m--I a~(t) q0 l[a k (t)] + bh (l) q% [a k(t)]-]- ai j ~--aa(t) k=O . s x. 'j[. ] d k~i_(t) ~ ---~ (/)Jq%. (T) d~c ~ -t- ~l. K (1) (t, ~) q?t (T) d'~ -t- ~. K (2) (t, T) .~2 .j_ 2~: -I dh (t) q)~ ...
APPROXIMATE WITH
A
VALUES V.
SOLUTION
CARLEMAN
SHIFT
OF
THE
D.
Didenko
I. Formulation plane of the complex In the present
OF
UNKNOWN
SINGULAR AND
THE
INTEGRAL
EQUATIONS
COMPLEX-CONJUGATE
FUNCTION UDC 518:517.948
of the Problem. Let 7 be the unit circle with center at the origin which variable z into two domains: the interior D + and the exterior D-. paper
we construct
an approximate
solution of the following
=
singular
divides the
integral equation
7
l S T - -~(T) -}- dh (t) --ai ct~ (t) dT} -}- ~ 1 ~KU) (t, ~) q) (~) gT -[- ~ '
SK~2)(t,T)q~(T)dT~---
g (t),
(1)
tE 7,
u n d e r the a s s u m p t i o n that a (t) h o m e o m o r p h i c a t i y m a p s 7 onto i t s e l f with p r e s e r v a t i o n o r change of o r i e n t a t i o n and s a t i s f i e s the C a r l e m a n condition a~(t)~--t;
ak(t)~t,
1~k~rn--
w h e r e m -> 2 is a n a t u r a l n u m b e r , ak(t) = a [ a k _ i ( t ) ] , a0(t) ~ t . H 6 t d e r condition with exponent ~t, 0 < # _< 1.
1,
(2)
We m o r e o v e r a s s u m e that aY(t) s a t i s f i e s a
We r e m a r k that a p p r o x i m a t e solutions of p r o b l e m s c l o s e l y r e l a t e d to (1) a r e c o n s i d e r e d by v a r i o u s m e t h o d s in [1-4]. In c o n s t r u c t i n g an a p p r o x i m a t e solution of Eq. (1) we use c e r t a i n r e s u l t s f r o m the t h e o r y of a p p r o x i m a t e solution of s y s t e m s of s i n g u l a r i n t e g r a l equations which a r e p r e s e n t e d without p r o o f in the p r e s e n t note. 2. A p p r o x i m a t e Solution of S y s t e m s of S i n g u l a r I n t e g r a l E q u a t i o n s . i n t e g r a l e q u a t i o n s with no shift
We c o n s i d e r the s y s t e m of s i n g u l a r
1 (K~) (0 -~ A (t) ~ (t) -r, B(t) ZF- j(' ~q)(T) - - t d~c.~_ ~ Y ~ K (t, ~) ~ (~) d~ = f (0, ?
(3)
7
w h e r e k(t, -r) = h(~ ~-)lt - "rU", 0 -< v < 1; A(t), B(t), and h(~ m a t r i x - v a l u e d functions of o r d e r m , and f(t) ~ R m .
~-) a r e R i e m a n n i n t e g r a b i e (E R m •
square
We a s s u m e that m a t r i x G (t) = [A (t) + B (t)]- I[A (t) - B (t)] a d m i t s the right f a c t o r i z a t i o n G (t) :
(4)
GZ ~ (t) A (t) G+ (t),
where G+(t) and G_(t) are matrices which are analytic and nondegenerate in the respective domains D + and D-; ]4'
m
A(t) = (t JSjk)j,k=1; zj, j = I, m are the right partial indices of the matrix G(t), and 6jk is the Kroneeker symbol. W e seek an approximate solution of Eq. (3) in the form of an interpolating vector polynomial 2n+l
x ~ (t) - -
2n 2-]- I Z
chjD ~ (s -
sj),
t = e ~,
k ---1, m.
(5)
O d e s s a State U n i v e r s i t y . T r a n s l a t e d f r o m U k r a i n s k i i M a t e m a t i e h e s k i i Z h u r n a l , Vo[. 32, No. 3, pp. 378382, M a y - J u n e , 1980. O r i g i n a l a r t i c l e s u b m i t t e d S e p t e m b e r 18, 1978; r e v i s i o n s u b m i t t e d A p r i l 10, 1979.
0041-5995/80/3203-0251507.50
9 1981 P l e n u m P u b l i s h i n g C o r p o r a t i o n
251
In the l a s t r e l a t i o n Dn(q0) is the D i r i c h l e t k e r n e l of o r d e r n , a n d t k , k = 1, 2n + 1 a r e the i n t e r p o l a t i o n n o d e s th - e
,s~ ,
2kr~ , 2n + 1
su--
k~
1,2n-}- I.
(6)
We f i n d the u n k n o w n c o e f f i c i e n t s {Ckj } f r o m the s y s t e m of l i n e a r a l g e b r a i c m
rn
2n-~- l
ArujCu~ "~ k~l
m
2n-l-:
B~u~a~pchp -+ k~l
tpho~,c~,~ = f~j,
p~I
equations
k~!
r = 1, m;
/ --~ 1, 2n -k 1,
(7)
c=l
in w h i c h A r k j = A r k ( t j ) , B r k j = B r k ( t j ) , f r j = f r ( t j ), a j p = 1 - i/?jp, /~jp = tan [(sj - S p ) / 4 ] o r c o t a n [ ( S p - s j ) / 4 ] for j - p even or odd, respectively, while
-~ sin -'r ~xI ] -
10/~ __~
{2
for I t j - - t v l ~ p
pl 4
for I O - - t p l < p .
~p-~, Let ei (n, P, P, m) = max
! 2~ ~ ~ 2a
1 1 P +~
--
~
1,
t=e
,
'
fr (t) hv~k (t, ~)
Es ,
"~=
eiO,
- -
h~(t,~)=
/ ~ ( t , ~ ) = { ~ _ ~ , ~ l -~,
R,i (t) ho~ (t, x)
~
,~(o) 1r
do
ds
,
(8)
(t,'0lp(t,'c),
't--~I>~O It---~l
