The Research Of Helmholtz Equation With Mixed Boundary Conditions On Annular Region

In this paper,we discuss the well-posed problem and numerical solutions of Helmholtz equation with Neumann-Robin conditions on annular region.LetD,Ω, (?) R~m (m=2,3) be a bounded region satisfying (?),we assume that both (?)and(?) are smooth enough as in C~2 class.Consider the following mixed boundary value problem for the Helmholtz equationWhere k>0is wave,the constantσ＞0 is boundary impedance coefficient.Form Green's formula and Holmgren uniqueness theorem,we can get the uniqueness of above problem:If we know the Cauchy data of (?)and(?) ,we can get the existence of the problem,for this ,refer to [3],by single- and double layer potential theories,we first reformulate the expression of the solution as a 2×2 system of boundary integral equational of the second kind about (?) aud(?),if the solution of boundary integral equation exist,the (*)exist a solution.We can get the stability from the expression of the solution and the properities of the operator of the second kind.We discrete the second kind boundary integral equation into a finite dimension system, and we can get the numerical solution of the system .We know the convergenceof the numerical solution by the standard argument from [5].