496 IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 81 THE CAUCHY PROBLEM FOR THE SYSTEM OF EQUATIONS OF THERMOELASTICITY IN E n I.E. NIYOZOV AND O.I. MAKHMUDOV Department of Mechanics and Mathematics, Samarkand State University Boulevard 15, 103004, Samarkand, Uzbekistan.. iqboln@mail.ru, olimjan@yahoo.com ABSTRACT: In this paper, we consider the problem of analytical continuation of solutions to the system of equations of thermoelasticity in a bounded domain. That is, we make a detailed analysis of the Cauchy problem regarding the values of thermoelasticity in bounded regions and the associated values of their strains on a part of the boundary of this domain. ABSTRAK: Di dalam kajian ini, kami menyelidiki masalah keselanjaran analitik bagi penyelesaian-penyelesaian terhadap sistem persamaan-persamaan termoelastik di dalam domain bersempadan berdasarkan nilai-nilainya dan nilai tegasannya bagi sebahagian daripada sempadan domain tersebut, iaitu kami mengkaji masalah Cauchy. KEYWORDS: Cauchy problem; system theory of elasticity; elliptic system; ill-posed problem; Carleman matrix; regularization 1. INTRODUCTION In this paper, we consider the problem of analytical continuation of the solution of the system equations of the thermoelasticity in spacious bounded domain from its values and values of its strains on part of the boundary of this domain, i.e., we study the Cauchy problem. Since, in many actual problems, either a part of the boundary is inaccessible for measurement of displacement and tensions or only some integral characteristics are available. Therefore, it is necessary to consider the problem of continuation for the solution of elasticity system of equations to the domain by values of the solutions and normal derivatives in the part of boundary of domain. The system of equations of thermoelasticity is elliptic. Therefore, the Cauchy problem for this system is ill-posed. For ill-posed problems, one does not prove the existence theorem: the existence is assumed a priori. Moreover, the solution is assumed to belong to some given subset of the function space, usually a compact one [1]. The uniqueness of the solution follows from the general Holmgren theorem [2]. On establishing uniqueness in the article studio of ill-posed problems, one comes across important questions concerning the derivation of estimates of conditional stability and the construction of regularizing operators. Our aim is to construct an approximate solution using the Carleman function method. Let x = (x1, ….., xn) and y = (y1, ….., yn) be points of the n-dimensional Euclidean space E n , D a bounded simply connected domain in E n , with piecewise-smooth boundary consisting of a piece ∑ of the plane yn = 0 and a smooth surface S lying in the half-space yn > 0. IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 82 Suppose ))(),(),...,((=)( 11 xuxuxuxU nn + is a vector function which satisfies the following system of equations of thermoelasticity in D [3]: 0,=)(),( xUB x ω∂ (1) where ,),(=),( 1)(1)( +×+ ∂∂ nn xjkx BB ωω and ,1,...,=,,)()(=),( 2 2 njk xx B jk jkxjk ∂∂ ∂ +++∆∂ µλρωµδω ,1,...,=,=),( 1)( 1)( nk x B n xnk + + ∂ ∂ −∂ γω ,1,...,=,=),( 1)( nj x iB j xjn ∂ ∂ ∂ + ωηω ,=),( 1)(1)( θ ω ω i B xnn +∆∂ ++ δij is the Kronecker delta, ω is the frequency of oscillation and λ, µ, ρ, θ its coefficients which characterize the medium, satisfying the conditions 0.>0,>0,>0,>230,> η γ θρµλµ + The system (1) may be written in the following manner:     ++∆ +−++∆ 0,= 0=)( 2 divuiv i v ugradvdivugradu ωη θ ω ρωγµλµ where ))(),((=)( xvxuxU . This system is elliptic, since, its characteristic matrix is , 1000 0)()()( 0)()()( =)( 2 2 1= 2 212 121 2 1= 2 1 Λ ΛΛΛΛΛ Λ Λ ni n i ni n i ξξµαξµξµλξξµα ξξµαξξµαξµξµλ ξχ ++++ ++++ ∑ ∑ and for arbitrary ),...,(= 1 n ξξξ with real components satisfying the conditions 1,= 2 1= i n i ξ∑ we have 0.)(=)( 2 ≠+ µλµξχdet Statement of the problem. Find a regular solution U of the system (1) in the domain D by using its Cauchy data on the surface S: IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 83 ,),(=)())(,(),(=)( SyygyUyRyfyU y ∈∂ ν (2) where ))(,( yR y ν∂ is the stress operator, i.e., =)),(,(=))(,( 1)(1)( +×+ ∂∂ nn ykjy yRyR γνν �� ���� � � ���… … … … … … …… … … … . . ����0 0 … … … � ��, ,),(=),(= nxn yjky TTT νν ∂∂ ,1,...,=,, )( )()(=),( njk yy y y T jk k j j kykj ν δλµνµνλν ∂ ∂ ++ ∂ ∂ + ∂ ∂ ∂ ))(),...,((=)( 1 yyy n ννν is the unit outward normal vector on D∂ at a point y, ),,,(= 11 +n fff Κ ),,(= 11 +n ggg Κ are given continuous vector functions on S. 2. CONSTRUCTION OF THE CARLEMAN MATRIX AND APPROXIMATE SOLUTION FOR THE CAP TYPE DOMAIN It is well known that any regular solution )( xU of the system (1) is specified by the formula −∂−Ψ∫∂ )}())(,(){,((=)(2 yUyRyxxU yD νω ,,))()},( ~ ))(,( ~ { * DxdsyUxyyR yy ∈−Ψ∂− ων (3) where the symbol − * means the operation of transposition, Ψ is the matrix of the fundamental solutions for the system of equations of steady-state oscillations of thermoelasticity: given by ,),(=),( 1)(1)( +×+ ΨΨ nn jk xx ωω +         ∂∂ ∂ −−−Ψ ++∑ jk lnl kj njnk l jk xx x 2 1)(1)( 3 1= 2 ))(1[(1=),( αδ πµ δ δδω , || )||(exp ])(1)(1 1)(1)(1)(1)(1)(1)( x xi xx i l lnjnk k nknj j njnkl λ γδδδγδδωηδβ ++++++ +         ∂ ∂ −− ∂ ∂ −+ 0,=1,2,3;=, 2))(2(2 )((11)( = 3 1= 2 3 2 1 2 2 21 11 l l lll l l l i α πρω δ λλµλπ δδλωθ α ∑− −+ +−− −− 0,=1,2,3;=, ))(2(2 )(1)( = 3 1= 2 1 2 2 21 l l ll l l l β λλµλπ δδ β ∑ −+ +− ,)2(=0,=1,2,3;=, )(2 ))((1)( = 122 1 3 1= 2 1 2 2 21 2 1 2 − + − +−− ∑ µλρωγ λλπ δδλ γ kl k l l lll l l IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 84 ),,(=),( ~ ,),( ~ =),( ~ 1)(1)( ωωωω xxxx kjkjnnkj −ΨΨΨΨ +×+ ΠΠ =))(,( ~ yR y ν∂ �� ����� � � ����… … … … … … …… … … … . . �����0 0 … … … � ��. Definition. By the Carleman matrix of the problem (1),(2) we mean an (n+1)×(n+1) matrix Π(y,x,ω,τ) depending on the two points y,x and a positive numerical number parameter τ satisfying the following two conditions: ),,,(),(=),,,(1) τωτω xyGyxxy +−ΨΠ where the matrix G(y,x,τ) satisfies system (1) with respect to the variable y on D, and Ψ(y,x) is a matrix of the fundamental solutions of system (1); ),(|)),,,(),(||),,,((|2) \ τετωντω ≤Π∂+Π∫∂ yySD dsxyRxy where 0,)( →τε as ;∞→τ here || Π is the Euclidean norm of the matrix ,||=|| 1)(1)( +×+ ΠΠ nnji i.e., .|=| 2 1 2 1 1=,       ΠΠ ∑ + ji n ji In particular, .|=| 2 1 2 1 1=       ∑ + m n m uU From the definition of Carleman matrix it follows that. Theorem 1. Any regular solution U(x) of system (1) in the domain D is specified by the formula −∂Π∫∂ )}(),(){,,,((=)(2 yURxyxU yD ντω ,,))()},,,(),( ~ { * DxdsyUxyR yy ∈Π∂− των (4) where Π(y,x,ω,τ) is the Carleman matrix. Using this matrix, one can easily conclude the estimated stability of solution of the problem (1), (2) and also indicate effective method decision this problem as in [4 - 6]. With a view to construct an approximate solution of the problem (1), (2) we construct the following matrix: ,),,(=),,( 1)(1)( +×+ ΠΠ nn kj xyxy ωω (5) +         ∂∂ ∂ −−−Π ++∑ jk lnl kj njnk l kj xx xy 2 1)(1)( 3 1= 2 ))(1[(1=),,( αδ πµ δ δδω ),,,(])(1)(1 1)(1)(1)(1)(1)(1)( llnjnk k nknj j njnkl kxy xx i Φ+         ∂ ∂ −− ∂ ∂ −+ ++++++ γδδδγδδωηδβ where , )()( =),,()( 22 2 0 su duku xysui ysuiK ImkxyxKC nn n n n +        −++ ++ Φ ∫ ∞ ψ (6) IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 85    ≥+ ≥ 1,1,2=,cos 1,,2=),( =)( 0 mmnku mmnkuuJ kuψ )(0 uJ -Bessel's function of order zero, 2 11 2 11 )(...)(= −− −++− nn xyxys and    +−−⋅− −−⋅− − − 1.2=,1)!(2)(21)( 2=,2)!(2)(21)( =,2= 2 mnmn mnmn CC n nm n nm n πω πω π K(ω),ω = u + iv (u, v are real), is an entire function taking real values on the real axis and satisfying the conditions: ,<)( ∞∞≠ uforuK 0,=)( /uK .,0,...,=,<),(|=)(exp|sup 1)( 1 RumpupMKImk p v ∈∞ ≥ ων The following theorem was proved in [7]. Lemma 1. For function Φ(y,x,k) the following formula is valid |,=|),,,()(=),,( xyrkxygikrkxyC nnn −+Φ ϕ where n ϕ - are fundamental solutions of the Helmholtz equation, gn(y,x,k) is a regular function that is defined for all y and x satisfies the Helmholtz equation: 0.=)( 2 nny gkg −∂∆ In (6) we put )(exp=)( τωωK . Then ),,(=),,( kxykxy −ΦΦ τ su duku xysui xysui Im s kxyC nn nn m m n +        −++ −++ ∂ ∂ −Φ ∫ ∞ − − 22 2 01 1 )()(exp =),( ψτ τ ,)( sin )(cos)(exp= 2 2 22 01 1 duku su su xyu s xy nnm m nn ψ τ αττ         + + −++− ∂ ∂ − ∫ ∞ − − (7) .=),( τ τ τ ∂ Φ∂ −Φ′ kxy ,)( sin )(exp=),( 2 2 01 1 duku su su s xykxyC m m nnn ψ τ τ τ + + ∂ ∂ −−Φ ′ ∫ ∞ − − ),,()(exp=),( 1 1 sk s xykxyC m m nnn ττ ψτ ′ ∂ ∂ −−Φ′ − −        − −′ kksJ mnks k >),)(( 2 1 2=,)(cos <0, = 22 0 22 ττπ τ τ ψ τ IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 86 Now, in formulas (5) and (6) we set ),(=),,( kxykxy −ΦΦ τ and construct the matrix ),,,(=),,( τωω xyxy ΠΠ From Lemma 1 we obtain, Lemma 2. The matrix Π(y,x,ω,τ) given by (5) and (6) is Carleman's matrix for problem (1), (2). Indeed by (5), (6) and Lemma 1 we have ),,,(),,(=),,,( τωτω xyGxyxy +ΨΠ where ,),,(=),,( ττ xyGxyG kj +         ∂∂ ∂ −−− ++∑ jk lnl kj njnk l jk xx xyG 2 1)(1)( 3 1= 2 ))(1[(1=),,( αδ πµ δ δδτ +         ∂ ∂ −− ∂ ∂ −+ ++++ k nknj j njnkl xx i )(1)(1 1)(1)(1)(1)( δγδδωηδβ 1.1,...,=,),,,,(] 1)(1)( ++ ++ njkkxyg lnlnjnk τγδδ By a straightforward calculation, we can verify that the matrix G(y,x,τ) satisfies the system (1) with respect to the variable y everywhere in D. By using (5), (6) and (7) we obtain ),(exp)(|),),,,(),(||),,,((| 1\ n m yySD xxCdsxyRxy τττωντω −≤Π∂+Π∫∂ (8) Where C1(x) is a bounded function inside of D. Let us set .))()},,,(),( ~ {)}(),(){,,,((=)(2 * yyyS dsyUxynRyURxyxU τωντω τ Π∂−∂Π∫ (9) The following theorem holds. Theorem 2. Let U(x) be a regular solution of the system (1) in D such that .\,|)(),(||)(| SDyMyURyU y ∂∈≤∂+ ν (10) Then for 1≥τ the following estimate is valid: ).(exp)(|)()(| 2 n m xxMCyUyU ττ τ −≤− By formulas (4) and (9) we have .))()},,,(),( ~ {)}(),(){,,,((|=)()(|2 * \ yyySD dsyUxyRyURxyxUxU τωνντω τ Π∂−∂Π− ∫∂ Now on the basis of (8) and (10) we obtain the required estimate. Next we write out a result that allows us to calculate U(x) approximately if, instead of U(y) and ),(),( yUR y ν∂ its continuous approximations )( yf δ and )( yg δ are given on the surface S: IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 87 1.<<0,|)()(),(|max|)()(|max δδν δδ ≤−∂+− ygyURyfyf y SS (11) We define a function )(xU δτ by setting ,))()},,,(),( ~ {)(),,,((=)(2 * yyS dsyfxyRygxyxU δδδτ τωντω Π∂−Π∫ (12) where 0.>,max=, 1 = 0 0 nn D n n xxx M ln x δ τ Then the following theorem holds: Theorem 3. Let U(x) be a regular solution of the system (1) in D satisfying the condition (10). Then the following estimate is valid: .,)(|)()(| 0 3 Dx M lnxCxUxU m n x n x ∈      ≤− δ δ δτ From all of the above results we immediately obtain a stability estimate. Theorem 4. Let U(x) be a regular solution of the system (1) in D satisfying the conditions: SDyMyURyU y \,|)(),(||)(| ∂∈≤∂+ ν and .1,<<0,|)(),(||)(| SyyURyU y ∈≤∂+ δδν Then ,)()(|)(| 0 4 m n x n x M lnxCxU       ≤ δ δ where Cds yx CxC ynD ~ , || 1~ =)( 4 − ∫∂ is a constant depending on λ, µ , ω. Corollary 1. The limits )(=)(lim),(=)(lim 0 xUxUxUxU δτ δ τ τ →∞→ hold uniformly on each compact subset of .D 3. REGULARIZATION OF SOLUTION OF THE PROBLEM (1), (2) FOR A CONE TYPE DOMAIN Let ),,(= 1 n xxx Κ and ),,(= 1 n yyy Κ be points in ρ DE n , be a bounded simply connected domain in n E whose boundary consists of a cone surface 1>0,>, 2 =,=,=: 2 1 2 1 2 11 ρ ρ π τατα ρρ nnn ytgyyy − ++Σ Κ and a smooth surface S lying in the cone. Assume ρDxx n ∈)(0,...0,=0 . IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 88 We construct Carleman matrix. In formula (5), (6) we set ( )[ ] 1.>0,>,=)( ρτωτω ρ n xEK − Then 0>),,(=),,( kkxykxy −ΦΦ τ su duku xysui xysuiE Im s kxyC nn nn m m n +        −++ −++ ∂ ∂ −Φ ∫ ∞ − − 22 2 01 1 )())(( =),( ψτρ τ (13) .=),( τ τ τ ∂ Φ∂ −Φ′ kxy [ ]{ } ,)((=),( 2 2 01 1 su duku xysuiEIm s kxyC nnm m n + −++′ ∂ ∂ −Φ′ ∫ ∞ − − ψ τ ρτ Where Eρ(w) is Mittag-Löffer`s entire function [8]. For the functions ),( kxy −Φτ hold Lemma 1 and Lemma 2. Further, we may show similar estimate for Uτ (x) and Uτδ (x) (in cone case) defined in (9) and (12), as Theorems 1, 2, 3, and 4. For the simplicity let us consider n = 3, since the other cases are considered analogously. Suppose that Dρ is a bounded simple connected domain in E 3 with boundary consisting of part ∑ of the surface of the cone 0,>1,>, 2 =,= 3 2 3 2 2 2 1 ytgyyy ρ ρ π ττ ρρ + and of a smooth part of the surface S lay inside the cone. Assume ρ Dxx ∈)(0,0,= 30 . We construct Carleman`s matrix. In formulas (5), (6) we take , cos)( )(4 1 =),,( 2 33 2 1 0 3 1 2 su duku xysui wE Im xE kxy +−++ Φ ∫ ∞ ρ ρ ρ ρ τ τ τπ (14) where 3 2 = ysuiw ++ . For the functions ),,( kxy τ Φ holds Lemma 1. It follows from the properties of Eρ(w) that for Σ∈y and ∞<<0 u the function ),,( kxy τ Φ defined by (13) its gradient and the second partial derivatives 1,2,3,=,, ),,( 2 jk yy kxy jk ∂∂ Φ∂ τ tends to zero as ∞→τ , for a fixed . ρ Dx ∈ Then from (5) we find that the matrix Π(y,x,ω,τ) and its stresses ),,,(),( των xyR y Π∂ also converge to zero as ∞→τ for all ,Σ∈y i.e., Π(y,x,ω,τ) is the Carleman matrix for the domain ρD and the part Σ of the boundary. IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 89 If U(x) is a regular solution of the system (1) then the following integral formula .))()},,,(),( ~ {)}(),(){,,,((=)(2 * yyy D dsyUxyRyURxyxU τωνντω ρ Π∂−∂Π∫∂ holds. For ρ Dx ∈ we denote by )(xU τ the following: .))()},,,(),( ~ {)}(),(){,,,((=)(2 * yyyS dsyUxyRyURxyxU τωνντω τ Π∂−∂Π∫ (15) Then the following theorem holds. Theorem 5. Let U(x) be a regular solution of the system (1) in Dρ such that .,|)(),(||)(| Σ∈≤∂+ yMyURyU y ν (16) Then for 1≥τ the following estimate is valid: ),(exp)(|)()(| 3 3 000 ρ ρτ ττ xxMCxUxU −≤− where 0,>,)(0,0,= 330 xDxx ρ ∈ .|,=|, 1 =)( 003 0 0 constantCxyrds r CxC y −−∫Σ ρρρ Let us take continuous approximations fδ(y) and gδ(y) of U(y) and )(),( yUR y ν∂ , respectively, i. e., 1<<0,|)()(),(|max|)()(|max δδν δδ ≤−∂+− ygyURyfyU y SS and define the following function ,))()},,,(),( ~ {)(),,,((=)(2 * yyS dsyfxyRygxyxU δδδτ τωντω Π∂−Π∫ Then the following theorem holds. Theorem 6. Let U(x) be a regular solution of the system (1) in the domain Dρ satisfying the condition (16), then ,)()(|)()(| 3 000 δ δ ρδτ M lnxCxUxU q ≤− where ,)(max=,)(= 3 ρρρ ρ δ ττ ysiReR M lnR S + − . 11 =)(,)(= 4 0 3 0 0 3 y ds rr CxC R x q       +∫Σρρ ρ The theorem is proved analogously as Theorems 3 and 4. Corollary 2. The limits )(=)(lim),(=)(lim 0 xUxUxUxU δτ δ τ τ →∞→ hold uniformly on each compact subset of . ρ D IIUM Engineering Journal, Vol. 15, No. 1, 2014 Niyozov and Makhmudov 90 REFERENCES [1] Lavrent'ev, M. M. “Some Ill-Posed Problems of Mathematical Physics.” , Springer- Verlag, (1967). [2] Petrovskii, I. G. “Lectures on Partial Differential Equations [in Russian].”, Fizmatgiz Press, Moscow(1962). [3] Kupradze, V. D., Burchuladze, T. V., Gegeliya, T. G. “Three-Dimensional Problems of the Mathematical Theory of Elasticity [in Russian]”, Nauka Press, Moscow(1968). [4] Makhmudov, O. I., Niyozov, I. E. “The Cauchy problem for the Lame system in infinite domains in Rm. 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