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

Pingrun Li

doi:10.1155/2017/6148393

Abstract

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.

Highlights

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
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 L2[−π, π]
By applying discrete Fourier transform and its properties, such equation can be transformed into a discrete jump problem depending on some parameter

Summary

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. 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]). This paper generalizes some results for [1,2,3,4,5]

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