Integral Equation Methods For Crack Problems With Mixed Boundary Conditions

The research of acoustic inverse scattering problem is a mainly research field in the study of mathematical physics inverse problem, it has extensive and practical applications in medical imaging, medical diagnosis,non-destructive testing, geophysical exploration, oil exploration, marine exploration, remote sensing, radar and sonar, and many other fields. Therefore, the acoustic scattering theory and its calculation methods have been widely studied. This paper mainly studies inverse scattering problems, which are about the reconstruction of two kinds of crack scatterers with different mixed boundary conditions by the linear sampling method. In this article, the main research content is as follows:In the second chapter, we present some mathematical models and the basic theories on acoustic scattering problem.In the third chapter, the existence and uniqueness of solutions of two kinds of crack scatterers with different mixed boundary conditions are proved via the integral equation method,and the integral equation method is established to solve the scattering problem. Then the singularities of the first derivative of the solution to the hypersingular integral equation are removed by using the cosine transformation. At last, numerical examples are given by using Matlab software;In the fourth chapter, the relevant theories of the linear sampling method are firstly put forward. Then two kinds of crack scatterers with different mixed boundary conditions are preliminarily reconstructed via the linear sampling method, and some cases of numerical simulation are presented;In the fifth chapter, we summarize the mainly studied content, significance and the shortage of this paper, and forecast the future research.