1. Introduction

The Nystrom-type methods are popular methods of numerical solution of boundary integral equations of mathematical diffraction theory ^{[1, 2, 3, 4]}. In the countries of the former Soviet Union, some variants of Nystrom-type methods are called methods of discrete vortices ^{[5, 6]} or methods of discrete singularities ^[7]. In the articles ^{[1, 2, 3, 4]} and many others (see. ^[8]) the solution of mathematical electrodynamics problems is carried out in two stages.

At the first stage the initial boundary-value problem for the Helmholtz equation is reduced to an equivalent system of boundary integral equations using the method of parametric representation of integral operators ^{[8, 9, 10]}. On the second stage the resulting system of integral equations are solved numerically by Nystrom-type methods. Using this approach the electrodynamics’ structures containing not perfectly conducting and superconducting elements ^{[12, 13, 14, 15, 16]} can also be investigated numerically.

In particular, the system of boundary integral equations of the problem of electromagnetic waves scattering by a system of superconducting band had been taken in the article ^[12].

The scheme for numerical solution of this problem and the results of numerical experiments had been proposed in the article ^[13]. A rigorous mathematical justification for the numerical solution of various electromagnetic problems by the method of discrete singularities ^{[6, 17, 18]} had existed at the time of publication of the articles ^{[12, 13]}. But the systems of boundary integral equations of the problem of electromagnetic waves scattering by a system of superconducting band are different from the systems of integral equations of other problems that were previously solved numerically by the method of discrete singularities.

Due to these differences rigorous mathematical schemes for the numerical solution of the problem of electromagnetic waves scattering by a system of superconducting band has not been given until now.

In the articles ^{[1, 2, 16]} the integral equations have the same properties as the boundary equations of the problem of wave scattering by superconducting tapes.

Therefore, the rigorous justification of schemes for the numerical solution of the problem of wave scattering by superconducting tapes is of interest.

In this article a scheme for the numerical solution of boundary integral equations of the problem of wave scattering by superconducting lattice has been justified. Also, the rate of convergence of the approximate solutions to the exact solution has been found.

2. The System of Boundary Integral Equations

The system of boundary integral equations of the problem of wave scattering by superconducting tape consists of integral equations of two different types.

The Equations of the first type are singular integral equations of the first kind:

(1)

with the additional condition

(2)

The equations of the second type are Fredholm equations of the second kind:

(3)

In the equations (1), (3) it is assumed that

(4)

(5)

(6)

Let are the Hilbert spaces of measurable functions with the inner product

(7)

and norm . Also we define the subspaces

We introduce the operators (see ^{[8, 19]}):

(8)

(9)

(10)

(11)

(12)

(13)

and mappings

(14)

(15)

With the preceding notation (8)-(15), equation (1) with the additional condition (2) and equation (3) can be written as

(16)

(17)

Proposition 1

The operator is invertible and operator is bounded.

Proof of Proposition 1

The operator А – is the sum of invertible operator (see ^{[8, 19]}) and a compact operator . Hence, by virtue of Nikolsky criterion (see ^[20], p.150),

From the uniqueness of the problem solutions (1)-(2) follows that

Hence,

Therefore operator А is bijective and bounded. So, by the Banach Isomorphism Theorem (see in ^[20], p.113) the operator A has the bounded inverse.

Proposition 2

The operator is invertible and the operator B^-1 is bounded.

Proof of Proposition 2

The operator is compact as the composition of the bounded operator and compact operator. This follows from the compactness of the operator and the Fredholm Theorem (see. ^[20], p.146), that

From the uniqueness of boundary integral equation solution (3) follows that

Hence

Therefore operator B is bijective and bounded. So by the Banach Isomorphism Theorem (see in ^[20], p.113) the operator B has the bounded inverse.

3. The Approximate System of Integral Equations and Its Properties

Let us consider sets of points

(18)

(19)

Here the points are the zeros of Chebychev polynomials of the first kind T_n(τ) and the points are the zeros of Chebychev polynomials of the second kind U_n-1(ξ).

Let us consider the basis polynomials

(20)

(21)

which has the properties

where is the Kronecker delta.

We introduce functions:

(22)

(23)

(24)

(25)

The functions f_n(τ) are the Lagrange interpolation polynomials of the function f(τ) with the nodal points and the functions g_n(ξ) are the interpolation polynomials of the function g(ξ) with the nodal points . The functions K_n(ξ,τ) are the Lagrange interpolation polynomials of the function K(ξ,τ) with the nodal points , with respect to both variables. The functions Q_n(ξ,τ) are the Lagrange interpolation polynomials of the function Q(ξ,τ) with the nodal points with respect to variable ξ and with the nodal points with respect to variable τ.

Let us consider the following system of integral equations:

(26)

(27)

(28)

The functions ϑ_n(τ) and ν_n(τ) are sought in the class of polynomials of degree n-1. Subsequently, the reasoning shows the existence of such solutions.

We define subspaces Π_n,α of spaces L_2,α. Elements of these subspaces are polynomials of degree n. Also we are taking under considerations the spaces

We introduce the operators:

(29)

(30)

(31)

(32)

(33)

In notation (29)-(33) the problems (26)-(27) and (28) have the form:

(34)

(35)

The following estimates hold true ^{[8, 20]}:

(36)

(37)

(38)

(39)

These estimates are the consequences of Jackson’s Theorems (see Corollary 1 of Th. 2 in ^[21], p.128). The constants M₁ and M₃ depend only upon µ and γ. Also the constants M₂ and M₄ depend only upon ψ.

Proposition 3

For all the following inequality holds true

(40)

In (40) constant M₃ depends only upon µ and γ. Besides,

Proof of Proposition 3

From (38) and equality

(41)

it follow that proposition 3 is valid.

Proposition 4

For all natural n the following inequality holds true

(42)

In (42) M^* depends only upon ψ. Besides,

(1)

Proof of Proposition 4

If the function s(ξ) is the polynomial of degree n then

From that fact and by corollary of ’s Theorems ^[21], p. 128 follow:

(43)

Hence, the estimation (41) shows that the operator

(44)

and operator

are bounded. Furthermore

(45)

The following inequality clearly holds

(46)

It is well-known ^[8], ^[19], that

(47)

(48)

It follows from (47)-(48) that

(49)

From (39), (45)-(46) and (45) follow that Proposition 4 is valid for

We introduce the variables:

(50)

(51)

(52)

In the monograph ^[22] you can see the following theorem.

Theorem 1

Let Х and Y be normed linear spaces and let and be finite-dimensional subspaces of the same dimension. We consider two equations.

The equation for exact solution of the problem

(53)

and the equation for the approximate solution of the problem

(54)

where and are the linear operators

(55)

Assume that:

1) the operator is invertible and the operator

(56)

is bounded,

2) the inequality holds

(57)

Then

1) for any function the equation (54) has the unique solution ;

2) let be the solution of equation (53) and let

then

(58)

The Propositions 1-4 and Theorem 1 lead us to the following result.

Theorem 2

For all the following statements hold true.

1) The problems (34) and (35) have a unique solution.

2)

3) The sequence converged to the exact solution of the problem (16) in the norm of space . Moreover,

(59)

4) The sequence converged to the exact solution of the problem (17) in the norm of space . Moreover,

(60)

Proof of Theorem 2

Let’s define numbers:

(61)

(62)

From the Propositions 3-4 and the existence of bounded operators A^-1 and B^-1 it follows that

(63)

Appealing to the Theorem 1 and (63) it concludes the uniqueness and the existence of the solution of the problems (16)-(17).

Also the estimations

(64)

(65)

follow from the statements of Theorem 1 and inequalities (63). By using (36), (37), (64), (65) and Propositions 3-4 we complete the proof of Theorem 2.

4. The Discretization of Approximate System of Integral Equations

As a result of substitution of functions ϑ_n and ν_n in the left-hand side of equations (26) and (28) we obtain polynomials of degree n-2 and, n-1 respectively. The right-hand side of equations (26) and (28) are polynomials of the same degrees.

It is well-known that the interpolating polynomial of the degree n for a given set of distinct n+1 nods is unique. From that follows the statement of Proposition 5.

Proposition 5

Let the statements of Theorem 2 hold true. In order that the integral equations (26) - (28) hold true on the set of the continuum |ξ|<1 is sufficient that:

1. the integral equation (26) is carried out on a discrete set of points ;

2. the integral equation (28) is carried out on a discrete set of points ;

3. the additional condition (27) is fulfilled.

By the Proposition 5, the problem (34) is equivalent to the system of equation:

(66)

(67)

and the problem (35) is equivalent to the equation

(68)

We take discretization of equations (66)-(68) using the interpolation-type quadrature formulas ^{[8, 19]}:

(69)

(70)

(71)

(72)

(73)

These quadrature formulas are exact for polynomials of degree n-1.

As a result of discretization the system of linear algebraic equations in the unknowns and had been obtained. This system has the form

(74)

(75)

(76)

where

(77)

(78)

The existence and the uniqueness of the solution of this linear algebraic equation system is a consequence of the existence and the uniqueness of the solution of system (66)-(68) and its equivalence to the problems (34)-(35).

After solving the system (66)-(68) of linear algebraic equations we obtain the solutions of the problems (34)-(35) by the formulas:

(79)

(80)

5. Conclusions

In this article the justification of the method of numerical solution of the boundary integral equation system (1)-(3) of the problem of wave scattering by superconducting tape had been done. This method is one of the modifications of method of discrete singularities. The convergence of the approximations to the exact solution is guaranteed by propositions proved in this article. Also, the rate of convergence of the approximate solutions to the exact solution had been found.

6. List of Abbreviations

- the Hölder space. It consists of functions defined on [-1,1], which have the properties:

1. the functions have continuous up to order µ;

2. the µ-th derivatives of the functions are Hölder continuous with exponent

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