Singular Integral Equations of Convolution Type with Cosecant Kernels and Periodic Coefficients

Abstract

We study singular integral equations of convolution type with cosecant kernels and periodic coefficients in class . Such equations are transformed into a discrete jump problem or a discrete system of linear algebraic equations by using discrete Fourier transform. The conditions of Noethericity and the explicit solutions are obtained by means of the theory of classical boundary value problem and of the Fourier analysis theory. This paper will be of great significance for the study of improving and developing complex analysis, integral equations, and boundary value problems.

1. Introduction

It is well-known that the boundary value problems for analytic functions have been widely used in many fields, such as engineering mechanics, physics, engineering technology, and fracture mechanics. Various types of boundary value problems for analytic functions and singular integral equations have been deeply studied and widely applied to practical problem (see [1–3]). In the theory of integral equations, the convolution type integral equations and singular integral equations are two important classes of equations, which had been studied by many mathematical workers and there were already rather complete theoretical systems (see [4, 5]). These theories have been widely used in practical applications, such as engineering mechanics, fracture mechanics, and elastic mechanics (see [6, 7]).

The main aim of this paper is to extend further the theory to singular integral equations of convolution type with periodic coefficients and cosecant kernel in class (see [8] for the definition of ). By using discrete Fourier transform, such equations are transformed into a discrete jump problem or a discrete system of equations. The explicit expressions of general solution and the conditions of solvability are obtained in class . Therefore, this paper generalizes some results for [1–5].

2. Definitions and Lemmas

In this section we present some definitions and lemmas.

Definition 1. Suppose is an infinite dimensional vector consisting of two-way infinite sequence; then all of which satisfy constitute a linear space; we denote it as

Remark 2. From Definition 1, we know that if , then for any constant . Moreover, if , , then , where . In this paper we mainly consider the case .

Definition 3. Let , be periodic functions with period , and , ; thenis called the convolution of and .

Remark 4. With regard to the convolution , we can extend the value of from to by Definition 3.

Definition 5. Let ; the transforms and defined byare called the discrete Fourier transform and the inverse transform, respectively. For simplification, in (2), we denote them as and , respectively.

Definition 6. Let be a number sequence and and be positive constants with . If holds for any integer , we say that belongs to class .

Definition 7. If , then we say .

The following lemma plays an important role, and it is proposed firstly in this paper.

Lemma 8. Let ,Then(1), ;(2), ,where , are singular integrals with cosecant kernels and

Proof. (1) Sinceby Definition 5, we haveIt follows from the extended Residue theorem (see [9]) thatThis impliesThe other equality can be proved similarly.
(2) By using Riesz’s theory (see [9]), we have , where is a constant. Therefore, it follows from that .
It is also obvious for the second conclusion in (2).

Lemma 9. If belongs to class , then , where is the class of Hölder continuous function.

Proof. By Definition 6, we know that converges; therefore, is convergent absolutely. For any two points , we havewhere ; thus .

Lemma 10. Let and ; then if and only if .

Proof. Since and , then is convergent absolutely and uniformly. It is easy to see that The proof of Lemma 10 is complete.

Lemma 11 is an obvious fact.

Lemma 11 (convolution theorem). If and , then and , where .

3. Problem Presentation and Methods of Solution

Consider the following singular integral equation of convolution type with periodicity and cosecant kernels:where is as in Lemma 8. are real constants with . The functions are periodic coefficients; is a positive integer. , , , , and are periodic functions with period . Once is obtained in , we can make the periodic extension of with period and then obtain the solution in . Therefore, in the following discussion we are restricted to .

Equation (11) is one important class of equations in the theory of integral equations, and it has important applications in physics, air dynamics, and electronic optical (e.g., see [10–14]). Hence, the study of (11) is meaningful not only in application but also in the method of solution.

Since , we write (11) aswhereand can be rewritten asLetthen,Applying to both sides of (16) and (17), respectively, by Lemma 8 we havewhereDenoteThen (18) can be reduced asSince , , then , , and or when . Therefore there exists () such that when for certain positive constant .

Consider first the case .

(1) When for , by (21) we know that must be fulfilled. Now take arbitrary constants.

(2) When , for , by (21) we have where

(3) When , for , by (21) we get where

(4) When , are given by (21) and this situation is similar to the situation .

Next we consider the case . Since for , then . By (21) we obtain thus where

Assume that where

Therefore, (28) can be written as the following discrete jump problem: Thus, we should only study (32) in place of (11). Since , , then , . Once is obtained, then can be given by (21). To solve (32), we first need to analyze the structure of . Assume that can be factorized; we can choose such that where ; is as the above. Provided that takes sufficiently small value, we can choose satisfying the above requirement (33). We now give an expression of , taking logarithms for both sides of (33) and denoting then, by (33) and (34) we obtain Note that we have taken a continuous branch of such that . Applying Fourier transform to both sides of (35), we can obtain where , , , and

From the above discussions, the following conditions of solvability follow and that is, where ; is the integer part of ). Therefore, From (38), we obtain and thus . Note (33) and denote Then (32) can be rewritten as

Since , and , we have and . By taking Fourier transform for both sides of (40), we have where , , , and

Since , we know that has finite number of zero points in .

Therefore the conditions of solvability , that is, must be augmented. In view of (41) we get

If (37) and (42) are satisfied, then by (43) we obtain Thus can be also obtained by (39).

Now we state our main result.

Theorem 12. Under (37) and (42), (11) has a solution. When , is given by (21). When , take arbitrary constants. When , , , are given by (23) and (24), respectively. When , , , are given by (25) and (26), respectively. But in (23) and (25), and are constants determined by (24) and (26), respectively. When is given by (21). Therefore, the solution of (11) is of the form and then

Proof. From the above discussion, we only need to prove that the function obtained by (45) belongs to . Obviously, (11), (13), and (21) are equivalent to each other. Since , then and It follows from (27) that is a bounded sequence, and is convergent. Thus, by Lemma 10, (11) has a unique solution in class .

According to Theorem 12, Definition 5, and Lemma 9, we obtain the following.

Theorem 13. In (11), if , , then (11) is solvable in class and all conditions of solvability and its solutions are similar to those in Theorem 12.

In order to illustrate that (11) has an explicit solution, we present an example and satisfy the above conditions (37) and (42). For example, suppose that then , , and (11) can be transformed into it is easy to prove that (47) satisfies conditions (37) and (42), and the solution of (47) was obtained in the literature [1]. Therefore, we can conclude that a solution set of (11) is not empty.

Remark 14. If , is a constant. This case is simple, and we do not discuss it here.

4. Results and Discussion

In this paper, we first proposed one class of singular integral equations of convolution type with cosecant kernels and periodic coefficients. By applying discrete Fourier transform and its properties, such equation can be transformed into a discrete jump problem depending on some parameter. Here, our method is different from the ones of the classical boundary value problem, and it is novel and simple. The exact solution, denoted by series, of (11) and the conditions of solvability are obtained in . We remark that our approach is also effective in other classes of equations, such as the equations of dual type with periodicity and cosecant kernel and Wiener-Hopf type equations. Thus, this paper generalizes the classical theory of boundary value problems and singular integral equations.

In this paper, we solved (11) in . Indeed, this class of equations can be also solved in Clifford analysis, which is similar to that in [15–18]. Further discussion is omitted here.

5. Conclusions

Equation (11) has important applications in practical problems, such as elastic mechanics, heat conduction, and electrostatics. Many problems, such as piezoelectric material, voltage magnetic materials, and functional gradient materials, can often attribute the problem to finding their solutions to (11). For the study of such equations, the present result is still rare due to lack of effective approaches. Our approach of solving the equations is novel, different from the ones in classical cases, and it is converted by using discrete Fourier transform into a discrete boundary value problem depending on some parameter; here we call it “a discrete jump problem.” The exact solutions of (11) and the conditions of solvability are obtained in class .

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

This work was supported by the Science and Technology Plan Project of Qufu Normal University (xkj 201606) and the author gratefully acknowledges this support.