Singular Integral Equation Of Convolution Type And Boundary Value Theory

Singular integral equations of convolution type and boundary value theory have many broad applications in physics, elasticity, mechanical engineering, aerodynam-ics, electron optics and engineering technology. In recent years, the researches in this field have been going deep into the extremely difficult settings involving higher-dimensional, variable coefficient, hyper-singular circumstances. For these hot issues, this paper makes a systematic and profound study.The main results and innovations are as follows:(1) For a class of dual type singular integral equation solution with convolution, we prove that the existence of the solutions with exponential growth or decay. Such a solution is of its great significance in physics, radiation balance theory for its describing of behaviors with exponential growth or decay as time going. The approach of solving the equations is novel, and it is converted by integral transformations into a Riemann boundary value problem with a complex boundary on the belt-shaped domain.(2) For discrete convolution type equations involving harmonic singular opera-tors, we solved the equations with new method different from that in classical discrete convolution type equations. The point is that the Fourier transforms of kernel functions appearing in this equations have discontinuous points on the unit circle.(3) So far, the results of the boundary value problems for analytic functions have been mostly confined to only one unknown function case. The article proposes to study the problems of the case with several unknown functions. The method is innovate, different from the ones in classical cases.(4) For the study of singular integral equations with variable coefficients, the result is still rare due to lake of effective approaches. In this paper the local theory is applied to study the solvability of convolution type singular integral equation with variable coefficients involving the boundary value for analytic function.