Spectral Galerkin Method For Cauchy Problem Of A Helmholtz Equation

In this paper, we consider the Cauchy problem for Helmholtz equation, whichis defned in an rectangle domain0≤x≤π,0≤y≤1. When Cauchy data isgiven for y=0, the solution for0<y≤1is sought.First, we show the ill-posedness of the problem by obtaining the analytic solutionvia separation of variables. At the same time, a conditional stability estimate isproved. We use spectral Galerkin method to get the stable regularization solutionfor the problem. By an a-priori bound and the appropriate regularization parameter,we get Ho¨lder type convergence estimates for0<y <1. However, for the casey=1, we get a logarithmic type stability estimates by introducing a stronger a-priori assumption. Finally, numerical experiments show that our proposed methodis feasible and efective.