Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane

: In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K ( w ) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K ( w ) .


Introduction
The paper studies the construction of the exact and approximate solutions of the illposed Cauchy problem for matrix factorizations of the Helmholtz equation. Such problems naturally arise in mathematical physics and in various fields of natural science such as electro-geological exploration, cardiology, electrodynamics and so on. In general, the theory of ill-posed problems for elliptic system of equations has been sufficiently formed by Tikhonov, Ivanov, Lavrent'ev and Tarkhanov in [1][2][3][4][5]. Among them, the most important and applicable topic is related to the conditionally well-posed problems, characterized by stability in the presence of additional information about the nature of the problem data. One of the most effective ways to study such problems is to construct the regularizing operators. For example, it can be done by the Carleman-type formulas (as in complex analysis) or iterative processes (the Kozlov-Maz'ya-Fomin algorithm, etc.).
The work is devoted to the main problem for partial differential equations, which is the Cauchy problem. The main aim of this study is to find the regularization formulas to find the solutions of the Cauchy problem for matrix factorizations of the Helmholtz equation. The question of the existence of a solution of the problem is not considered-it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classic sense. Moreover, for explicit regularization formulas, the optimal solution can be obtained. In this sense, exact regularization formulas are very useful for real numerical calculations. There is good reason to hope that numerous practical applications of regularization formulas are still ahead. In [6][7][8] some applications of the Cauchy problem and the regularization technique for solving different kinds of integral equations have been presented.
The Cauchy problem for matrix factorizations of the Helmholtz equation is among ill-posed and unstable problems. It is known that the Cauchy problem for elliptic equations is among unstable problems which by a small change in the data the problem will be incorrect [1,4,[9][10][11][12][13]. Tarkhanov [14] has published a criterion for the solvability of a large class of boundary value problems for elliptic systems. In some cases of unstable problems, we should apply some operators for solving the problem. But the image of these operators are not closed, therefore, the solvability condition can not be written in terms of continuous linear functions. So, in the Cauchy problem for elliptic equations with data on part of the boundary of the domain the solution is usually unique and the problem is solvable for everywhere dense a set of data, but this set is not closed. Consequently, the theory of solvability of such problems is much more difficult and deeper than theory of solvability of Fredholm equations. The first results in this direction appeared only in the mid-1980s in the works of Aizenberg, Kytmanov and Tarkhanov [5].
The uniqueness of the solution follows from Holmgren's general theorem [10]. The conditional stability of the problem follows from the work of Tikhonov [1], if we restrict the class of possible solutions to a compactum.
Formulas that allow finding a solution to an elliptic equation in the case when the Cauchy data are known only on a part of the boundary of the domain are called Carleman type formulas. In [13], Carleman established a formula giving a solution to the Cauchy-Riemann equations in a a special form of a domain. Developing his idea, Goluzin and Krylov [15] derived a formula to determine the values of analytic functions from known data. A multidimensional analogue of Carleman's formula for analytic functions of several variables was constructed in [11]. The Carleman formula to find the solution of the differential operator with special properties can be found in [3,4]. Yarmukhamedov [16][17][18][19] applied this method to construct the Carleman functions for the Laplace and Helmholtz operators for special form and domain. In [5] an integral formula was proved for the first order elliptic type system of equations with constant coefficients in a bounded domain. In [20], Ikehata applied the presented methodologies in [16][17][18][19] to consider the probe method and Carleman functions for the Laplace and Helmholtz equations in the three-dimensional domain. In [21], a formula for solving the Helmholtz equation with a variable coefficient for regions in space where the unknown data are located on a section of the hypersurface {x · s = t} has been presented by Ikehata.
In many well-posed problems of the system of equations of the first order elliptic type with constant coefficients that factorize the Helmholtz operator, calculating the values of the vector function on the entire boundary is not possible. Therefore, the problem of reconstructing the solution of system of equations of the first order elliptic type with constant coefficients and factorizing the Helmholtz operator [29][30][31][32][33][34][35][36][37][38][39][40][41] are among the more challenging problems in the theory of differential equations.
Additionally, the ill-posed problems of mathematical physics have been investigated by many researchers. The properties of solutions of the Cauchy problem for the Laplace equation were studied in [3,4,[16][17][18][19] and subsequently developed in [5,14,15,.
Let R 2 be the two-dimensional real Euclidean space, G ⊂ R 2 is a bounded simply-connected domain with piecewise smooth boundary consisting of the plane T: y 2 = 0 and some smooth curve S lying in the half-space y 2 > 0, i.e., ∂G = S T. We introduce the following notation: Let D(ξ T ) be a (n × n)−dimensional matrix with elements consisting of a set of linear functions with constant coefficients of the complex plane which is satisfied in the following condition where D * (ξ T ) is the Hermitian conjugate matrix D(ξ T ) and λ is a real number. We consider the following system of differential equations in the region G where D ∂ ∂x is the matrix of first-order differential operators.
We denote the class of vector functions in the domain G by A(G) which is continuous on G = G ∂G and satisfy in the system (1).

Construction of the Carleman Matrix and the Cauchy Problem
Formulation of the problem: suppose U(y) ∈ A(G) and where f (y) is a given continuous vector-function on S. It is required to note that the vector function U(y) is in the domain G, based on f (y) on S.
If U(y) ∈ A(G), then the following Cauchy type integral formula is valid and where t = (t 1 , t 2 ) shows the unit exterior normal which is drawn at point y on curve ∂G and ϕ 2 (λr) denotes the fundamental solution of the Helmholtz equation in R 2 , which is defined in the following form Here H (1) 0 (λr) is the the Hankel function of the first kind [12].
We introduce K(w) as an entire function which takes real values as real part of w, (w = u + iv, u, v−real numbers) and satisfies in the following conditions: We define the function Φ(y, x; λ) at y = x by the following equality where I 0 (λu) = J 0 (iλu)−is the zero order Bessel function of the first kind [10].
In the Formula (6), choosing we get in Equation (3) instead of ϕ 2 (λr), the formula will be correct where g σ (y, x) is the regular solution of the Helmholtz equation with respect to the variable y, including the point y = x.
Then the integral formula can written in the follwoing form: where

The Continuation Formula and Regularization
then the following estimations are correct: where C(λ, x) shows the bounded functions on compact subsets of the domain G.
Proof. Let us first estimate inequality (13). Using the integral Formula (10) and the equality (12), we obtain Taking into account the inequality (11), we estimate the following For this aim, we estimate the integrals T |Φ σ (y, x; λ)|ds y , T ∂Φ σ (y, x; λ) ∂y 1 ds y , and T ∂Φ σ (y, x; λ) ∂y 2 ds y on the part T of the plane y 2 = 0.
Separating the imaginary part of (8), we obtain Given (16) and the inequality we have To estimate the second integral, we use the equality Given equality (16), inequality (17) and equality (19), we obtain Now, we estimate the integral T ∂Φ σ (y, x; λ) ∂y 2 ds y .
Taking into account equality (16) and inequality (17), we obtain From inequalities (17), (20) and (21), bearing in mind (15), we obtain an estimate (13). Now the inequality (14) can be proved. To do this, we take the derivatives from equalities (10) and (12) with respect to x j , (j = 1, 2) then we get: Taking into account the (22) and inequality (11), we estimate the following To do this, we estimate the integrals T ∂Φ σ (y, x; λ) ∂x 1 ds y and T ∂Φ σ (y, x; λ) ∂x 2 ds y on the part T of the plane y 2 = 0.

Corollary 1.
For each x ∈ G, the equalities are true We define G ε as Here ψ(x 1 )-is a curve. It is easy to see that the set G ε ⊂ G is compact.
We should note that the set E ε = G\G ε serves as a boundary layer for this problem, as in the theory of singular perturbations, where there is no uniform convergence.

Estimation of the Stability of the Solution to the Cauchy Problem
Suppose that the curve S is given by the equation where ψ(y 1 ) is a single-valued function satisfying the Lyapunov conditions.
We put Theorem 2. Let U(y) ∈ A(G) satisfies in the condition (20), and on a smooth curve S the inequality Then the following relations are true Proof. Let us first estimate inequality (28). Using the integral formula (10), we have We estimate the following Given inequality (27), we estimate the first integral of inequality (31).
Given equality (16) and the inequality (17), we have To estimate the second integral, using equalities (16) and (19) as well as inequality (17), we obtain To find the integral S ∂Φ σ (y, x; λ) ∂y 2 ds y , using equality (16) and inequality (17), we obtain From (33)- (35) and applying (32), we obtain The following is known Now taking into account (36)-(37) and using (31), we have Choosing σ from the equality we obtain an estimate (28). Now let us prove inequality (29). To do this, we find the partial derivative from the integral formula (10) with respect to the variable x j , j = 1, 2: Here We estimate the following Given inequality (27), we estimate the first integral of inequality (42).
To do this, we estimate the integrals S ∂Φ σ (y, x; λ) ∂x 1 ds y , and S ∂Φ σ (y, x; λ) ∂x 2 ds y on a smooth curve S.
Assume that U(y) ∈ A(G) is defined on S and f δ (y) is its approximation with an error We put Theorem 3. Let U(y) ∈ A(G) on the part of the plane y 2 = 0 satisfies in the condition (11). Then the following estimates is true Proof. From the integral formulas (10) and (50), we have Using conditions (11) and (49), we estimate the following: and Now, repeating the proof of Theorems 1 and 2, we obtain From here, choosing σ from equality (39), we obtain an estimates (51) and (52). Thus Theorem 3 is proved.

Conclusions
The article obtained the following results: Using the Carleman function, a formula can be obtained for the continuation of the solution of linear elliptic systems of the first order with constant coefficients in a spatial bounded domain R 2 . The resulting formula is an analogue of the classical formula of Riemann, Voltaire and Hadamard, which they constructed to solve the Cauchy problem in the theory of hyperbolic equations. An estimate of the stability of the solution of the Cauchy problem in the classical sense for matrix factorizations of the Helmholtz equation was presented. This problem can be considered when, instead of the exact data of the Cauchy problem we have their approximations with a given deviation in the uniform metric and under the assumption that the solution of the Cauchy problem is bounded on part T, of the boundary of the domain G.
We note that for solving applicable problems, the approximate values of U(x) and ∂U(x) ∂x j , x ∈ G, j = 1, 2 should be found.
In this paper, we have built a family of vector-functions U(x, f δ ) = U σ(δ) (x) and ∂U(x, f δ ) ∂x j = ∂U σ(δ) (x) ∂x j , (j = 1, 2) depend on a parameter σ. Also, we prove that under certain conditions and a special choice of the parameter σ = σ(δ), at δ → 0, the family U σ(δ) (x) and ∂U σ(δ) (x) ∂x j are convergent to a solution U(x) and its derivative ∂U(x) ∂x j , x ∈ G at point x ∈ G.
According to [1], a family of vector-functions U σ(δ) (x) and ∂U σ(δ) (x) ∂x j is called a regularized solution of the problem. A regularized solution determines a stable method to find the approximate solution of the problem.