THE CAUCHY PROBLEM FOR THE SYSTEM OF EQUATIONS OF THERMOELASTICITY IN E

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.


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 = (x 1 , ….., x n ) and y = (y 1 , ….., y n ) 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 y n = 0 and a smooth surface S lying in the half-space is a vector function which satisfies the following system of equations of thermoelasticity in D [3]: δ ij is the Kronecker delta, ω is the frequency of oscillation and λ, µ, ρ, θ its coefficients which characterize the medium, satisfying the conditions The system (1) may be written in the following manner: This system is elliptic, since, its characteristic matrix is , is the unit outward normal vector on D ∂ at a point y, ), , , ( = are given continuous vector functions on S.

CONSTRUCTION OF THE CARLEMAN MATRIX AND APPROXIMATE SOLUTION FOR THE CAP TYPE DOMAIN
It is well known that any regular solution ) (x U of the system (1) is specified by the formula 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 , ) , ( = ) , ( 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: ), , , ( ) 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); 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 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: , is an entire function taking real values on the real axis and satisfying the conditions: The following theorem was proved in [7].
where n ϕ -are fundamental solutions of the Helmholtz equation, g n (y,x,k) is a regular function that is defined for all y and x satisfies the Helmholtz equation: Now, in formulas ( 5) and ( 6 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 ), , , 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 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: hold uniformly on each compact subset of .D We construct Carleman matrix.In formula ( 5), (6) we set
and a smooth surface S lying in the cone.Assume ρ

For
the simplicity let us consider n = 3, since the other cases are considered analogously.Suppose that D ρ is a bounded simple connected domain in E 3 with boundary consisting of part ∑ of the surface of the cone smooth part of the surface S lay inside the cone.Assume ρ the properties of E ρ (w) that for Then from(5) we find that the matrix Π(y,x,ω,τ) and its stresses) .e., Π(y,x,ω,τ) is the Carleman matrix for the domain ρ D and the part Σ of the boundary.

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: