f Nsuyen Thj NgSn Tap chi KHOA HOC & CONG NGHfi 15I(06):211-2I4
\ ^^I'^ZT^^JJl^^'^^ OF DUAL INTEGRAL EQUATIONS
O T H E S Y S T E M O F 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 N g u y e n T h i N g W , CoUege of E d u c a t i o n - T N U S u m m a r y The aim nf t-ii
sanations involving F e m e ; t r ^ Z T l ? " " " " : " " " " " ^ ' " " * ' " ' ' ' " " ^ " ^ " " 'O^ts-M of dual integral B t t e system of a C w I ^ t e S " " ' ° """""^ " " ^ ' " * ™ fc' ^ » P ' - « ' " ' " " ' t a
1 I n t r o d u c t i o n
defined by the formulas
/(f) = ^l/l({) = / ' f(xy(dx.
JlT.)e~'-(dx.
(1.4) (1.5) Dual integral equations and syslems of dual integral
equations arise when integral transforms are used to solve mixed boundary value problems of math- ematical physics and mechanics Formal technique fot solving such equations have been developed ex-
tensively, but their solvability problems have been ^be Fourier transforms of tempered generaUzed comdered comparatively weakly |5, 10, n ] , Solv- '^»«i°'«' e™ be found, for example, in |2 12]
ability and solution of a system of dual integral T h . f 1 . .. v,
equations involving Fourier transforms occurring in ™ ' ™ ' ' " = ' ' P"l>lem (1.1)-(1.3) is reduced to lalxed bonndar,, ...1.,..—!.i . . . ^ ''"''^°'^o^uig system of dual integral equations *= ^""^ ""^""lorms occurrmg in imxed boundary value problems for Laplace equa-
tion were considered in (9], The aim of the present we propose a method for reducing tbe systems el dual mtegral equations involving Fourier trani^
forms occurring iu mixed boundary value problems te Laplace equation to the systems of singular in-
tegral. ^ Consider the foUowing problem' find a solution of
the Laplaee equation
-_s.nh(ft) s,({)
•ecosh(eft)""^' S i h ( f k ) j W = - f ' W ' X e (o,6),
_aiU)_ fm±{(b) eosh(f/i) + cosh({/.)""'f'i<^' - Mx),
Te(g.b),
" i ( i ) = 0 , isM\(a,b), Mx) = 0, x e K \ ( a , f c ) ,
MO
dx' dy-
subject to the boundary conditions (-00 < j ; < o c , 0 < y < / t )
(1,1)
f - * ( i , 0 ) = A ( i
| ^ f e O ) = 0 ,
. i e ( a , 4 ) ,
• € R \ (o, i).
* ( i , / i ) = o, l e : xiHg.b], ' \ ( a . * ) ,
(1.2)
(L3)
where / ] , / 2 are given functions.
We WiU solve the problem (1.1)-(1.3) by the method of Fourier transformation. For a suitable func- tion / ( i ) , x e K = (-oo,oo)(for example, f{x) e M R ) ) , direct and inverse Fourier transforms are
'Tel: 0f)r'r!)1223
where '
"•(x) = F-'[il,(OJ(x) = » £ ! £ ) ,
dy us(j:) = f - ' [ i , ( e ) ] ( x ) = #(i,ft).The system (1.6) may be rewritten in the form pf-'[A({)n(flJ{i) , f(i), ^ g f j , . („^ JJ . P ' f '[2(01(1) = 0 . xsSl'-=IL\n, where f(i) = (fiM.Mxn-T. (i({) = p r J l [MO.MOr. the operator f - „ u„dc,.,to„!i m tne generalized, and
/lanh(eft) 1 .^
MO = f cosh(f/i)
P and p' denote restriction operators to 11 and n ' respectively. Dcnotea = ( a i . o j ) ' " . ( - i , i j r n .j
Ngu>en Thi Ngan Tap chi KHOA HOC & CONG NGHE 151(06): 211 -214
iioi difheiiJt to show tliat A(^) £ I^^(E) Wc make Wc shall need some lelatioiis for Clicbyiishev poly- ihe.t,juniptionbf(3-) e >]-'^ -in)and:=hallfind the nomials beluw Let Ti,(x) and L\(J:) be the solution u{x) = F~^\u][x'i m the space B^''^{Q) ChcbyLishc\' polynomial hrst and second kind re-
spectively We have the follow-iiig relations [4|
^Hl(» + 1)&
2 Reduction t o t h e following r„(cos(9) = coi,77^, f',.(c system of singular integral
equations /•'' n[.;(r)]T_,[^(r)]^
I Ml) '•' =
111 this section ue propose a method for reduciiii' f , , , , .,-., ,
the „ « c m > „f dual integral equations (1 6) to the j(, U,-l,M]UA„U'M-'H, - J**;, M following system ot singular integral equations
(-'.21
(2!!)
Jo {^' -y)p[y) b - a
2.1 Some preliminary considerations
We introduce the following definlt^oll^
Definition 2 . 1 . Denotr />i/ C^fo b) lhe c/nw '^ ^ ^ ^ ' of canhmioiis functions V{J:) € S'(R). such thai i, . „ , ,
uir) € C^-'ia.b], i/l^-'(.r) = 0 (A- = 0, 1, ,m - ' ' ^ " - ' ^'^ ' ' ' ^''"^ I^ronecker symbol and 1) J ? ( / , 6). a''-">{,ij<EL^[a,i) ' f^ i - -
Definition 2.2. Let p[r) = ^(r - a)(h - x)(<, < " ^ = ![' ^ ^ ^' ^ • ^ = — -r < 6j Ife (/ff!o?e hy I-l^.ia.b) Hilbert spaces of - '
fiUiLlioni, with iLtpnt to Ihe .-.ailai iiivdiict mid the
- > ' - [ •
p-'\.i)„{c],lx},Ii.Now we Uun lo the system (1,6) We shall find the huirtioi, « , ( r ) = F - " | , 7 | | ( r ) and ,,,(,) = l!"[lj;., - \ ' ' l " ' " ) t ; ^ , < + ^ f " ' I ' J ( J ) lli the Ion
The lollowing Icrania holds
L e m m a 2 . 1 . te( ^- £ i > ( „ , i , Dc„gt, tg .,:„ ' " ' ' ' ' " ^ I ^ ' "> ^ " . ( " . ' ) n t;,-. (o, S],
,l,f z,:To-ETtn,^,on of Ike f„nct,on ^ on R Then (2,7) , ; . e H . - ' " ( , i , ( . )
II. (he .p.,ees !,;;.,(„ « „ eon^der Ihe singular ° 2 / . " ' ' " " « " l ' " " " ' • I " ' iiitegial operator
S i i M U ) = J - ^ ^ ^ , / , r e ! ! . ( a . l , ) wlic.c . , e 0 , ( , i , [,), i e
l.rmeip,rl v.ihie The followmg t h e o r e m " i s ' ^ C t o / " ' ' ' ' ' ' ' ° "
klnedehdze and Duduchaia [l3l
l u e o r e n .1 -^ n Takmg the Founer translorm ivilh r ol the fniit- .l„„-,.L' ' ' • ' " " " • " " ^'" I' b';,„ded ,n lhe Uoiisu,, u , and i r , ej in virlue „1 (2 7) ami (2 ! |
,'---,"- ', %ve get 1 2.2. Tl, , _ _ „„„...„
e get l i ^ ' i H l i t - : . , , . . SCI,-I (2 1)
S i t t ) = ( - i ( ) i ; i ( 5 ) : , ( { ) . — - f . ( £ J (21)1
, iNiUyer Thi Ngan Tap chi KHOA HOC & CONG NGHE 15l(06):21l-214
P ^ 1 ' ° " ° * " " l " " ' " " (2.9) in to (1.6), after som, I •onafaim. » , get the syOem:
5<»ah(|e|ft)J'*' - • / i W , 16(0.1,)
„ _ , , 01,(0
' l::ri;7777rT-«lS.i(€)-tanh(|£|/,)S,(f)](.r) osh(|£|ft)
= >/2(ai), i€(a.b).
Using tbcmula
f^'IsSsnlO^WKi) = i / ' s t o *
Jrr 7„ x-t« e i j . . ( a . l . ) ,
t«d,(|f IA) . S S 2 i ) _ , _ 2e-'lil*
COsh(|^|/l) 1 J 21^1ft (2.10)
trauafijnns of left of the firet of the system (2.10) F-'[slgn(J). t«nh(|{|ft)i;,(f) .
fcoshdijl.) 1(.) - F-[sign(OB,(f) - s l g „ ( J ) ^ ^ - ! ! ! | l _ r , ( j |
. MO , , ,
{cosh(|£|),)Il''-
f-' [si6n(«li;,({)](!) = i y ' i l l l l j , , 2e^=K'''
1 (•- 2e-=l!l'' _ ,
" 2 ? / - , l + e - ' W ' ' • " " • K ) 8 i E n ( « «
Put
t „ ( i - 0 = | y 1° "^.sin{(»-O.I{,
- 2 | ( i h
jnijsrsisn(«Li({))
-J^v.(l)k„(x-tWe hav^ following:
T h e o r e m 2 . 3 . TTie system 0/ dual intetpal equa.
Hani, (1.6) wah (S,({), Bi,(4)) is ojuiealenl tie sy, tem of singular integrgl equatums \a,b) :
1 f''viit)dt /*
+ y " ! ( l ) * i ! ( i - ( ) d < - i / i ( r ) .
— / -rrrr'^ »i(i)42i(i - i)di
+ y_ " 2 ( l ) t a ( i - ' ) d ( = - i A ( i ) , ii(.r) e i ; - . ( a , 6 ) n f f i " ( a , ( , ) , t W € IJto.l.) C H^"'(a.b). g<i<b.(2-11) until 1-2 e Oi[a.b). t.e
J r^(T)dr = 0.
M^) = ^J Mt)sienlx-I)dt, x€R,
- t)dl.
Similarly, we get
£5i(e) F - . [
'mshdfl/i) :](x)
=^/-:
'"'[^s
' • - M
!i(£)e-<'<i5
^ coshJIf |/i)
f~'[sisn(Ow(f)l(ir) = ^ f ~ d ' .
^^,ijij,^sign({)ii({)J(i)
l,2,{r)-'-J' S™(Ei) coslKSA)"
213
X g u v l n T h i N a a r T a p chi K H O A H O C & C O N G N G H E 151(06).211 - 2 1 4
DUA M O T H E P H L " 0 \ G TRJN'H C A P TICH P H A N \'E H E P H U D N G T R I N H T I C H P H A N K Y DI iM htOn [rung bai iiMn T o m lat. Tron^ bat bju
bien hon hop ^uj phuang irinh Lapl.i Ttr khoa; Hi phuang u
References
inh bay phuong phJp Jir.i he phugng Irinh t a p tit.h phaj
• ~ ' n hinh ddi ve he phuong irliili i k h phan ky di cip tich phan bai loan bien hi5n hop, he phuong irinh I
; C i a u r n 0 ' , G a i d a l u p e J o s e J a n d Varoiia ,Juaii L [ 2 0 0 0 ) , •'Soh tegTd.! e q u a t i o n s o n L e b e s g u e s p a ( .Math , 1 4 2 N 2 pp. 253-267
' E - k i i i G I {]97.i]. Boiuidnri/ Ialiie elliptic p-,eiido-difTercntial equalior -N'aiika 2311 p (in R i t i s i a i i )
' e r e z M a r i o ing d u a l m -
!-s Stud! a
m g s y i u l i o l s "
p p 3 0 5 - 3 1 8 ch phan ky di
.•\Lta .Math ]'ict!iaii, 34(3)
!3| G ^ F D anti C h e r s k i i tion, uf lonculidion l^p,. ] p (m Rub-fkiii)
[41 K a u t o i o - itli L \- K i y l o v "•
)(//f Method', in Hujhri o s « n v (,n R u s s i a n ) ' I B N (199<l) .\drnu,
•IT,. C h a p m a n tL- Hall , 220 p
I (1978), Equa- ow X a i i k a 296
•'" A (1962) Ap- Anrilij,,. F i z i n a t -
1 Ngok , I N'guvei
• D u a l
Foiiiier transfoMi.-.' Cli ' ski, Zhin-iial. 38 ^i'i ]ip
N'giiwn Van Xgijc ()9,SS) d u a l m U - f j r a l e q u a l K i n . ,n.
I " n i r \,tu.\Julh \;,tno 1 Popo- C Ya (1!)S6)
^iociated w i t h
' O n t l K ' ^ o l v a h l l l t y o l (iKing F o u n e r Irans.^
• " " " 1 3 X2, pp J i -
ll \ ' a i i N f ; o i (2U0U) Dik u v o l v i u ^ F o n n e r t r a n > f u n
m l o g r a l equa-
i9j N g u v e n T h i N g a n a n d N g u v e n T h i iMinh (2012),
•'Solvability uf a s y s t e m of d u a l i n t c g r a j equa- t i o n s of a m i x e d b o u n d a r y v a l u e p r o b l e m for tlie L a p l a c e e q u a t i o n *', Joumal of iLientc and tethnnlcNj'ij. ThaiXgnyen I'rnferiity, 9 3 ( 5 ) , pp 117-122,
[10] Ufiiand l a S t l 9 ( i 7 ) , Transform methods m problems of da-,licily. L e u m g i a d N a u k a AQ2 p (in R u s s i a n )
[11] Ufliand Ll S ( 1 9 7 7 ) , Method nf dual ,qna- twns m problems of 'mathematical pbymcu.
L c i i m g i a d N a u k a , 2211 p (in R u s s i a n ) [12] V l a d i n n r o ^ ' V S {n)79}. Generahzedfunclmm
inalhemaUtal phy.-,i,s M o s c o w M u , J i g p ,11)
[13] VolevK s p a c e s ( 3-74 (
. R , P a n e k h B P (1965), ' S o m e
•ncralizcd l u n e t i o i w a n d imbedding Usprkhn Math Naiik. 2 0 N l , pp, 1 R u s s i a n )
[ I 4 | W a l t o n J R ( 1 9 7 5 ) , ' A d i s t r i b u t i o n a l ap- pro,ich t o d u a l i n t e g r a l p q u a t i o n s of Titehniarsli type--. ,SLU! I Math .Aval. 6 N 4 , p p , G2B- fi.l3
5i Zen i l " A H ( 1 9 6 8 ) . Ct'iieiahzed Inteyml matioii-., I n t e r s c i e n c e N e w Yoik
/ ' f t X ' A ^ ^ ; ; ; . ; : " 7 ^ 1 , ^ f ^ ^ ^ ! ' ' " ' ^ ^ ^ ' V / i / 2 < 9 / 6 , ^gay duyet dang 30/5,2016 - ^ ' ^ 'iui Ihc Uung- TrmmgDai hoc Su pham - DHTN
