Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green's formulation, the problem can be transformed into a moment problem. Then we propose a numerical algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimate and convergence analysis have also been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.

[1]  Dimitri E. Beskos,et al.  Boundary Element Methods in Dynamic Analysis , 1987 .

[2]  Y. Hon,et al.  Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators , 2007 .

[3]  C. DeWitt-Morette,et al.  Mathematical Analysis and Numerical Methods for Science and Technology , 1990 .

[4]  R. Gorenflo,et al.  A Multi-Dimensional Hausdorff Moment Problem: Regularization by Finite Moments , 1999 .

[5]  W. S. Hall,et al.  A boundary element investigation of irregular frequencies in electromagnetic scattering , 1995 .

[6]  Derek B. Ingham,et al.  An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation , 2003 .

[7]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[8]  Dimitri E. Beskos,et al.  Boundary Element Methods in Dynamic Analysis: Part II (1986-1996) , 1997 .

[9]  G. Talenti.  Recovering a function from a finite number of moments , 1987 .

[10]  Bangti Jin,et al.  A meshless method for some inverse problems associated with the Helmholtz equation , 2006 .

[11]  Daniel Lesnic,et al.  The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations , 2005 .

[12]  Michael Slavutin,et al.  Boundary infinite elements for the Helmholtz equation in exterior domains , 1998 .

[13]  F. C. Wong,et al.  DUAL FORMULATION OF MULTIPLE RECIPROCITY METHOD FOR THE ACOUSTIC MODE OF A CAVITY WITH A THIN PARTITION , 1998 .

[14]  Derek B. Ingham,et al.  Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations , 2003 .

[15]  G. Viano.  Solution of the Hausdorff moment problem by the use of Pollaczek polynomials , 1991 .

[16]  Bangti Jin,et al.  Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation , 2005 .