1. Introduction

Many problems of sciences and technics is naturally reduced to singular integral equations. Moreover (see. ^[1]) plane problems are reduced to one dimensional singular equations. The theory of one domensional singular integral equations is given in ^{[2, 3]}. It is known that the solutions of such integral equations are expressed by singular integrals. Therefore approximate calculation of the singular integrals with high exactness is actual problem of numerical analysis.

For the singular integral of Cauchy type we consider the following quadrature formula

(1.1)

with the error functional

(1.2)

where are the coefficients, are the nodes, , is the characteristic function of the interval , is the Dirac delta function, is a function of the space . Here is the Sobolev space of functions with a square integrable th generalized derivative.

The difference

(1.3)

is called the error of the formula (1.1).

By the Cauchy-Schwarz inequality

the error (1.3) of the formula (1.1) on functions of the space is reduced to finding the norm of the error functional in the conjugate space .

Obviously the norm of the error functional depends on the coefficients and the nodes of the quadrature formula (1.1). The problem of finding the minimum of the norm of the error functional by coefficients and by nodes is called the S.M. Nikol’skii problem, and the obtained formula is called the optimal quadrature formula in the sense of Nikol’skii. This problem was first considered by S.M. Nikol’skii ^[4], and continued by many authors, see e.g. [5-10]^[5] and references therein. Minimization of the norm of the error functional by coefficients when the nodes are fixed is called Sard’s problem and the obtained formula is called the optimal quadrature formula in the sense of Sard. First this problem was investigated by A. Sard ^[11].

There are several methods of construction of optimal quadrature formulas in the sense of Sard such as the spline method, -function method (see e.g. ^{[5, 12]}) and Sobolev’s method which is based on construction of discrete analogue of a linear differential operator (see e.g. ^{[13, 14]}). It should be noted that, using the Sobolev method, in works [15-21]^[15] optimal quadrature formulas and interpolation splines are constructed in the Hilbert spaces.

The main aim of the present paper is to construct of optimal quadrature formulas in the sense of Sard of the form (1.1) in the space by the Sobolev method for approximate integration of the Cauchy type singular integral.

To the problem of approximate integration of the Cauchy type singular integrals are devoted many works (see, for instance, [1,22,23,24.25,26] and references therein).

In the work ^[27] in the space for the coefficients of the weight optimal quadrature formulas of the form

(1.4)

the following system of linear equations is obtained

(1.5)

(1.6)

where is a weight function, are unknown coefficients, is a polynomial of degree .

In particular, from (1.4) after a linear transformation we get quadrature formula (1.1) when and correspondingly from system (1.5)-(1.6) when for the weight function we have the following system of linear equations for the coefficients of optimal quadrature formula in the sense of Sard in the form (1.1) in the space

(1.7)

(1.8)

(1.9)

where

(1.10)

(1.11)

(1.12)

and , are unknowns.

The rest of the paper is organized as follows. In section 2 we give the discrete analogue of the operator and its properties. Section 3 is devoted to calculation of the optimal coefficients i.e. we solve system (1.7)-(1.9). In section 4 some numerical results are presented and compared with numerical results of the work ^[26].

2. The Discrete Analogue of the Operator and its Properties

In the process of solution of system (1.7)-(1.9) the Sobolev method of construction of optimal quadrature formulas (see ^{[13, 14]}) are used. In addition, here we use the concept of discrete argument functions and operations on them. The theory of discrete argument functions is given in ^{[13, 14]}. For completeness we give some definitions.

Assume that and are real-valued functions of real variable and are defined in real line.

Definition 2.1. The function is a function of discrete argument if it is given on some set of integer values of .

Definition 2.2. The inner product of two discrete functions and is given by

(2.1)

if the series on the right hand side of the last equality converges absolutely.

Definition 2.3. The convolution of two functions and is the inner product

(2.2)

Moreover, we need the discrete analogue of the differential operator , which is defined by the following formula (see ^[28])

(2.3)

where .

We give some properties of the discrete function from ^[13]:

(2.4)

(2.5)

where is the discrete delta function and

3. The Coefficients of the Optimal Quadrature Formula (1.1)

In the present section we solve system (1.7)-(1.9).

Suppose when and Then, using Definition 2.3, we rewrite system (1.7)-(1.9) in the following convolution form

(3.1)

(3.2)

(3.3)

where , and are defined by (1.10), (1.11) and (1.12).

We denote

(3.4)

(3.5)

Using the properties (2.4), (2.5) of the operator from (2.3) and (3.5) we have

(3.6)

But for calculation of the convolution (3.6) we need to define the function in all integer values of . When from (3.1) we have . Therefore it is sufficient to determine the function when and .

Furthere, we define the form of the function for and . From (3.4) taking into account (3.2), (3.3) we have

where and are unknown constants.

Taking into account the last equality, from (3.5) for we obtain

(3.7)

where , , , are unknowns and

(3.8)

Thus, keeping in mind (2.3), (3.6) and (3.7) we get the following problem

Problem. Find the solution of the equation

(3.9)

in the form (3.7).

If we find unknowns , , , , then from (3.8) we have

(3.10)

Unknowns , , , will be find from (3.9), using (2.3) and (3.7). Then we get the explicit form of the function and from (3.6) we find optimal coefficients . Moreover, from (3.10) can be found . Thus, the stated problem will be solved completely.

The following theorems hold

Theorem 1. Assume , then the coefficients of the optimal quadrature formula (1.1) in the space have the form

(3.11)

(3.12)

(3.13)

where

(3.14)

(3.15)

(3.16)

(3.17)

Now we consider the case when the singular point coincides with the nodes of the optimal quadrature formula of the form (1.1). In this case from (1.10), (1.11) and (1.12) we respectively get the following

(3.18)

where

Theorem 2. Assume then the coefficients of the optimal quadrature formula of the form (1.1) in the space have the form

(3.19)

(3.20)

(3.21)

(3.22)

(3.23)

where

Here , and have the form as in Theorem 1, but and are defined by the following formulas

Theorem 3. Assume , . Then the coefficients of the optimal quadrature formula (1.1) in the space have the form

(3.24)

(3.25)

(3.26)

(3.27)

(3.28.1)

(3.28.2)

(3.29)

where

here , and have the form as in Theorem 1, but and are defined by the following formulas

Theorem 4. Suppose . Then the optimal coefficients of the quadrature formula (1.1) in the space have the form

(3.30)

(3.31)

(3.32)

(3.33)

(3.34)

where

here , and have the form as in Theorem 1, but and are defined by the following formulas

Proof of Theorem 1. From (3.7) for and we get (3.16) and (3.17), i.e.

(3.35)

(3.36)

From (3.9), using (2.3) and (3.7) for and we have

(3.37)

(3.38)

Thus, for unkowns we obtained the system of linear equations (3.35)-(3.38). Solving this system we obtain (3.14)-(3.17). This means we obtained the explicit form of the function .

Furthere, using (3.7), from (3.6) calculating the convolution for for the optimal coefficients we have

Hence, using (2.3), for and we respectively get (3.11), (3.12) and (3.13). Theorem 1 is proved.

Proof of Theorem 2. From (3.11) when we get (3.19). And the equalities (3.20), (3.21) are obtained from (3.12) for and when . From (3.12), (3.13) when putting instead of we get (3.22), (3.23). Theorem 2 is proved.

Proof of Theorem 3. Suppose , . Then coefficients , , , i.e. the coefficients (3.24), (3,25), (3.28.2) and (3.29), are respectively obtained from (3.11), (3.12), (3.13) when . And coefficients , and , i.e. the coefficients (3.26)-3.28.1), we get from (3.12) when . Theorem 3 is proved.

Proof of Theorem 4. Suppose . Then from (3.11) and (3.12) for we get (3.30), (3.31). And the equalities (3.32) and (3.33) are obtained from (3.12) when and . Finally, from (3.13) when we obtain (3.34). Theorem 4 is proved.

4. The Numerical Results

In the work ^[26] the singular integral with the Cauchy kernel is considered. New quadrature formulas (QFs) based on the modification of discrete vortex method to approximate singular integral are constructed. Convergence of QFs and error bounds are shown in the classes of functions and . In numerical experiments the authors considered two examples for the functions and , respectively.

Here we also consider these functions. To the singular integral when and we apply optimal quadrature formula of the form (1.1) with the coefficients defined by Theorem 1.

In Table 1 and Table 2 we give the error bounds obtained for the case by our optimal quadrature formula and compared with the corresponding error bounds of the work ^[26] at different values of singular point (see Table 1 and Table 2 of ^[26]). In Tables we denote our optimal quadrature formula as OQF.

Table 1. The error bounds of the optimal quadrature formula (OQF) of the form (1.1) and QF (15) of [26] for the function φ(x)=2x²-5x+10, t≠hβ-1 with h=0.02

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Table 2. The error bounds of the optimal quadrature formula (OQF) of the form (1.1) and QF (15) of [26] for the function φ(x)=√x+2, t≠hβ-1 with h=0.02

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5. Conclusion

The present paper is devoted to construction of the optimal quadrature formula in the sense of Sard in S.L.Sobolev space for approximate calculation of the Cauchy type singular integral. For construction of the optimal quadrature formula we use the Sobolev method which is based on the discrete analogues of the differential operators. Applying the discrete analogue of the differential operator we obtain new optimal quadrature formulas. It should be noted that the explicit formulas of the optimal coefficients for the cases (when the singular point does not coincide with nodes) and for the cases (when the singular point coincides with nodes) are obtained, respectively in Theorem 1 and Theorems 2-4. Finally, in numerical examples, we give the error bounds obtained for the case by our optimal quadrature formula and compared with the corresponding error bounds of the quadrature formula (15) of the work ^[26] at different values of singular point . The numerical results show that our quadrature formula is more accurate than the quadrature formula constructed in the work ^[26] (see Table 1 and Table 2).

Acknowledgment

We are very grateful to the referee for remarks and suggestions, which have improved the quality of this paper.

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