ON THE CORRECTNESS OF A NONLOCAL PROBLEM FOR THE SECOND ORDER MIXED TYPE EQUATION OF THE SECOND KIND IN A RECTANGLE

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1 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov ON THE CORRECTNESS OF A NONLOCAL PROBLEM FOR THE SECOND ORDER MIXED TYPE EUATION OF THE SECOND KIND IN A RECTANGLE SIROZHIDDIN ZUHRIDDINOVICH DJAMOLOV * The Insue of Maheacs a he Naonal Unversy of Uzbeksan, 115, Tashken, Acadegorodok, Duron yul sree, 9. * correspondng auhor: sro@al.ru (Receved: 15 h Apr. 16; Acceped: 1 s Jul. 16; Publshed on-lne: 3 h Nov. 16) ABSTRACT: In hs work, he correcness of a nonlocal proble fro he Sobolev spaces s proven under soe resrcons o he coeffcens of he consdered second order xed ype equaon of he second knd. The proof s accoplshed wh ulple ehods naely, "ε-regularzaon", prory esaes, and he Galerkn ehod. ABSTRAK: Dala karya n, keepaan sau asalah bukan seepa dar ruang Sobolev elah dbukkan d dala beberapa sekaan kepada koefsen persaaan ens bercapur pernah kedua darpada ens yang kedua yang dperbangkan. Buknya dcapa dengan pelbaga kaedah au, "ε-regularzaon", anggaran pror dan kaedah Galerkn. KEYWORDS: second order xed ype equaon of he second knd; Sobolev space; Galerkn and ε-regularzaon ehods 1. INTRODUCTION The basc conceps of he heory of dfferenal equaons n paral dervaves were fored n he sudy of classcal probles of aheacal physcs and are currenly well undersood. However, he curren probles of naural scences lead o he necessy of seng and qualavely sudyng new asks, whch s a srkng exaple of a class of nonlocal probles. As nonlocal probles we call such probles ha deerne relaonshps beween he value of he soluon or s dervaves a he boundary and neror pons of a consdered doan. In recen decades, nonlocal probles for paral dfferenal equaons are beng acvely suded by any aheacans. Aong he frs sudes of non-local probles, we noe a work by Bsadze and Saarsk [4]. In hs work, spaal and non-local probles for a ceran class of ellpc equaons were forulaed and suded, whch led o he sudy of non-self-adon specral probles. Subsequenly, he proble forulaed n [4], has been naed a Bsadze- Saarsk proble. The nvesgaon of nonlocal probles s caused by boh heorecal neres and praccal necessy. Ths s due o he fac ha he aheacal odels of varous physcal, checal, bologcal, and ecologcal processes are ofen probles n whch, nsead of he classcal boundary condons, defne connecon values of he unknown funcon (or s dervaves) on and whn he boundary are gven. Probles of hs ype ay arse n he sudy of phenoena relaed o plasa physcs, he spread of hea, he 95

2 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov process of osure ransfer n capllary porous eda ssues, deographcs, aheacal bology, and soe echnologcal processes. Nonlocal probles also have praccal value n solvng he probles of sold echancs. They allow o conrol he sress-sran sae and hese are slar o he conrol asks [4, 8-1, 16, 17, 19]. For he frs e, non-local probles n ceran wegh and negave spaces for equaons of xed ype were nvesgaed by funconal ehods n he works [1, 6, 8, 1, 11, 17].. FORMULATION OF A PROBLEM In a recangle (,1) (, T ) we consder a second order dfferenal equaon Lu K ( x, ) u u ( x, ) u c( x, ) u f ( x, ). (1) xx l Proble: To fnd a generalzed soluon of Eq. (1) fro he Sobolev space W ( ), ( l s neger), sasfyng boundary condons here cons. u( x,) u( x, T ), () u(, ) u(1, ), (3) In he prevous works [6, 7, 11, 17] he correcness of proble Eqs. (1)-(3) n he case where K( x,) K ( x, T ) was proven under coparavely srong condons o he coeffcens of Eq. (1). In he presen work, we consder he case where, K( x,) K ( x, T ). Noe ha Eq. (1) s he second knd of xed ype equaon, snce here s no resrcon nsde he doan o he sgn of he varable of he funcon K( x, ) [5]. Frs, we consder he case where l. Assung he coeffcens of Eq. (1) are sooh enough and ( x, ), c( x, ) are perodc n funcon wh respec o varable. 3. UNIUENESS FOR PROBLEM Theore 1: Le aforeenoned condons o he coeffcens of Eq. (1) are fulflled, oreover le K K 1 ; c c, where ln, such ha (1, ). Then for T any funcon f ( x, ) L ( ) a generalzed soluon of proble Eqs. (1)-(3) fro he space W ( ) s unque and he followng nequaly holds rue Here (,) l and l u f. 1 are respecvely regular scalar produc and nor of he Sobolev space l ( ) W, l s an neger nuber and s known ha a l, W ( ) L ( ) [, 5, 13-15]. Here, and furher, by we desgnae posve consans, he exac values of whch are no of neres. 96

3 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov Proof: For any funcon by pars: u W, one can easly ge he followng equaly by usng negraon ( ) Lu exp( ) u dxd exp( )( K K) u u ( c c ) u dxd x exp( ) Ku u u u cu ds x 1 x where ( cos(, ); 1 cos(, x)) s a un vecor of he nner noral o he. Condons of Theore 1 guaranee non-negavy of he negral over he doan. Le u W ( ) sasfy boundary condons Eqs. ()-(3), on condon ha Theore 1, wh respec o funcon K( x, ), us ee he condon of K( x, ) K ( x, T ). Wh T respec o funcon c( x, ), he perodcy of varables and e sasfy condons Eqs. ()-(3), hereby leadng o posvy of he followng boundary negral. x x x J exp( ) Ku u u u c u ds exp( ) Ku u cu ds T T x 1 x u u ds [ K( x, T ) e K( x,)] u ( x,) dx [ e 1] u ( x,) dx 1 T [ (, ) T (,)] (,) [ x(, ) (, ) x(1, ) (1, )] c x T e c x u x dx u u u u d 1 1 [ K( x, T ) K( x,)] u ( x,) dx [ c( x, T ) c( x,)] u ( x,) dx. Ong non-posve es fro Eq. (4) we ge Lu exp( ) u dxd exp( ) ( K K) u ux ( c c ) u } dxd. (5) (4) Applyng Young's nequaly o Eq. (5) [5, 13, 14] he followng esae s obaned u f, 1 fro whch he unqueness of he soluon of proble Eqs. (1)-(3) follows. 4. EXISTENCE FOR PROBLEM In order o prove he exsence of he soluon of proble n Eqs. (1)-(3) n W ( ) he "ε-regularzaon" ehod s used wh he Galerkn ehod [7, 13-15]. Consder a nonlocal proble for a copose ype equaon L u u Lu f ( x, ) (6) q q T D u D u ; q,1, (7) u (, ) u ( 1, ) ( 8) 97

4 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov p p u u u Here D u, D,,1,, p u u p u s he Laplace operaor n a plane, x s a sall posve nuber, and cons such ha (1, ). Eq. (6) acs as an -regularzaon of Eq. (1) [5, 7, 13, 14]. The class of funcons u ( x, ) W ( ), u L ( ) are desgnaed hrough V, sasfyng condons n Eqs. (7), (8). Defnon: As he regular soluon of proble n Eqs. (6)-(8), funcon u ( x, ) V sasfes Eq. (6). Theore : Le he followng condons holds rue n : K K ; c c, 98 1 where ln such ha (1, ). Then for any funcon f, f L ( ), T f ( x,) f ( x, T ) here exss a unque regular soluon of proble n Eqs. (6)-(8) and he followng nequales are rue: Proof: I) ( u u ) u f ; 1 x II) u u f f. The proof of nequaly I) wll be done slarly o he proof of Theore 1, fro whch he unqueness of he regular soluon of proble n Eqs. (6)-(8) follows. Now, he second pror esae s proven. Le ( x, ) are egenfuncons of he followng proble:, ( 9) x D D ; p,1, (1) p p T (, ) (1, ). (11) Due o he general heory of lnear self-adon ellpc operaors [, 13, 15], s known ha all egenfuncons of proble n Eqs. (9)-(11) for a fundaenal syse n C ( ), whch s orhogonal n L ( ). Solvng probles n Eqs. (9)-(11), s ( x, ) cos sn x, where ; ; ;,1,,... T Usng hese sequences of funcons, a soluon of he auxlary proble s consruced

5 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov e, (1) ( x,) ( x, T ). (13) Obvously, proble n Eqs. (1)-(13) s unquely solvable and s soluon has he for T T exp( ) exp( ) sn cos cos 1 1 d d x e d e d e sn sn cos 1 x 4 I s clear ha he funcons ( x, ) C ( ) are lnearly ndependen. Really, f c for soe se of sequences of funcons 1,,...,, hen acng on hs su 1 by he operaor resuls n c 1 1 c, Fro whch follows ha c for any 1,. Noe, fro he consrucon of funcon ( x, ) he followng condons o he funcons ( x, ) C ( ) follow: D D, q,1, ( 14) q q q (, ) (1, ) (15) Now an approxae soluon of Eqs. (6)-(8) s searched for n he for N N w u c 1 where coeffcens c ; 1, are defned as soluons of he algebrac syse. N L u exp( ) dxd f exp( ) dxd. (16), The unque solvably of he algebrac syse n Eq. (16) s proven. Mulplyng every equaon of (16) by c and suarzng wh respec o fro 1 o N, consderng proble n Eqs. (9)-(13) resuls n L wexp( ) w dxd f exp( ) w dxd. (17) Fro whch, by vrue of Theore, by negrang he deny n Eq. (17) an approxae soluon of proble n Eqs. (6)-(8) s obaned ha esaes I).e. u u u f 1 x Ths ples he solvably of Eq. (16) [6, 8, 13, 14]. In parcular, fro he esae I) we oban a weak soluon of proble n Eqs. (6)-(8).. 99

6 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov Now he second prory esae II) us be proven. Thanks o Eqs. (9)-(13), fro he deny n Eq. (16) coes 1 1 L wexp( ) dxd f exp( ) dxd. (18) Mulplyng each equaon of (18) by c (13),(14) no accoun fro Eq. (18), he followng s derved and sung fro 1 o N, akng Eqs. w w L we w w dxd fe w w dxd (19) 4 4 Inegrang Eq. (19) accordng o he condons of Theore and he boundary condons n Eqs. (14)-(15), we oban he followng nequaly w ( ) ( ) x xx f f e K K w K K w w dxd e K ( w w w ) ds e Kw w w w w w ds J () x xx x x x 1 1 where J 1 s negral along he doan and J,,3 are negral along he boundary. Consderng he condon of Theore, usng Young's nequaly, resuls n: w J1 w w w dxd cons exp( ) x xx. (1) 3 Based on boundary condons n Eqs. (14),(15) and he condon of Theore, J, 1,. () c cons does no depend on N, hence fro Eqs. (17)-() he second esae for he approxae soluon of Eqs. (6)-(8) follows. An esae of Eq. (17) ogeher wh Eq. u converges N (1) allows o pass o l a N and conclude ha a subsequence k n vew of he unqueness (Theore 1) n L ( ) ogeher wh a frs-and second-order o he desred regular soluon u ( x, ) of Eqs. (6)-(8) possessng he properes saed n Theore [6, 8, 1, 13, 14]. For u ( x, ) by vrue of Eq. (1), he nequaly holds rue u u f f (3) Theore s hus proven. Now usng he ehod of "ε-regularzaon" he solvably of Eqs. (1)-(3) s proven. Theore 3: Le all condons of Theore be fulflled. Then he generalzed soluon of he proble fro W ( ) exss and s unque. 1

7 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov Proof: The unqueness of he soluon of proble n Eqs. (1)-(3) fro W s proven n ( ) Theore 1. Now he exsence of a generalzed soluon of Eqs. (1)-(3) fro W ( ) s proven. For hs, consder Eq. (6) n doan wh boundary condons n Eqs. (7), (8) a. Snce all condons of Theore are fulflled, hen here exss a unque regular soluon of proble Eqs. (6)-(8) a and for hs soluon esaes I), II) are rue. Fro here follows, under he known heore of weak copacness, ha fro a se of funcons u, s possble o ake a poorly convergng sub-sequence of funcons n V such ha u u a. I wll be shown ha lng funcon u( x, ) sasfes he equaon Lu poorly n ( ) W and he sequence f (1). In realy, as he sequence u converges 3 u,( ) s n regular nervals led n 3 L ( ), and he operaor L - lnear s had u u Lu f Lu Lu ( ) (4) L u u Fro here, he followng nequaly wll be perfored u u Lu f L( u u ) u u (5) Fro (5) approachng a l a we ge he unque soluon of Eqs. (1)-(3). Thus, heore 3 s proven. 5. SMOOTHNESS OF THE SOLUTION Now, he ore general case of l 3 us be proven. Furher, he coeffcens of Eq. (1) are assued o be nfnely dfferenable n a closed doan. Theore 4: Le he condons of Theore 3 be fulflled and le ( K ) K K. Then, for any funcon f ( x, ) such ha f W ( ), D f L ( ), 1 D f D f, here exss a unque generalzed soluon o proble (1)-(3) fro he space W where,1,,3,... Proof: T ( ) Fro he soohness of he soluon o proble n Eqs. (9)-(13), he followng condons for he approxae soluon of Eqs. (6)-(8) follow: w u C ( ); D w D w ; q,1,,3,4,..., N q q T w(, ) w(1, )., 11

8 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov Consderng he condons of Theore a, nonlocal condons a, T and fro he equaly hen T T T (, ) e L u e u e Lu e f x u ( x,) u ( x, T) cons. Hence, follows ha funcon v ( x, ) u ( x, ) belongs o V and sasfes he followng equaon P v L v K v f u c u F (6) Fro Theore, follows ha he se of funcons F s unforly bounded n he space L ( ),.e. F f f Furher, fro he condons of Theore 3, one can easly see ha operaors P,( ) sasfy he condons of Theore 4. Fro here, based on esaes of I) and II) for a funcon, he followng analogcal esaes are obaned v v v v f f 1 x v v f f 1 (8) (7) Funcon u sasfes a parabolc equaon wh condons n Eqs. (),(3) u u u xx f u Ku ( 1) u cu, (9) here L ( ). The se of funcons s unforly bounded n f f (3) 1 1 f W 1 ( ).e. Based on prory esaes for parabolc equaons [4,6] and nequaly n Eq. (3), hen Slarly, one can prove ha where,3,... u u f. 3 f, 1

9 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov 6. CONCLUSION Solvably of he proble consdered can be forulaed n he ers of soohness of K (x, ) n he case where, K( x,) K ( x, T ). In hs he second knd of xed ype equaon suded, snce here s no resrcon nsde he doan. Under he assupon ha he coeffcens of he equaon are sooh enough and ( x, ), c( x, ) are perodc wh respec o varable. REFERENCES [1] Alov NA. (1983) On a nonlocal boundary value proble for a non-classcal equaon. The heory and ehods for solvng ll-posed probles and her applcaons. Novosbrsk, [] Bereznsky YM. (1965) Expanson n egenfuncons of selfadon operaors. Kyev. [3] Berdyshev AS. (1993) Nonlocal boundary value probles for equaons of xed ype n a devaon fro he characerscs. Dfferensal.Uravn., 9(1): [4] Bsadze AV, Saarsk AA. (1969) On soe sple generalzaons of lnear ellpc boundary value probles. DAN SSSR, 185(4): [5] Vragov VN. (1983) Boundary probles for non-classcal equaons of aheacal physcs. Novosbrsk, 16pp. [6] Glazaov SN. (1985) Nonlocal boundary probles for xed ype equaons n a recangle. Sberan Mah. J., 6(6): [7] Daalov SZ. (1989) On correcness of nonlocal boundary probles for any-densonal xed ype equaon. Applcaon of ehod of funconal analyss o he non-classcal equaons of aheacal physcs. Novosbrsk, [8] Yogorov IE. (1995) On soohness of a soluon o a nonlocal boundary value proble for an operaor-dfferenal equaon wh varable e drecon. Ma. Zaek YaGU, (1): [9] Il'n VA, Moseev YI. () On he unqueness of soluon of xed proble for wave equaon wh nonlocal boundary condons. Dfferens. Uravn., Ç6(5): [1] Karakopraklev GD. (1987) Nonlocal boundary probles for xed ype equaons. Dfferens. Uravn., 3(1) [11] Karaoprakleva MG. (1991) A nonlocal boundary-value proble for an equaon of xed ype. Dfferens. Uravn., 7(1): [1] Kalenov TS. (199) On he Drchle proble and nonlocal boundary probles for wave equaon. Dfferens. Uravn., 6(1):6-65. [13] Kozhanov AI. (199) Boundary probles for equaons of aheacal physcs of odd order. Novosbrsk, 13 pp. [14] Kuzn AG. (199) Non-classcal xed ype equaons and her applcaons o he gas dynacs. Lenngrad 71pp. [15] Ladyenskaya OA. (1973) Boundary probles of aheacal physcs. Moscow 47 pp. [16] Moseev YI. (1) On a solvably of a nonlocal boundary proble. Dfferens. Uravn., 37(11): [17] Terekhov AN. (1985) Nonlocal boundary probles for equaons of varable ype. Nonclasscal equaons of aheacal physcs. Novosbrsk, [18] Nakhushev AM. (198) On an approxae ehod for solvng boundary value probles for dfferenal equaons and s applcaon o he dynacs of sol osure and groundwaer. Dfferens. Uravn., 18(1):7-84. [19] Beln SA. (6) On a xed nonlocal proble for a wave equaons. Elecronc J. Dff. Equaons, 13:1-1. [] Berdyshev AS, Karov ET. (6) Soe-non-local probles for he parabolc-hyperbolc ype equaon wh non-characersc lne of changng ype. Cenral European J. Mah. (CEJM), 4():

10 IIUM Engneerng Journal, Vol. 17, No., 16 Daolov [1] Rassas JM, Karov ET. (1) Boundary-value probles wh nal non-local condon for parabolc equaons wh paraeer. Eur. J. Pure Appl. Mah. (EJPAM), 3(6):