..,..,.,

Transcription

1 «..» 7, 9,,. 3 DOI:.459/mmph7..,..,., E-mil: yshr_ze@mil.ru -,,. -, ( ). -., -. ( - ) , - -., - -, -., -. -., - - -, -., -. : ; ; - ;., -,., - -, []., -, [].,, - [3, 4]. -. 3

2 ( ) u (, ) + ( ) u(, ) = u (, ) + f (, ), D = {(, ) :, }, () : u(,) + δu(, ) = ϕ( ) ( ), () 4 u (, ) = ( ), (3) du(, ) + u(, ) d = ( ), (4) u(, ) = h( ) ( ), (5) d >, δ, ( ) >, f (, ), ϕ ( ), h ( ), u (, ) ( ).. { u(, ), ( )} u(, ) ( ) () (5), : ) u(, ) u (, ), u (, ), u (, ) D ; ) ( ) [, ] ; 3) () () (5)..., δ, ( ) [, ], ϕ( ) [,], f (, ) D, h( ) C [, ], h( ) ( )., ( ) ϕ() = h() + δ h. (6) () (5) u(, ) C, D ( ) C [, ], () (4) ( ) h ( ) + ( ) h( ) = u (, ) + f (, ) ( ). (7) ( ). { (, ), ( )} u () (5)., h( ), (5) : u (, ) = h ( ) ( ). (8) = (), : ( ) u (, ) + ( ) u(, ) = u (, ) + f (, ) ( ). (9), (5) (8),, (7). u(, ), ( ) () (3), (7), - { } (6). (7) (9) : d ( ) ( u(, ) h( )) + ( )( u(, ) h( )) = ( ). d y( ) u(, ) h( ) ( ) () : ( ) y ( ) + ( ) y( ) = ( ). () (), () (6),, y() + δ y( ) = ϕ() h() + δ h( ) =. () ( ), () : ( ) y( ) = cep d,( ) ( ). (3), (), : ( ) c+ p d = ( ). (4) Bullein of he Souh Url Se Universiy Ser. Mhemics. Mechnics. Physics, 7, vol. 9, no., pp. 3

3 ..,.. δ (4) c =, (3),, y( ) = ( )., (), (5).. [6]: y ( ) + y( ) =,, (5) y () =, y () = d y(), d > (6) y ( ) = cos( ), =,,... - g = d., () (4), -, (5) y () =, d y() + y( ) d =, d >. (7) { y ( )} =,.. (5), (6), =. [6],, N, π / ( ) π <. (8) ( dπ ) { ( )} v, v ( ) cos( µ ) y y ( ) π { ( )} =, µ = + π ( ), =,,..., - L (,). [5], N, (8), y ( ) v ( ) <. L (,) 3( dπ ), y ( ) v ( ) <, (9) L (,) = N 9d.. z ( ), =,,... - [6]. { } [6]. { } z ( ) = (cos( ) cos ) /( + d cos ). y ( ), =,,... L (,). η ( ) = sin( ), ξ ( ) = sin( µ ), =,,.... (9) η ( ) ξ ( ) <. () L (,) = N 9d, g( ) L (,). (9), () : ( g( ) y ( ) d) M g( ), () L (,) = ( g( ) η ( ) d) M g( ), () L (,) =. «..» 7, 9,,. 3 5

4 ϕ( ) C[,], g ( ) L (,), (), M = N( + N) d J ( g) dg() + g( ) d =. g = ( g( ), z ( )) = g ( )sin( ) d, (3) α α = + d cos >. g M g L (,) ( ) ( ). (4) =, ϕ( ) C [,], g ( ) L (,), J ( g) =, g () =. (3) (), (5), α g = ( g ()cos( ) g ( )cos( ) d ). (5) = 3 g m g + M g L (,) ( ) () ( ). (6), g( ) W (,), J ( g) =, g () =, g () + dg () =. (4), (), g = g ( )sin( ) α d. g M g L (,) ( ) = ( ). (7) : 3. B, [7] u(, ) u(, ) u ( ) y ( ) =, = D, u ( ) C[, ], 3 B, ( ( ) ) u C[, ] < +. = : u(, ) 3 = ( ( ) ) B u, C[, ]. =. 3 3 B, C[, ] E, B 3, z = u(, ) + ( ). 3 3 E B, C[, ] 3 E.. 6 Bullein of he Souh Url Se Universiy Ser. Mhemics. Mechnics. Physics, 7, vol. 9, no., pp. 3

5 ..,.. (, ) : u { (, ), ( )} u () (3), (7) u(, ) = u ( ) y ( ) (8) = ( ) u ( ) + u ( ) = F ( ;, u) ( =,,...; ), (9) u () = ϕ, ( =,,...), (3) u ( ) = u(, ) z ( ) d, F ( ; u, ) = f ( ) ( ) u ( ), f ( ) = f (, ) z ( ) d, ϕ = ϕ( ) z ( ) d ( =,,...), y ( ) = cos( ), z ( ) = (cos( ) cos( )) /( + d cos )., (9), (3),, ( ) > δ : ϕe F ( ;, u) u ( ) = + e d ( ) + + (, ) ( ) s F ( ;, u) ( ) u { (, ), ( )} e d ( =,,...). (3) u () (3), (7) - u ( ) ( =,, ) (3) (8): ϕe F ( ;, u) ( ) s F u(, ) e ( ;, u) = + d e d y ( ). = ( ) (3) ( ) + + (8), (7), ( ) = h ( ) h ( ) f (, ) u ( ) =. (33), u ( ) (3) (33) - ( ) { (, ), ( )} u () (3), (7) : ϕe ( ) = h ( ) ( ) h ( ) f (, ) + = + ( ) s F ( ;, u) F ( ;, u) + e d e d. ( ) ( ) +. u(, ), ( ) () (3), (7), 3. { } (34). «..» 7, 9,,. 3 7

6 .. (, ) (3). u ( ) = u(, ) z ( ) d ( =,,...), u { }., { (, ), ( )} y ( ), =,,... [, ] u () (3), (7).-, d u (, ) z ( ) d = u(, ) d = u ( ) d ( =,,...), (35) o u ( ) C[, ]., (3),, : ( ) u (, ) z ( ) d = u (, ) cos( ) cos d = α = ( d u (, )cos + u (, )cos( ) d ) = α = ( du(, ) + u(, ) d)cos α + + u(, ) ( cos( ) cos ) d = u ( ), (36) α = + d cos >. () z ( ) ( =,,...), - (35), (36), (9). (), (3)., u ( ) ( =,,...) (9), (33). -, u ( ) ( =,,...) [, ] (3) () (3), (7), (3), (34). 3 E Φ ( u, ) = { Φ ( u, ), Φ ( u, )}. Φ = = ( u, ) u (, ) u ( )sin, Φ ( u, ) = ( ), u ( ) ( =,,...) ( ) (3) (34)., : / / ( u C[, ] ) ( ) ( ) 3 ϕ + ( + δ ) 3 ( ) = = C[, ] ( f ( ) ) d + ( ) [, ] ( u ( ) [, ] ), C C (37) = = 8 Bullein of he Souh Url Se Universiy Ser. Mhemics. Mechnics. Physics, 7, vol. 9, no., pp. 3

7 ..,.. ( ) ( ) [ ] ( ) ( ) (, ) [, ] h h f + ( ) C[, ] C, C ϕ + = = + ( + δ ) ( f ( ) ) d ( ) ( ( ) ) C[, ] u C[, ] ( ) +. C[, ] = = (38), () (3), (7) : (3). ϕ( ) W (,), d ϕ() + ϕ( ) d =, ϕ () =, ϕ () + dϕ () = ;. f (, ), f (, ), f (, ) C( D ), f (, ) L ( D ), df (, ) + f (, ) d =, f (, ) =, f (, ) + df (, ) = ( ); 3. h( ) C [, ], h( ) ( ). (5) (7), (37) (38) : u (, ) A ( ) + B ( ) ( ) u(, ), (39) 3 3 B, C[, ] B, ( ) B ( ) + B ( ) ( ) u (, ), (4) 3 C[, ] C[, ] B, A ( ) = 3 M ϕ ( ) + ( + δ ) f (, ) L (,) L( D ) ( ) C[, ], B ( ) = 3 ( + δ ), ( ) C [, ] ( ) ( ) ( ) ( ) ( ), ( A = h h f + ) ( ) [, ] C[, ] M ϕ + C L (,) = ( δ ) f (, ) + +, L( D ) B ( ) ( ) = h ( ). C[, ] C[, ] = A( ) = A ( ) + A ( ), B( ) = B ( ) + B ( ) (39), (4), u (, ) + ( ) A ( ) + B ( ) ( ) u (, ). (4) 3 3 B, C[, ] C[, ] B,,., 3 ( A( ) + ) B( ) <. (4) K = K ( z 3 R = A( ) + ) R E 3 E () (3), (7).. (3), (34) z = Φ z, (43) z = { u, }, Φ z = { Φz, Φ z}, Φ i ( u, )( i =,) (3), (34). 3 Φ( u, ) K = K ( z 3 R = A( ) + ) E. R E (4), z, z, z KR :. «..» 7, 9,,. 3 9

8 Φ z A( ) + B( ) ( ) u(, ), (44) 3 3 E C[, ] B, Φ z Φ z 3 B( ) R ( ) ( ) u (, ) u (, ) 3 E + C[, ] B,. (45) (44) (45), (4),, Φ K = K -., Φ K = K { u, }, (43)., (3), (34) R K = K. 3 B,, u(, ), u (, ) u (, ) - D. (4) (9), ( u ( ) ) 3 ( ( ) ) C[, ] u [, ] ( ) C + = C[, ] = M f (, ) ( ) u(, ) + +. C[, ] L [, ] (,) C, u (, ) D., () () (4) (7)., () (3), (7) { (, ), ( )} 3 B, R R u. 3 K = KR.. () (5). 3., ϕ() = h(), ψ () = h ( ). K = K ( z 3 R = A( ) + ) R E. 3 E () (5).,.. /..,.. //..., ,.. /.., , ,.. /.. // ,.. - /.., , ,.. /..,.. //... 36, ,.. /..,.. // , ,.. - /..,... :, Bullein of he Souh Url Se Universiy Ser. Mhemics. Mechnics. Physics, 7, vol. 9, no., pp. 3

9 ..,.. Bullein of he Souh Url Se Universiy Series Mhemics. Mechnics. Physics 7, vol. 9, no., pp. 3 DOI:.459/mmph7 ON ONE NONLOCAL INVERSE BOUNDARY PROBLEM FOR HE SECOND- ORDER PARABOLIC EQUAION Y.. Mehrliev, A.N. Sfrov Bu Se Universiy, Bu, Azerbijn E-mil: yshr_ze@mil.ru he pper is focused on solvbiliy of n inverse boundry problem wih n unnown coefficien which depen on ime for second-order prbolic equion wih non-clssicl boundry condiions. he ide of he problem is h ogeher wih he soluion i is required o deermine he unnown coefficien. he problem is considered in he recngulr re. he pper inroduces clssicl soluion of he se problem. A firs, n uiliry inverse boundry problem is emined nd he equivlence (in some sense) of he originl problem is proved. Firs, we pply mehod of vrible seprion o nlyze he uiliry inverse boundry problem. hen, we emine specrl problem for n ordinry secondorder differenil equion wih inegrl condiions. Hving used forml scheme of he mehod of vrible seprion, he soluion of direc boundry problem (in cse of specified unnown funcion) resolves iself ino soluion of Cuchy problem. Afer h he soluion is limied o he soluion of counble sysem of inegro-differenil equions in Fourier coefficiens. In is urn, he ls sysem regrding unnown Fourier coefficiens is recorded in he form of n inegro-differenil equion in he desired soluion. Using relevn ddiionl condiions of he uiliry inverse boundry problem, we obin sysem of wo nonliner inegrl equions for defining unnown funcions. hus, he soluion of he uiliry inverse boundry problem comes down o he sysem of wo nonliner inegrodifferenil equions in unnown funcions. he specific Bnch spce is designed. hen, in he sphere mde of he Bnch spce we wih he help of conrced mpping prove he solvbiliy of he nonliner inegro-differenil equions se, which is unique soluion of he ddiionl inverse boundry problem. Using he equivlence of problems, i is concluded bou eisence nd uniqueness of clssicl soluion of he originl problem. Keywor: inverse boundry problem; prbolic equion; Fourier mehod; clssicl soluion. References. Gordezini D.G., Avlishvili G.A. Memichesoe modelirovnie,, Vol., no., pp (in Russ.).. Bisdze A.V., Smrsiy A.A. DAN SSSR, 969, Vol. 85, no. 4, pp (in Russ.). 3. Gordezini D.G. Seminr Insiu prildnoy memii pri bilissom universiee (Seminr of he Insiue of Applied Mhemics bilisi Universiy), 97, no., p (in Russ.). 4. Gordezini D.G., Dzhioev D.Z. O rzreshimosi odnoy revoy zdchi dly nelineynogo urvneniy ellipichesogo ip (On he solvbiliy of boundry vlue problem for nonliner equion of ellipic ype). Soobshch. AN GSSR (Communicion of AN GSSR), 97, Vol. 68, no. 4, pp (in Russ.). 5. Kpusin N.Yu., Moiseev E.I. Convergence of specrl epnsions for funcions of he hölder clss for wo problems wih specrl prmeer in he boundry condiion. Differenil Equions,, Vol. 36, Issue 8, pp DOI:.7/BF Moiseev E.I., Kpusin N.Yu. Doldy demii nu,, Vol. 385, no., pp. 4. (in Russ.). 7. Khudverdiev K.I., Veliev A.A. Issledovnie odnomernoy smeshnnoy zdchi dly odnogo lss psevdogiperbolichesih urvneniy re'ego poryd s nelineynoy operornoy prvoy chs'yu (he sudy of one-dimensionl mied problem for one clss of qusi-hyperbolic hird-order equions wih nonliner operor righ side). Bu, Chshyogly Publ.,, 68 p. (in Russ). Received November 5, 6. «..» 7, 9,,. 3