Approximate Methods for Calculating Singular and Hypersingular Integrals with Rapidly Oscillating Kernels

: The article is devoted to the issue of construction of an optimal with respect to order passive algorithms for evaluating Cauchy and Hilbert singular and hypersingular integrals with oscillating kernels. We propose a method for estimating lower bound errors of quadrature formulas for singular and hypersingular integral evaluation. Quadrature formulas were constructed for implementation of the obtained estimates. We constructed quadrature formulas and estimated the errors for hypersingular integrals with oscillating kernels. This method is based on using similar results obtained for singular integrals.


Introduction
Recent years have shown the importance of evaluating singular and hypersingular integrals with rapidly oscillating kernels in mathematical modeling of wave processes in many areas of physics and technology: electrodynamics (waveguides, gyrotrons), aerodynamics, geophysics (transformation of gravity and magnetic fields), etc.
Today, there are very few manuscripts devoted to approximate methods for evaluating singular integrals with rapidly oscillating kernels.We are unaware of papers dealing with approximate methods for evaluating hypersingular integrals with rapidly oscillating kernels.
In this paper, we construct an optimal with respect to order quadrature formulas for evaluating singular and hypersingular integrals on Hölder functions and differentiable function classes.
The paper is organized as follows: Section 1 contains the review of publications evaluating singular integrals with rapidly oscillating functions.In this section, we give the definitions of singular and hypersingular integrals and optimal algorithms for their evaluation.
In Section 2, Levin's method is extended to singular and hypersingular integrals.
In Section 3, we introduce an optimal with respect to order quadrature formulas for calculating singular integrals with oscillating functions.
In Section 4, we present methods for evaluating the hypersingular integrals with rapidly oscillating functions.
In Section 5, we give the conclusions of our study.
Here, we give a brief overview of the manuscripts considering singular integrals with oscillating kernels.
The chapter "Oscillatory Singular Integrals" in [18] is devoted to the study of oscillatory singular integrals.The authors consider singular integrals of the form R n e ip(x,y) K(x − y) f (y)dy, where (i) K is a C 1 function away from the origin; (ii) K is homogeneous of degree -n; (iii) the mean value of K on the unit sphere vanishes; (iiii) p(x, y) is a real-value polynomial on R n × R n .The boundedness of the operator T is investigated in a number of function spaces.Integrals of the form (1) are widely used in the Radon transform.
The paper [11] deals with approximate methods for evaluating the integral where f (x) is an analytic function in [−1; 1], w ∈ R \ {0}.The integral ( 2) is converted to the form Philon's method is used for the integral I 0 ( f ).
The second method proposed in [11] consists in approximating the function f (x) in I 0 ( f ) by segment of the Taylor series.The well-known methods are used to approximate the integral I( f ) [19].
Approximate methods for evaluating singular integrals with oscillating kernels of the form have been studied in [9].In [16], quadrature formulas are constructed for evaluating singular integrals of the form where w is a large positive number, −1 < a < 1.
To construct a quadrature formula, the function f (x) is approximated by interpolation polynomial P n ( f , x) on nodes x 1 , x 2 , . . ., x n , a, and a does not match x 1 , x 2 , . . ., x n .As a result, the integral (5) is approximated by the quadrature formula The estimate for R n ( f , a) is given in [16].

Definitions of Singular and Hypersingular Integrals
Recall the definitions of function classes.
Let γ be the unit circle centered at the origin in the plane of the complex variable.Let Definition 2. The class W r (M; A), r = 1, 2, . . ., consists of functions f ∈ C[a, b] which have absolutely continuous derivatives of orders j = 0, 1, . . ., r − 1 and a piecewise continuous derivative f (r) satisfying f (r) (x) ≤ M. Definition 3. The class W r H α (M; A) consists of functions f (x) belonging to the class W r (M; A) and satisfying the additional condition f (r) (x) ∈ H α (M, A).

Consider the integral
Definition 4. The Cauchy principal value of the singular integral (6) is called the limit Recall the definitions of hypersingular integrals.Hadamard [23] introduced a new type of integral, hypersingular integrals: for an integer p and 0 < α < 1 defines a value of the above integral ("finite part") as the limit of the sum as x → b if one assumes that A(x) has p derivatives in the neighborhood of point b.Here, B(x) is any function that satisfies the following two conditions: (i) The above limit exists; (ii) B(x) has at least p derivatives in the neighborhood of a point x = b.
An arbitrary choice of B(x) does not depend on the value of the limit in (i).Condition (ii) defines the values of first (p − 1) derivatives of B(x) at point b.An arbitrary additional term in the numerator is infinitesimal, of order (b − x) p .Notation 1. Hadamard [24] gave a fascinating report of various aspects of the creative process in solving mathematical problems and, in particular, on his discovery of hypersingular integrals.
Chikin [25] introduced the definition of the Cauchy-Hadamard type integral that generalized a singular integral in the Cauchy principal and Hadamard sense.
where ξ(v) is a function chosen so as to provide the existence of the limit above.
In some cases, it is more convenient to use the following definition of hypersingular integrals, which is equivalent to Definition 6.
A hypersingular integral with order of p + 1 singularity, p ≤ r, is defined by 1.3.Optimal Quadrature Formulas for Calculating Singular and Hypersingular Integrals Formulation of the problem of constructing the best quadrature formula belongs to Kolmogorov.Bakhvalov introduced [26] the concepts of asymptotically optimal and optimal with respect to order passive algorithms for solving problems in numerical analysis.Other approaches to determine optimal passive algorithms are given in [27][28][29].
We give now the definition of optimal quadrature formulas for singular integrals.Consider the quadrature rule The error (8) is The error of (8) on Ψ class is We introduce the functional where the lower bound takes over all the nodes t k , −1 < t k < 1, and the coefficients p kl , k = 1, 2, . . ., N, l = 0, 1, . . ., ρ.
The quadrature Formula (8) is defined by a set of nodes t * k , k = 1, 2, . . ., N, and coefficients p * kl , k = 1, 2, . . ., N, l = 0, 1, . . ., ρ, called optimal, asymptotically optimal and optimal with respect to order if , respectively.In a similar way, the concept of optimal, asymptotically optimal and optimal with respect to order quadrature formulas for evaluating hypersingular integrals is introduced.
).Let the integral Cϕ be evaluated with quadrature formula with fixed nodes t k , k = 1, 2, . . ., N, and fixed coefficients p kl (t), k = 1, 2, . . ., N, l = 0, 1, . . ., ρ.In this case, the functional R N (Ψ, p kl , t k ) is equivalent to the Peano constant.Theory of the Peano constants is a very important part of classical numerical theory (see [30]).Comparing the definitions of the Peano constant and optimal quadrature formulas, one can observe that the Peano constant theory is a special case of optimal algorithms theory.

Levin's Method for Evaluating Singular and Hypersingular Integrals with Rapidly Oscillating Kernels
We present an application of Levin's method for evaluating hypersingular integrals with rapidly oscillating kernels.
Consider the integral The integral ( 9) is associated with the differential equation where t is a parameter.Differentiating the left-hand side, we have moreover, it is enough to consider the equation If it is possible to find an analytical solution of Equation ( 10), then Note that when solving the differential Equation ( 11), the singularity can be avoided for τ = t.
Indeed, by the definition of the hypersingular integral, we have The function ϕ(τ) has continuous derivatives up to p − 1 order in a neighborhood of zero and is chosen such that the limit exists.
Taking the integrals separately on the right-hand side of ( 12) and applying the formula (11) to each of them, we have The functions x(t ± η)e iωg(t±η) can be represented as a sum: where the first term tends to infinity as η → 0, and the second term tends to the finite limit.Obviously, 1) .( 13) ) ) = 0 and from ( 13) the final formula follows: 1 1) .
Thus, the analogue of the Newton-Leibniz formula for hypersingular integrals has been obtained.The application of the Newton-Leibniz formula for hypersingular integrals for certain function classes has been shown in [31].
It follows from the above that for evaluating hypersingular integrals with rapidly oscillating kernels, one can use numerical methods for solving ordinary differential equations.

Quadrature Formulas for Evaluating Singular Integrals with Rapidly Oscillating Functions
In this section, we study methods for evaluating the following types of singular integrals with rapidly oscillating functions where m is a natural number.Note that integral ( 14) is reduced by the Hilbert transformation to integral (15).Therefore, in this section we can restrict ourselves to considering the integral (15).

Lower Bound Estimates for Quadrature Formula Errors
First, we find a lower bound estimate for the quadrature formula errors using N values of integrands.
The integral (15) will be evaluated using the quadrature formula We find a lower bound estimate of the error for (16) provided that ϕ ∈ H 1 ([0, 2π], 1).
In doing so, we generalize the method for constructing optimal quadrature formulas for evaluating singular integrals proposed in [1,3].
It follows from below that when some nodes coincide, the lower bound error of the quadrature formula does not decrease.Thus, we assume that the number of nodes v j , j = 0, 1, . . ., n − 1, is equal to n = 2m + 2N.
It is easy to see that in the marked segments Thus, for N ≤ m/2, we have the estimate From this estimate and the inequality (18), we have Here, and below C, are the constants independent of N and m.For m 2 ≤ N ≤ 2m, we must change the proof.Let {v i }, i = 0, 1, . . ., n be a union of node sets {w k }, k = 1, 2, . . ., N and {s l }, i = 0, 1, . . ., 2m. Let Each node s l is associated with the function Then, Averaging the previous inequality over l, l = 0, 1, . . ., 2m − 1, we have Estimate from below the integral The integral takes the smallest value if in each interval (s i , s i+1 ) there is at most one node w j , j = 1, 2, . . ., N.
It was shown above (19) that if there are no nodes w j , j = 1, 2, . . ., N in (s i , s i+1 ), then Next, consider the case when the interval (s i , s i+1 ) contains more than one node from {w j }, j = 1, 2, . . ., N. Without loss of generality, we assume N = 2m.
Repeating the above arguments yields From the inequalities ( 24) and ( 25), the next statement follows.
Theorem 1.Let ϕ(t) ∈ H 1 (1).For all possible quadrature formulas of the form (16) using N nodes, the following estimate holds where C 1 , C 2 are constants independent of N.
Making the proof more difficult yields the following statement.

Quadrature Formulas
Let us construct quadrature formulas for evaluating integrals of (15).We start by considering singular integrals with the Hilbert kernel: where f ∈ W r ([0, 2π], M), M is an integer.First, we consider the integral The function f (s) is approximated by the interpolation polynomial Above, we used ( [32], p.36) Next, consider the integral As above, the function f (s) is approximated by the interpolation polynomial f n (s).
Obviously, for m > n, Now, we study error estimates for constructed quadrature formulas.It is enough to consider the quadrature Formula (29).It is easy to see that the error of ( 29) is estimated by the inequality Here, ψ n (s) = f (s) − f n (s).Evaluate each term separately, where E n ( f ) is the best approximation in the uniform metric for the function f by nth-order trigonometric polynomials The following statement is well known.
Setting β = 1 ln n , we have From the estimates I 1 and I 2 , we have and, therefore, on the function class The final estimate is valid for any m ≥ 1.Now, we consider the following quadrature formula for H f evaluation.We approximate the function f (s) by the polygon f N (s), constructed on the nodes t k = 2kπ/N, k = 0, 1, . . ., N.
The integral H f will be evaluated using the quadrature formula The error of ( 32) is estimated by Consider two cases: (1) 2m ≤ N, (2) N < 2m.
Start with the first one.Let be s ∈ Estimate the integral Let s ∈ ∆ j , j = l.Estimate the integral Represent the previous integral as Estimate each of the integrals J 21 , J 22 separately.Obviously, Estimate the integral J 21 .We have Estimate J 211 , j 212 .Obviously, From inequalities (36)-(40), it follows From inequalities ( 33)-(41), it follows that for 2m < N the inequality holds The inequality is proved in a similar way.Consider the second case, Estimating (34), we again consider two cases, j = l − 1, l, l + 1 and j = l.
For the first one, after making some calculations, it can be shown that the largest error is yielded for functions of the form ψ N (s) = (min(s Then, Now, let j = l, s ∈ [t j , t j+1 ] and s ∈ [s v , s v+1 ].
We represent the integral J 1 as Obviously, where L = 2m/N .Making calculations similar to those above, we obtain the estimate where constants C 1 , C 2 are independent of N.
Theorem 5.Among all quadrature formulas of the form (16) using N nodes, the optimal with respect to order on the function class H 1 (1) turns out to be Formula (32).The estimate is valid: where constants C 1 , C 2 are independent of N.
Similarly, we can prove the following.Theorem 6.Among all quadrature formulas of the form (16) using N nodes, the optimal with respect to order on the function class H α (1) turns out to be Formula (32).The estimate holds: where constants C 1 , C 2 are independent of N.
Using the Hilbert transformation, we obtain To evaluate the integral C f , we use the quadrature formula where P n is a projection operator onto a set of interpolating trigonometric polynomials on nodes s k = 2πk/(2n + 1), k = 0, 1, . . ., 2n.
The error of ( 46) is estimated by where ψ n (t) = f (t) − P n [ f ](t).
The estimates I 1 and I 2 have been obtained above (see (31)).
The estimates hold: The inequalities (31), ( 48) and (49) yield the estimate Transform the integrals It was shown above that Thus, the following quadrature formula is valid: Estimates ( 48)-(50) hold: where E n ( f ) is the best uniform approximation of the function f by trigonometric polynomials of order n.
Let us take a look at an illustration of the quadrature formulas we have discussed.Consider an integral where m is integer.Let us apply the quadrature Formula (29) for evaluation of the integral: where f (s) = s(s−2π)(s−π) 12 and s k = 2kπ 2n+1 for k = 0, 1, . . ., 2n.We present the results of evaluation of the integral by series summation and by quadrature formula in Figure 1.We observe rapid convergence of the quadrature formula to the exact value of the integral.We also show that the amplitude of the oscillations is determined by the function f (s), as is suggested by Equation (29).

Approximate Evaluation of Hypersingular Integrals with Rapidly Oscillating Functions
In this section, we study approximate methods for evaluating hypersingular integrals of the form Here, γ = {z : |z| = 1}− is a unit circle centered at the origin in the complex plane, m is a natural number.To obtain a lower bound estimate for the error of the quadrature formula, we use the Hilbert transformation from the integral (56) to the hypersingular integral with a Hilbert kernel.We change the variables in (56): τ = e iσ , t = e is , σ ∈ [0, 2π].Now, we have f (e iσ )(cos mσ + i sin mσ)e iσ dσ (e iσ − e is ) p .
Converting the fraction yields Thus, to estimate from below the error for evaluation integrals of the form (56) by quadrature formulas constructed on N nodes, it is enough to study the integrals of the form on the function class W r ([0, 2π], 1), r ≥ p.
When estimating the error of the quadrature formula from below, two cases should be considered: (1) p is an even natural number; (2) p is an odd natural number.
We introduce the function The constant A is chosen such that ϕ * ∈ W r ([0, 2π], 1).
To each node t k , k = 0, 1, . . ., N − 1, we assign the function When constructing quadrature formulas for hypersingular integral evaluation, we will use Definition 7, which allows us to construct methods for hypersingular integral evaluation based on well-known methods for evaluating singular integrals.
From Definition 7, it follows that   (n + m) p−1 n r+α ln 2 n.

Conclusions
We studied approximate methods for evaluating Cauchy and Hilbert singular and hypersingular integrals with rapidly oscillating kernels.In the case of periodic integrable functions, lower and upper bound quadrature formula estimates have been obtained.Optimals with respect to order quadrature formulas for certain classes of functions have been constructed.We developed a method for constructing and estimating quadrature formulas for hypersingular integrals, based on similar results for singular integrals.
Finally, we point out a few key points of our study presented in this paper: (1) We introduced a method to estimate below quadrature formulas for evaluating singular and hypersingular integrals with rapidly oscillating kernels (in this paper, a method to obtain lower bound estimates by functional ζ N [Ψ] in the class of functions Ψ).Moreover, these estimates can be obtained from any set of N nodes located in the range of integration and N values of integrand function.
The method can be extended to singular and hypersingular integrals defined on other varieties, to polysingular and polyhypersingular integrals and to many dimensional singular and hypersingular integrals.The existence of lower bound estimates of functional ζ N [Ψ] allows us to construct an optimal with respect to order (to accuracy) passive algorithms for evaluating corresponding integrals in the classes of functions Ψ.

Definition 6 .
The Cauchy-Hadamard principal sense of the following integral b a ϕ(τ) dτ (τ − c) p , a < c < b is defined as the limit of the expression b a