p.45):
M(t)K(t,tJcK J
t-z
- =0,
VzeD ,
(2.16)
r w h e r e D~ i s t h e c o m p l e m e n t o f D*u Proceeding
to
the
limit
at
T t o t h e whole
z-»t
(zeD~)
33
complex p l a n e .
and
applying
Sokhotzky-
Plemelj's
formula ([18],p.66)
we
obtain
D(t)K(t,t ) d t +
- ^ ' . i f t ' o - V
lib-
-
t
0
;
t
r
2
V -
' -
1 7
'
r Comparing
(2.9)
and
(2.17),
we
find
t d ( t ) =0, t e r . Q
Lemma
o
2.1
is
proved. Let
us c o n s i d e r t h e f u n c t i o n a l l (f)=
(see
(2.14)):
| o(t)f(t)dt,
j
j = l
m;.
(2.18)
r The
functionals
I
Holder's class.
We
as f u n c t i o n a l s
over
are
shall
linearly
prove t h a t
the class
independent
as
functionals
they are also l i n e a r l y
of analytic
boundary v a l u e s b e l o n g i n g t o t h e Holder
over
independent
vector-functions
i n D*
with
class.
Let
C
i V
f
)
+
C
A
(
f
)
+
---+
5
W ^
Substituting
from
(2.IB) i n t o
|
0
"
G
'
V f
-
(2-19)
0
( 2 . 1 9 ) , we
have:
U>(C)f(t)dt er,zeD*
by
o
z^t (zeD*)
in
o
(2.24)
and
i(t ). o
applying
obtain:
r Comparing
(2.23)
and
(2.25)
one
*(t )
- u(t )
o
Hence, ( J ( t )
finds: , t er.
Q
i s t h e boundary
(2.26)
o
value of a n a l y t i c vector-function * ( z ) . 1
According
t o lemma
independent, condition
then
(2.14)
2.1 u ( t ) = 0
C =...-C^-0
(k=m^).
i
are linearly
k
a n d s i n c e d>(t) , . . . , u ( t ) Thus,
we
are
proved
linearly that
the
independent.
2.3. A n a l y s i s o f t h e E q u a t i o n ( 2 . 2 )
Equation here of
(2-2) should
we w i l l
be
analyzed
formulate the results
similarly
(2.1),
so
moments
proofs f o r the statements. Here,
linear
(2,2)
and t h a t
(2.2)
will
(2.2)
independence
t o the limit
and a p p l y i n g
E»
of the solutions
of solvability
conditions
0
of real
(zeD*)
i n both
S o k h o t z k y - P l e m e l j ' s f o r m u l a , we
1 tK(t .t ))Kt ) 0
a t z^tt^r
0
1 + £j
f K(t ,t)V»(t)dt —
[
suppose
2T7I |
equation
numbers. sides
of
equation
obtain:
0
c
r +
o f homogeneous
f o r non-homogeneous e q u a t i o n
be a s s u m e d t o be o v e r t h e f i e l d
Proceeding
We
t o equation
and p o i n t o u t t h e e s s e n t i a l
M(t.E)V(t)dt -t^l
= < >' f
that:
35
C
0
+ i«(t ,t ) (t ) o
o
P
0
+
det (E+K(t ,C >)*0, o
This (Cf.
condition
C er.
o
i s the condition
of
(2.28}
o
normality
f o r equation
(2.27)
[ 1 4 ] , p.275).
Equation f(t )
(2.27)
will
i s the right
o
The
be
solved
i n Holder's
side of equation
normal-type
equations
classes,
assuming
that
(2.2).
(2.27)
i s completely
analyzed
i n
[ 14 ]
(pp.273-278). As f o r e q u a t i o n ( 2 . 1 ) , are
one c a n p r o v e t h a t
e q u i v a l e n t a n d t h e o r e m 2.1 i s s a t i s f i e d
(2.1)
being
replaced
by
the equation
equations
(2.2) and
(2.2) and
t h e e q u a t i o n (2.5)
b e i n g r e p l a c e d b y t h e e q u a t i o n ( 2 . 2 7 ) when f o r m u l a t i n g We
consider
together
with
the
(2.27)
f o r t h e them, t h e e q u a t i o n
equation
this
(2.27)
theorem.
the
following
equation: ,
,
i
f u(t)K(t,t )dt 0
f a(t )
[ u(t)«(t, t j d t
Q
2ni
I
= ° .' t„eT, a
t-t
(2.29)
r where
a ( r ) = l i m ^-^—, t«=r i-iZ
and
u ( t ) = (u ( t } ,...,u ( t ) ) t
(vector-line),
The e q u a t i o n ( 2 . 2 9 ) equation If
is
n
satisfying
(2.27)
([14],
vector-function
Holder's condition
i s called
(2.30)
t-T
to
be
found
on r .
associated equation t o the
p.275).
the normality condition
(2.28)
i s satisfied,
i t i s shown i n ( [ 1 4 ] ,
p.276) t h a t one can p r o v e t h e f o l l o w i n g t w o s t a t e m e n t s : 1)
The
equation of
number
k
(2.27)
and t h e number
o
of linearly
independent k'
of
o
solutions
linearly
t h e a s s o c i a t e d e q u a t i o n (2.29) a r e f i n i t e
The
non-homogeneous
equation
(2.27)
36
homogeneous solutions
and
where x i s t h e i n d e x o f t h e f u n c t i o n d e t (f?+K(t, t ) ) 2)
of
independent
o n F.
i s solvable
i f and
only i f
the vector
function
f(t)
satisfies
the condition:
j
Be
fc)(t)/(t)dt-0,
(2.31)
r i where
u ( t ) (J=l,...,k' )
are
g
associated The
equation
linearly
independent
solutions
of
the
(2.29).
equivalence
of
equations
(2.2)
and
(2.27) ,
and
the
above-mentioned statements r e s u l t i n : Theorem
2.4.
The
number
homogeneous e q u a t i o n solutions
of
the
non-homogeneous satisfies
2.4.
Let
equation
equation
the condition
Efficient
D'
be
containing
h
of
Q
(2.2) and
linearly
independent
t h e n u m b e r k'
(2.27)
(2.2)
are
is
of
finite,
solvable
solutions
linearly
k -k'=-2x-
and
i f and
of
independent
only
The
i ff ( z )
(2.31).
Method f o r S o l v i n g E q u a t i o n ( 2 . 1 )
a
simply
the origin
connected
domain
with
o f c o - o r d i n a t e s . L e t us
smooth
boundary
consider the
T
following
integral equation:
. 2Si
z
*< >-
f K ( z , t ) (p{£)dt t-«(z)
- i, hi J K (Z,t)«(t)l (l- f ) d t = f ( z ) , 2
r where
K (z,t)
complex
function
with
logarithmic
We
JC (z,t) ?
z
and
respect
t
are at
Holder's function,
a t every f i x e d
(2.32)
condition which
and
functions teD
variables
and on
D*ur,
t o be f o u n d , a n a l y t i c on
D*ur,
i s analytic
In(1-2) with
with
respect
satisfying
is
to
Holder's
f(z) is a
given
i n d o m a i n D* a n d that
respect t o z
branch
of
( i n domain
t e r and e q u a l t o z e r o a t z=0.
impose f o l l o w i n g
D* a n d s a t i s f i e s
analytic zeD
t o both
and (>(z) i s a f u n c t i o n
satisfying
D*)
and
variables
condition
n
r
restrictions
Holder's condition
on c t ( z ) :
ot(z) i s a n a l y t i c
i n domain
on D*u f , a n d t h e v a l u e s a ( z ) a r e :
a(z)eD'
at
zeD*u T,
(2.33)
[^|^-ja7(t)dt= g i j J K ^ t ^ t J p f t J I n f l -
|)dt.
(2.37)
| K ( z , t ) i o ( t ) d t - f ( z ) , z«D*,
(2.38)
2
o Hence,
case of
that
shall
unique
particular
obtained.
and K ( z , t ) * ( t )
t
K^z,t)p(t)=
limits
we
equation of
is a
i t follows
equation
Here
c o e f f i c i e n t s t o be
(2.32) (2.33)
i s equal
have reduced
method f o r s o l v i n g
sufficient
equation
equation
and i t s i n d e x
we
i n Holder's
efficient
that
the condition
equation
section
equation
and
to verify
(2.1).
normal-type In
constant.
i s easy
r
equation
(2.32)
may
be r e w r i t t e n
as:
i
ip(z)-B(z)p(a(Z))-z"'
2
0
where S[z) It
i s clear
The s o l u t i o n
= R (z,«(Z)).
t h a t |3(z) i s a n a n a l y t i c of equation
(2.38)
(2.39,
(
function
i s sought
38
i n domain
D*.
i n the form o f :
E
C J
jfz
+ z"u>(z) ,
(2.40)
] =o w h e r e C ,.-.,C o
satisfying chosen
ui
are constants,
Holder' s
u ( z ) i s an a n a l y t i c o n D*<J I " , m i s a
condition
( i n D*)
natural
function,
number
t o be
later.
Substituting
tp(z)
from
(2.40)
i n t o equation
(2.33)
we
obtain:
z z ( J ( z ) - S ( z ) (n(z))° (a(z) ) - z"' JjC ( Z , t ) c'u (t) d t = t l ( z ) , N
tl)
z
(2.41)
where
£![z)=f(z)-^
j f z ' + f3(z)^
J -O
In
equation
C ,...,C
-jllatzD'+z-'^
J-0
(2.41)
a r e t o be
From t h e c o n d i t i o n
J =0
the analytic
Performing
(2.34)
(2.42)
0
function
u ( z ) and
the
constants
i t follows: | '-*(C))dt —
+
r +
3H(t ,t )(p(t )-X(t )) o
o
o
0
+
f w(t ,t) ( (t)-JC(t))dt P
+
2WT
—
E=E
- f < V -
r
2
V "
< -™>
IL e t us
* ( z ) ^ ( z
denote:
)
+
I
;
F
I
, K(z,t) (?(t)-X(t))dt j ^ —
j
+
r Proceeding
, M(z,t) ( ( j ( t ) - X ( t ) ) d t ^ (2.74)
r
to
the
Sokhotzky-Plemelj's 1
limit
at
z^t sr, o
zeD*
in
(2,74)
and
applying
f o r m u l a , we o b t a i n : ,
i
44
f K(t
t) ( (t)-X(t))dt V
!
,
.
S ^ W ("'V-^^D
+
Hence, e q u a t i o n
(2.73)
Similarly,
+
c
5ffi J_ —
f=£;
2
as :
*(t )-Jt(t )-tr(t ),
t
o
(x,t)=0.
+
(3.46)
(x,t)).
from
(3.41)
we show
that
into
(3.46)
and t a k i n g
equation
W(x,t)
the vector-function
i s
the
(3.46).
(3.41)
and (3.44),
i t follows
that
W{x,t)
belongs t o
M. formula
(3.44)
into
account,
formula
( 3 . 4 1 ) may b e r e w r i t t e n
as: 3 Bjx.fj-
»n
1 2
Y
a
| (s -4«f )
2
(*,(€) ( * j ( € ) + f l ) -
-iS^mlexpt^Ot-iSxldf;, 2
, *™
J = 1
(3.47)
- I
-oa
+A (f)0 (sT))ex (X (i:)t-iex)d :J
where
2
a
0 (?) and ^ I C )
respectively Therefore,
r
e
2
P
t r i e
J
(3.48)
1
Fourier
transforms
/ ( x ) and t
f (x) 2
( c f . (3.34)). i f f (x)ec"(R ) ( j - 1 , 2 ) ,
55
thesolution
o f Cauchy's
problem
(3.16),
In
(3.17)
(3.16),
(3.17).
(3.23), is
i s defined
(3.24)
defined
Applying
we
functions g
0
and
(-6)
Let
the
(£)
T
and g
functions
(3.18),
(3.19)
0
and
2
(EJ
are the Fourier
j
f (x) ]
(J-1,2)
g {x)
and
}
i n R' and s a t i s f y
boundary may
be
belonging
Harmonic
In
this
data
solved
t o the class
C™(R
we
shall
construct
]
i
and
t h e boundary
A^,
u,
are
be c o n s i d e r e d shall
consider
(3.13) given
and
of
and these
case t h e f o r m u l a e f o r
Waves a l o n g
the
the
solution and
x-axis
of
(3.19)
Maxwell's
o f t h e form : ,
(3.49)
0 ) = j ) C o s (ux+(p ) ,
(3.50)
2
real
(3.17)
i n t h e case
( x , 0 ) =A cas{ux+
> ) 3
Thus,
+ - f ^ e x p f - i ^ ) , 0 =argd .
(
i fthe initial
(3.13),(3.15),(3.49)
data
and
3
B(x,t)
(3.50)
and E ( x , t )
one
can
satisfy
conclude
(3.86)
3
the conditions
from
the
derived
E
of the
formulae: 1)
At
every
fixed
t>0 the projections
B
. B 1
vectors zero 2)
B and E a r e harmonic
2
2
I fe E4(J a,
then
with
t h e phases o f these
t h e phases o f these
Harmonic O s c i l l a t i o n s along
Maxwell's takes
E
. y
amplitudes
x
decaying
to
a t t->+™,
otherwise
3.4.
oscillations
and x
oscillations
oscillations
are stabilized,
are unstable.
of Electromagnetic
waves
t h e axes x, y and z
equation
system
( 3 . 4 ) , ( 3 .5)
i n the Cartesian
t h e form:
SB ' st 3B
at
z
8E
BE ay
az x
dB i y ay ' at aE
8E
y 3E
y
ax'
at
60
dE X ax SB
as » SB
By
az
az
I
x u
y
coordinates
BE
SB
dB 1
dE
dB
SB -SE ,
S
J~ V
ax SB Let
us
consider
SB
Cauchy's
W -
SB
problem
y_ »
Sir
f o rthis
(3.89)
a
Sy problem
with
initial
conditions: ,
B(x ,y,z,0)=a exp(ju x)+
a^expfiu^y) + a^expliu^Z) ,
(3-91)
E ( x , y , z , 0)^h^expfiu^x) + i > e x p ( i t > y ) + b ^ e x p l i w ^ z ) ,
{3.92,
i
i
2
a -la
where
constants real
,a
a n d b = (£>
)
with,
constants,
a
i ]
u^O. S u b s t i t u t i n g
It
iu exp(icj x, i
further
be s a t i s f i e d .
uniquely
equation
u {j=l,2,3) are (3,90)
and t a k i n g
3
+ a ^ i ^ e x p f i c j ^ z ) =0,
(x,y,z,eR .
solution.
=0, a =0, a =0.
we assume
We s h a l l
3.1 (theorem
We s e e k
J t
uniqueness
for this
+
^z)
+ 0 (t,exp{iu y) 2
(t) *l )
2
(t))
J 3
3
and
(Sj f t ( * ) . # ,
a
f
£
(3.94,
,
(3.95)
]
•1
,
t
i
t h e v e c t o r - f u n c t i o n s t o be f o u n d a n d :
(P (t)s , n
0
,> (t)s0, 2 2
(p (t,30.
The v e c t o r - f u n c t i o n B d e f i n e d b y e q u a t i o n equation
( 3 . 9 0 ) . I t i seasy t o see t h a t HjftJexpUc^*:)
(3.96)
3 3
(3.94)
obviously
equality:
+ U ( t ) e x p ( i u y ) + fi^ ( t ) e x p ( i u z ) 2
3
61
3
satisfies
is
ijt)exp(iw x)
+ iJt)exp(i 0 , ( x , y , z ) e f i " ,
,
3
3
(3.97)
i f and o n l y i f (3.98)
Substituting
B
and
E
from
(3.94)
and
(3.95)
into
the
system
( 3 . 8 7 ) - ( 3 . 8 9 ) and t h e boundary c o n d i t i o n s (3.91) and ( 3 . 9 2 ) , and u s i n g the
equivalence
the
following
of the of the equalities f o r ip^lt)
Cauchy p r o b l e m
0^ (t)=-S0 j
p; (t)=iu i/ 2
|
(t),
j )
(3.97) and
( 3 . 4 8 ) , we
(3.99)
( t ) , j-1,2,3,
(3
100)
(3
101)
(3
102)
0 (t)=«i J P (t)-Bil; (t),
(3
103)
(t) , 0 (t)=ai J i(i (t)-Si(; (t),
(3
104)
(3
105)
I ] 3
0; (t)=aiu 3
] (
p
(t)-B!l(
] z
l 3
(t),
r Jt)=- ,(t)=-iL> 0 (t), 3 :
3
2(
2 l
( t ) -BK( (t) , 2 3
E3
3
2!
3 ]
=-ai(J ip
3 i
•
12
=-oiw ip
Z3
M
P
obtain
tp^t):
and
3
3 2
Z 3
( t ) -Bif,
(t) ,
3i
(3 106) «', , ' 0,
that
i s proved s i m i l a r l y .
2
-co
£>ER*.
s h a l l evaluate the following
integral:
7
ijt)= From
the
|
inegua1ity
Lagrange's
mean
value
From f o r m u l a
( 3 . 1 3 1 ) we
P
(3.134) theorem
account of t h e i n e q u a l i t y exp(- ! ? * t ] -
:
[e*p(- |e t]-ejf (x (i:)t)jce.
i t follows to
the
( 3 . 1 3 4 , we
(3.136)
2
1^ ( t ) ^ 0 .
that
function
e*
and
Applying
taking
into
obtain:
e x p ( A ( S ) t ) a e x p ( - | i l t j (2
2
| e t - X (?) t ] . i
j
(3.137)
have:
(3.138)
From t h e i n e q u a l i t i e s
(3.137) and
( 3 . 1 3 8 ) we
obtain:
ID
°« (t)i f ^ —Bp s
-
p
f
where
66
- ^ t l d C -
(3.139)
We r e p r e s e n t t h e f u n c t i o n
I ( x , t ) as: 3
I (*,t)=r (*-,t)+I (jf,tJ+r (x-,t) , 3
6
7
(3.140)
B
where
r
j e y p ( A ( i ; ) t ) - e x p ( - j ^ t ] ] exp ( - i * ? ) d£_,
r {r,t)=Vt"| a
(3.141)
?
-vl: I
I (X,t)7
2
exp(- |? t-ixt:]d?,
(3.142)
r ( X , t ) = v T f e x p ( - l ^ t - i x s j d ? - vWHIT e r p ( - f ^ j.
(3.143)
B
From
(3.136) and t h e e s t i m a t e
(3.139) i t f o l l o w s :
| I ( X , t ) |S / E J ( t ) 5 6
From e q u a t i o n
|
7
i s clear
(3.144)
exp(- |€'t)d«3 v T
e X
p ( - ^ f / t ) J exp (-
f ^ t j d e .
that:
expj-
From
|.
( 3 . 1 4 2 ) we h a v e :
| X ( X , t ) |*
It
s
(3.135),
|^-
i
x
S
j= , [e
P
« (t t
+
i y ] - p ( -
(3.143) and (3.146) i t f o l l o w s
ggj.
(3.146,
immediately that:
I (x,t)s0.
(3.147)
a
From
the relations
the
function
at
t-H-».
IJx.t)
(3.140),(3.144),(3.145),(3.147) i t follows uniformly
I t i s proved
converges
similarly
67
that
with
respect
the function
t o xeR
1
I {x, t ) t
that
t o zero tends
uniformly
3.7.
w i t h r e s p e c t t o xeR
1
t o z e r o a t c-»+». Lemma 3 . 1 i s p r o v e d .
Asymptotic Formulas f o r S o l u t i o n s for
In
Maxwell's
this
solution
noted
System
we
obtain
section
shall
o f t h e problem
of t
values
Equations
that
(3.16),
o f Cauchy's
the asymptotic
(3.17)
and
w h e n f {x)ed^(S )
a n d g ( x ) eC^fJ? )
Cauchy's
at
1
1
}
;
problem
(3.10)
f o r Maxwell's
equations
(3.16),
(3.17) and ( 3 . 1 8 ) ,
Problem
formulae
(3.18),
(3.19)
f o r the at
great
( J - 1 , 2 , 3 ) . I t s h o u l d be
initial
condition
system
i s
(3.8),
reduced
to
( 3 . 9 ) and
the
problem
(3.19).
There i s t h e f o l l o w i n g : Theorem the
3.2.
problems
The
(3.17) and ( 3 . 1 8 ) ,
Bjx,t)=
^ [ ^ x p ( -
V ' = x
EJx,t)
solution B(x,t),
(3.16),
t J
exp
(3.19)
fil)
a n d B (X,t),
+ ^(X,C)],
( " f=r)
E(x,t)
of
a r e represented as:
(3.150)
- CO
w h e r e t h e f u n c t i o n s A^x.t) w i t h r e s p e c t t o xsR Theorem A
1
3.2 s t a t e s
I J
(J-1,2/3,4)
tend uniformly
t o z e r o a t t-*+«
a t t-»+o>. that a t great
v a l u e s o f t i t c a n be a s s u m e d
( X , C ) B O (J'=1,2,3,4) i n t h e f o r m u l a e
that
(3 . 1 4 8 ) - ( 3 . 1 5 1 ) , i . e . :
(3.152)
68
The
formulae
asymptotic (3.5)
(3.20),
formulae
(3.21),
w i t h Cauchy's boundary
Proof
of
Theorem
representation
(3.152),
(3.153)
f o r s o l u t i o n s o f Maxwell's
3.Z.
(3.148).
conditions
(3.13),
In
the
first
From
the
formula
will
equations (3.15),
place
we
be
(3.17),
shall
(3.47)
called
system (3.4) (3.19).
find
i t follows
the that
B ( x , t ) may b e r e p r e s e n t e d a s : B ( 3 f , t ) - J > ( X , t ) + ( x , t ) + «> (x,t) ,
1
3
(3.154)
4
where
VA >L)=-
hi
X
'• -I
*> (X,t)- ^
z
/ S -4oe
j
2
2
)£|*7 r
/ B -4«P.
expt^ieit-ixfld?,
(3.155)
exp(X (5)t-ixOde,
(3.156)
2
a
2
#,(e)(i-(€)+Pi-i#,(€:) —
#, ( 0 ) e x p f A ^ t J t - i x e j e l S ,
2
(3.157)
iS, ( o ) ?
-fcr
-
