Numerical Method Of Cauchy Problem For Helmholtz Equation

the Helmholtz equation ariscs in many physical applications especially in wave and vibration phenomena,such as the acoustic cavity problem,the scattering of a wave,neumann problem,dirichlet problem or mixed boundary problems have been studied,but sometimes we only have a part of the boundary,so we come to a inverse problem.But the Cauchy problem s of the two equations are ill-posed.This means the solution does't depend on the boundary data contin-uously.So far,several methods have been applied to solve this problem,such as boundary element method,method of fundamental solutions and so on.letΩis a simply connected and bounded domain inR2,and the boundary(?)Ωis sufficiently regular,and r is an part of aQ.Consider the following problem△u(x,y)＋k2u(x,y)=0, (x,y)∈Ω, u(x,y)=f(x,y), (x,y)∈r, where f∈H(?)(r),g∈H(?)(r),n is out vecter of (?)Ω,k>0,in the following,we assume一k2is not an eigenvalue of Laplace operatorSuppose the cauchy problem(1)一(3)have a solution in H2(Ω),then for (?)∈H1(Ω),we know u satisfy the following:For any(?)q∈L2(r),letvq∈H1(Ω)is a weak solution of the following problem:△v(x,y)＋k2v(x,y)=0, (x,y)∈Ω, thenvqsatisfyDenote H={v∈H1(Ω)|v.For(?)q∈L2(r)satisfy 1.8}. ForVv∈H.In(4)take(?)=vbut in(8)take(?)=v,vq=v(4)minus(8),we havethe following equationProm this idea,we take the following problem:letΩis a Unit disk inR2,and the bondaryaQis Unit circle,and r is an part of aQ.Consider the following problemwhere f∈H(?)((?)),g∈H(?)(r),n is out vecter of (?)Ω,k>0,in the following,we assumc -k2is not an eigenvalue of Laplace operator.Suppose the cauchy problem(1)一(3)have a solution in H2(Ω),then for (?)∈H'(Q),we know u satisfy the following:letvn∈H1(Ω)is a weak solution of the following problem:△u(x,y)＋k2v(x,y)=0, (x,y)∈Ω, Apply the Green formula,we have the boundary integral equation: Put u=(?) in the above boundary integral equation and then we get a system of linear equations. Then approach to solve this system of linear equations.