A Spectral Regularization Method for a Cauchy Problem of the Modified Helmholtz Equation

The Cauchy problem for Helmholtz equation arises from inverse scattering problems. Specific backgrounds can be seen in the existing literature; we can refer to 1–6 and so forth. A number of numerical methods for stabilizing this problem are developed. Several boundary element methods combined with iterative, conjugate gradient, Tikhonov regularization, and singular value decomposition methods are compared in 6 . Cauchy problem for elliptic equations is well known to be severely ill-posed 7 ; that is, the solution does not depend continuously on the boundary data, and small errors in the boundary data can amplify the numerical solution infinitely; hence it is impossible to solve Cauchy problem of Helmholtz equation by using classical numerical methods and it requires special techniques, for example, regularization methods. Although theoretical concepts and computational implementation related to the Cauchy problem of Helmholtz equation have been discussed by many authors 8–11 , there are many open problems deserved to be solved. For example, many authors have considered the following Cauchy problem of the Helmholtz equation 8–11 :