On The Solution To The Helmholtz Equation With The Third Mixed Boundary Value Problem

In this paper, we consider the third mixed boundary value problem of Helmholtz equation in annular region, that is to find u∈C2(D1\D2%)∩\C(D1\D2) satisfying the following problem: where k is a positive wave number,λ1λ2(λ1<0,λ2> 0) is the boundary impedance coefficient. D1 and D2 are boundary close regions in Rs(s=2,3), we assume both boundaries (?)D1 and (?)D2 are belonging to C2, where v is the unit normal vector direct into the exterior of (?)D1 and (?)D2.By applying potential theory, the problem (*) can turn into an equivalent boundary integral equation. After carefully calculated, the existence and unique-ness of solution to the boundary integral equation is established employing Fredholm theorem, so we can solve the existence and uniqueness of the solution to problem (*). In order to obtain numerical solution of the boundary integral equation, it is necessary for us to discrete the boundary integral operators for this equation, finally the discrete form of the boundary integral equation can be reformulated. Thus we can get numerical solution of the original problem (*) with computer.