I. Spinei

V. Zolotarevschi

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REVUE D'ANAI,YSE NUMÉRIQUE ET DE THÉORIE DE L'APPROXIMATIONTome 28, Nu l, 1999, pp. 97-104ON THE. APPROXIMATE SOLUTION OF THE EXCLUSIVECASE OF SINGULAR INTEGRAL EQUATIONS WITHCOMPLEX CONJUGATION VALUES ON THE UNKNOWNFUNCTION ON THE CLOSED LIAPUNOV CONTOURV. ZOLOTAREVSCHT and I. SpINEIl. The formulation of the problem. Let f be a closed Liapunov curve, thatlimits the simple connected D+ on the complex,plane. Let us consicler that thePoint¡=0belongstoD*.we examine the singular integrar equation (sIE) with conjugation on thecontour I(Rq =) c,(r) <p(¡) * d,Ø*[*dt + r:2 (r)r0(r ) + ,trØ-/-Jg; -(l) r' ¡**lr,(r,r)9(t) ar+fi,1n,(r,t)e(t) dr= f (t), re r,I rwhere coQ), doQ),h¡(r,r), k=1,2 and f(t) aregiven function on l-and <p(r)is unknown function.Let us suppose, that equation (l) is one of non elliptic type, icl est rhefunctiondefa(r ¡ = fc, e ¡ +dr (r)l [q Ø -,1 t(, l] - [., (, I + ct, ç ¡)lti1r I - a.ç ¡fhas zeroes of integer order on l.The last conditioil is equivalent ro thatdet lC(t) t D(l)l has zeroes of integer order on l, wherec(. lc,(t) c2Ø'l D(ù=ld,(rl -drØl'" = Lrrr,, r (,)l "r\t ) - lîrç¡ -¡,,,,¡ l' The publications il-51 are dedicated to the approximate sorution ofequation (l) either in elliptic and non elliptic cases. In all these works thel99l AMS Subject Clasification: 45G0-5,98V, Zolotarevschi and l. Spineisg(r) = -(s<pX¡l + (?i<pXr) ,o, = -* I n1r, r¡@(r) Colt'¿, ,J'dr+23 Singular Integral Equationstheoretical foundation 'f approximate methods was obtained for the case, when'6 of complex plane. But the case of more2' The computing schenles of the methods. Let us introduce new unknownfunctions gr(r) = g(r) and ez|) =e(¡) . Then, fiom rhe represenrarion99andsmoothness, and K(o, s) satisfies the Hölder condition for both variables withexponent p.The equation (1.¡ and the system (z) at'e equivalent in the sense (see Ir])that for every solution g(t) of equation (l) there is a conesponding solutiono={rp(¡),þ(r)} of the system (2) expressed by rhe funcrion q(Ð, andconversely, if O(/)={q,(¡), qzØ} is rhe solution of sysrem (2), then thecorresponding solution ofequation (l) can be found by the formula:(3) e(r) = |[w,f,l+ezØ].Further we cohsider, that functions c¡(t) and d¡,(t) (k = 1,2) belong to thespace Hþ*t)(f), år Q,r)c n9)[),by borh variables, l(ùe n#+tt(f), whereq is determined below. Moreover, we consider the following representation to betrue:(4) C(t)+ D(t) = A(t)' A)(t)D-(t)R-(r)'C(1) - D(t) = B(t) = B,(t)D *(t)R* (r ) ,where detA'(r)detBl(r) +0, t e f, R*(r) are polynomiar matrix-functions bytlt with constant and nonzero determinants, and the matrix-functions D*(l)and D_.(l) have the following form:D*1r¡ = {|],,-00)u1"ôro }r_= ,D-(,¡ ={¡Jf+-+l' u,- };_=,,where ôi¿ is the Kroneker symbol, d¡,, k=Lp, þ0, k=l¡ are contour,s.fpoinrs, ¡.Í',>pto,>0, k=1,p, vlkt>vyt >0, k=ill ur.naruralnumbers.Let us denote(5) Q = max(tÍ',, r!r,, uÍ,,, u!r,).For such functions, as c¡(r) and, d¡(t), * = \i the equation ( I ) isdegenerated. Further we consider that left particular indexes of matrix-functionBlt (t) A,(t) are equal ro zero and the equation fl ) has the unique solurion.The approximate solution of the, equat,ion ( r ) are found using theapproximate solutions of the system (2). The approximate solution of the system(2) is seeking in rhe formhl 'u,x) q(;r)F(2)where '5 is the singurar operator on r, and ?", is the singurar integral operator;ith the kerner 4#!, 0=arg(r- t), r=.ror ro. ;;;:;,,,."*,n orr) isthe contour I-equation, o is the arch,s absciss, the equation (l) is recluced to theequivalent SIE system without "";j;g;;;(M@ =¡ cQ) aØ. DØ/-lyI**[of,,r) @(t) dr = F(t). r€ t,where- c(¡) and D(t),a1e.¿"t"lrin"¿ above, H(t, r)is the maffix.function ofsecond order' F(r) and o(¿) are the u".,or-"ron.rions of the second order:{Þ(r) =iqirrl, çzØ}, FG)={.f{t),Tr,l}In rhe sysrem (2) the vecror-funcion Õ(r)={gr(¡), rpz(t)} is unknown.According to [], p. 3g], the funcrion q(o.,.ï)= uu5î,r, i, giu.n in the formn(o,r)=Í(o'l). n.r\-'vl l"-"jr' u'( o"r < /, À € (l - ¡r; I), ¡r is the exponent of contour.s Ir00 V. Zolotarev.schi ancl l. Sprnet4 Si ngular Irrtegt al Equations l0r(6) or(¡)- 2ro,r, /€1, (Me.r)(/) = c (t ) xot + !!!- [fl o'T *fH,r(t,'c).r(¡)dt=F(/), ¡€ f,where c* ={rÍ",r1''} ,r. unkn.wn vectors of dimension 2. Then, according to(3), the approximate sorution of the equation (r) has the foilowing form(7) e,,(t) =if Gl,,- +n[2)r r), ¡e r.k=-tt( l0)+2ni JIa) colbc'ation nrcthor, The approximate sorurion of the systern SIE (2) isseeking ln the form (6), and the unknowfls c¡, k =-l,n ur" å*æ.*ined fromcondirion of rurning inro zero of rhe error (nra)çl-f(t) inpoinrs ,, (€ f),j = 0.2n:(ua)ç ¡¡-.f(r,'¡=0, j =o2r.and apply the quadrature method. According to this method, the approximatesolution of system o1'SIE (10) is seeking in the form (6), and rlre u¡knowrrs r:¡,k = -n, n are obtained from SLAE (8), where function H(t, r) is substituted byH,,(t,r), and integral, fitnofr¡.r)'ck dt, .l= olu; k=-n,n are subsrituredIby some quadrature formula. As a quadrature formula, we will chose thefollowing interpolation quadrature formula(r1) fi,lrr srrtar=fi,tu,,fs(t;rltk-r dt, k = -n,n ,fIt is easy to check, that the rast c'nditions turn to the foilowing systenr oflinear algebraic equations (SLAE):Here U,, is the interpolation operator by points t¡, .i=0,2n(8),i [0r,,)sign(Å) + B(r i)sign(-k).]rra + (r2)2tt(u,,s )1t) = Ð x{,, )t,.(t ),r=0-"iî., ;+=å,n,",''ITri .II'H )rk,,,t,=(?) r = (1.2¡t+ Í, f dr ct,'F(t¡), j =02n;here sign(À) = l, for k > 0 and sign(k) = 0, ,( < 0.nto consìderation that matrix_functionf = T, the quaclrature method can not bes introduce a ne.w system of SIE, çlosedkernel without singularities. So, thepositìve number. we,r denore by rrp(o, ,;#îr.tli;lon' Let p'be an arbitrarv(9) , r,,{o,.r¡={ 1:.r,' whenlo-,1>p I "';,\: [p "K(o.,.r), when lo,_.rl.p,and by H,rQ,r) the matrix-ttunction rhat is obtainecl f,.om,natrix_function H(t,.c)substituting rhe funcrions # ,, funcrion 4p(o, s;. : ,.Now let us examinc thc system of SIETaking into consideration the.formulas2ni^tJ t'dt=]¡t0, when r+ -l,,l. whenr'=-1,and the definition (12). it is easy to show that the quadrature formula (l i) turnsinto fbllowing(l3) *lr* rr", u =!stt,¡ r* ngi,,. r=0k=-n,nSo. the SLAE for quadrature method is the following2¡c<, ,Vj+ sign(k)D(r¡)rf +il =-l2nlH,,{r,,t,)r, A!/]r=0(t4¡u^ =.f(t¡), .i=0,2nt02V. Zolorarevschi and I. SpineiQ,,,p(t) obtainerJ b1: the forntula6 7 Singular lntegral Equations t03su,p = i ,f (rr',0 +clyt k),k=_n,n.,r,#í, ':;":{:{:!í::;:,.1;,i1,."ïJ,lîn n."u, that SLAE (8) of coilocarion(18) (tJ,,lvt(J,,ø,, =)U,,(Ap + Be+ K)U,,e,, = U,, F,examined as an equation in a limiçdimensional space u,,[Hp(f)]2, concerningthe unknown two-dimensional vector-function ,Þr(l); ,f is integrational operatorwith kernel H(t.r).Do to the representations (4), the equation (18) is equivalenr ro rheoperatorial equation (in the same space)(19) u,,{(rc-+ec;t)v +r,v +K,}u,,o,, =(J,,F1,where matrix-functions C*(t) , C_(t) are the coefficients of left canonicalfactorizarion of matrix-fïnction Bt t (t) A(t):Blt (t) A(t) = C* (r) C- (¡), I E f.,V =pD_R_l +eD+R+l , T=eC_p+pc;te,\=CIIBIIK +C_QD_R_P +C;IPD*R*Q, FI=C;IBIIF ,where P and,Qare Riesz projectors, =*,Q+S), 0 = I - P.Following 12,p.4424431 let us introduce rhe spacexy ={zu)e[autrt], ; Uz)e[r,!')rrl], ez\].The norm of this space isqlzl -- lv zl, = Ð llr v. r,', ll. * lltv, l,', llui=0This space become a Banach space, ZcXr. It is easy to check, thatkerV= 0. That's why the contraction i =VlXv is one-to-one continuousmapping Xy on Z. Let further E:Xy -+[É1p(f)Jz be the embedding operator.Then. for the contraction M of the operator M on the space Xy we obtainrepresentation ú = M0 +T0, where M s = (PC- + QC;t )ü, To = (T + Kt)E.As the operator PC-+QC;t :tHp(f)Jz --sZ is continuously inversible,then the operator M0:Xy -+Z is also continuously inversible and the operatorTg : Xy -+ lu[ø+rt r, is continuously inversible. Let denote by Ù ,, rhecontraction U,, on Xy .Repeating now the reasoning from [3, gl0], we obtain that for sutïcientlylarge values of numbers r¡ ) l I the operato r IJ ,,M6Û,, : t,,Xr, -> U ,,2 isinversible, and moreover:3. The .ormuration of the basic theorems. The theorems that give thetheorericar foundation or.orpriìn; ;;.r", elaborated in p.2are given berow.THEOREM l. Ler ¿.r(/) and ,trØ, * = l, 2 belong þ space Hås+r)(I-).!'îi"i,,i;,7!i,:';k = I' 2' r(t)eÍty)Q)' 'n' 'u^i'", q is deterntined bt,t e t, the ,"r, ,",,,,1,1í';::;::::i:i"i,ä,',,ï.r,!1",::::';:,î,,r),lii,l,:::,);,:to zerr¡, the equailon ( I ) has a un|lstem of Feier nodes on l, then.fo pomt"; I ¡' i = 0,2n form a(B) of cottocation meiltod can be so values of numbers n, sl,AEsolutions tç,,(t) .fi-om (7), cot,erge in the order and the approximatusolutir¡ngG)rÍ'theequarion(l)withrr"rnlr'"o'" Hþ[)' 0 < P < õ b the exac't(ls) ilq-q,,llp ='(#),where õ = min (cl' þ), p ¿s the exptnent of trrc contour,.s r smooîhness.THEOREM 2, ¿,n. r,,,",,"'" J í^ií',i,",":,;*'T;:,,,"rr:l:",:,ff",:,i;rrî!:{:,,i,î!,îí,i,",,Íí,,sfficientry sma' varues.of p, the su, (14) of quadralure-method has a uniquesoturion "[or = {r11'; .1gr},1-he app rox imate s o I ut i t¡ nsr04 V. Z,olotarevschi antl l. Spinei Ill{u,, tt't .,Û,,1-' ll = of r l.Then, clue to rhe total continuity of ?"¡: Xv,à¡¡7[v*'l1l¡1r, rhe operarorLl ,,(M ¡ + T)Ú ,, :Û ,,X1, ) U ,,2 is also inversible for n > nt () n), and moreovcr(20) llq,,tu,, +r)û,,J-, ll - nf U.Consequently, both operatorial equation (19) and equation (18) haveunique solutions. This is the proof of the uniqueness of the solution of SLAE (8).Let us determine the estimation of the convergence (15) to approximatesolution of the exact solution of equation (l)"ln accordance with (20) and withestirnation.lls_u,sllu =r(#),that take place according to the chosen interpolation kernel [3, $6], it is easy toobtain thatINSTRUCTIONS FOR CONTRIBUTORSRevue d,Analyse Numérique et de Théorie de l'Approximation w|r| consider for publicationimatìon, uniform nu-also their aPPlica ces'ublication are str heirmanuscript in a LATEX file (or AMS TEX or TEX), or in a file using word 6'0 (or highe¡) underwindows. The hgures must bà either translated into the picture environment of LATEX, or sent intoa PCX file format.Theproperpositionofeachtableandfiguremustbeclearlyindicatedinthepaper.The first Page should begin with:Titler)Author2)Abstract-Summary in English (at most 20 printed lines' approx' 200 words)'The footnotes should contain:(1) Work suPPorted bY...(2) Affiliation...AMS Subject Classification:Received. '. (date inserted by the editor)'References should be given ai the end of the paper according to the rules adopted by MathematicalReviews: author, title of pupi., title of journal, number of series, volume number, year of publication'first and last pages. For instance: D. D. Stancu' Evaluation of the remainder term in approximationfo r mul as by B ei ns t e in p o ly no nú aI s, Math Comp " l7 (19 63)' 2'1 0-2'7 8'Fòr books there sháuld be given: author, title, publisher, place and year of publication' Forinstance:J.F.Traub, IterativeMethodsfortheSolutionofEquations,Prenr'ice-Hall, Inc.'EnglewoodCliffs, N.J., 1964'Three copies of the paper and a floppy disk containing the coffesponding file(s) should be sentto the following address:ROMANIAN ACADEMY,,Tiberiu Popoviciu" Institute of Numerical AnalysisGh' Bilaçcu' 37c.P.683400 Cluj-NaPoca 1RomaniaAII OrdCrS ShOUId bE SCNI tO EDITURA ACADEMIEI ROMÂNE' tO RODIPET SA OT tOORION PRESS IMPEX 2OOO SRL:EDITURA ACADEMIEI ROMÂNE, ]alea l3 Septembrie, nr. 13' sector 5, P.o. Box 5-42;Bucureçti, România, Ro-76 117; Tel. 4ll-4ll90 08; T;l./Fax. 401-410 39 83' 401-410 34 48;e-mail: edacad@ear ' ro.RODIPETSA,Pia[aPreseiLibere,nr'1'Sectorl'P'O'Box33-57'Bucure$ti'România;Fax 401-222 64 07; Tel. 401-618 51 03,401-222 4t 26'oRIoN PRESS IMPEX 2000 SRL, Sector 3, P.o. Box 77-19, Bucu¡esti, Româniallo lìIn¿o"l),I,rô-[JRemains to use the tbrmulas (3) and (7).Theorem I is proved.Theorem 2 is proved in accordance with results that are obtained whileproving the theorem l, and that are used for the general theory of direct, but notprojection methods determined in [3, $l], and using the accuracy of quadratureformula (l l) obtained in [3, $8].REI.'ERENCESI l G. Litvincittc, Marginal Problems and Sittgulur Integral Equatious With Renoval, Moscow,Mir, 1977, p. 448.[2 | S. Prösdort, Some Clasrcs of Singular Equclion.r. Moscow Mir, I 979, p. 49ó.[3] V. Zolotarevschi, Fittite Dintensir¡nal Methods of Solving Integral Equuti<trts t¡tt (llosetlConkurs ol lntegratkn, Kishinau, Stiintsa, I 99 | , p. I 36, I 36 c.Received: Dcccnrber 8, 1998

On the approximate solution of the exclusive case of singular integral equations with complex conjugation values on the unknown function on the closed Liapunov contour

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On the approximate solution of the exclusive case of singular integral equations with complex conjugation values on the unknown function on the closed Liapunov contour

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