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

We study singular integral equations of convolution type with cosecant kernels and periodic coefficients in class L2[−π, π]. 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.


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][2][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  2 [−, ] (see [8] for the definition of  2 [−, ]).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  2 [−, ].Therefore, this paper generalizes some results for [1][2][3][4][5].
The following lemma plays an important role, and it is proposed firstly in this paper.
(1) Since by Definition 5, we have It follows from the extended Residue theorem (see [9]) that This implies The other equality can be proved similarly.
where  is the class of Hölder continuous function.
Proof.By Definition 6, we know that where The proof of Lemma 10 is complete.
Lemma 11 is an obvious fact.

Problem Presentation and Methods of Solution
Consider the following singular integral equation of convolution type with periodicity and cosecant kernels: where The functions   () ( = 1, 2) are periodic coefficients;  is a positive integer.  (),   (), Φ(), Φ(), and () are periodic functions with period 2.Once Φ() is obtained in [−, ], we can make the periodic extension of Φ() with period 2 and then obtain the solution Φ * () in R. 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][11][12][13][14]).Hence, the study of ( 11) is meaningful not only in application but also in the method of solution.
(31) Therefore, (28) can be written as the following discrete jump problem: Thus, we should only study (32) in place of (11).Since Once   is obtained, then   can be given by (21).To solve (32), we first need to analyze the structure of 1 +   .Assume that 1 +   can be factorized; we can choose   such that where  < |  | <  −1 ;  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 ln(1 +   ) such that ln(1 +  k ) ∈  2 .Applying Fourier transform to both sides of (35), we can obtain where () = , () = ,  = {  } +∞ =−∞ , and  = {  } +∞ =−∞ .
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  1 () =  1 sin (/2),  2 () =  1 cos (/2), 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.
This case is simple, and we do not discuss it here.

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  2 [−, ].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  2 [−, ].Indeed, this class of equations can be also solved in Clifford analysis, which is similar to that in [15][16][17][18].Further discussion is omitted here.

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  2 [−, ].