The Fourier regularization for solving the Cauchy problem for the Helmholtz equation

Authors:
Chu-Li Fu;Xiao-Li Feng;Zhi Qian
Affiliations:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China;School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China;School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China and Department of Mathematics, Nanjing University, Nanjing 210093, PR China
Venue:
Applied Numerical Mathematics
Year:
2009

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Abstract

The Cauchy problem for the Helmholtz equation in an infinite ''strip'' is considered. The Cauchy data are at the boundary x=0 given in an approximate manner and the solution is sought in the region {(x,y)|0=1}. This problem is severely ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. In this paper we use the Fourier regularization method to solve the problem. The method is independent of the interval length and wave number. Some sharp error estimates between the exact solution and its regularization approximation are given and numerical examples show that the method works effectively.