Reconstruction of the electric field of the Helmholtz equation in three dimensions

In this paper, the paper is devoted to study the Cauchy problem for the inhomogeneous Helmholtz equation that arises naturally in many physical applications related to the ra-diation field, vibration phenomena. Under specific assumptions, we prove the ill-posedness of the problem in a particular situation. Applying the truncation regularization method, we establish the regularized solution and show that it converges to the exact solution as the perturbation in given data tends to zero. Besides, the error estimate and regularization parameter are formulated. Our theoretical result is guaranteed by a numerical example