The wave scattering problem by a crack F in R2 with impedance type boundary is considered. This problem models the diffraction of waves by thin two-sided cylindrical screens. A numerical method for solving the problem is developed. The solution of the problem is represented in the form of the combined angular potential and single-layer potential. The linear integral equations satisfied by the density functions are derived for general parameterized arcs. The weakly singular integrals and the Cauchy singular integral arising in these equations are computed using a highly accurate scheme with a truncation error analysis. The advantage of the scheme proposed in this paper is, in one hand, the fact that we do not need the analyticity property of the crack and we allow different complex valued surface impedances in both sides of the crack. In the other hand, we avoid the hyper-singular integrals. Numerical implementations showing the validity of the scheme are presented.
Consider the determination of Dirichlet-to-Neumann(D-to-N) map from the far-field pattern in inverse scattering problems,which is the key step in some recently developed inversion schemes such as probe method.Essentially,this problem is related to the reconstruction of the scattered wave from its far-field data.We firstly prove the well-known uniqueness result of the D-to-N map from the far-field pattern using a new scheme based on the mixed reciprocity principle.The advantage of this new proof scheme is that it provides an efficient algorithm for computing the D-to-N map,avoiding the numerical differentiation for the scattered wave.Then combining with the classical potential theory,a simple and feasible regularizing reconstruction scheme for the D-to-N map is proposed.Finally the stability estimate for the reconstruction with noisy input data is rigorously analyzed.
WANG HaiBing 1,2 & LIU JiJun 1 1 Department of Mathematics,Southeast University,Nanjing 210096,China
