Regularization of the Cauchy problem for the Helmholtz equation by using Meyer wavelet

In this paper, we investigate a Cauchy problem associated with Helmholtz-type equation in an infinite strip. This is a classical severely ill-posed problem, i.e., the solution (if it exists) does not depend continuously on the data (or Cauchy data), a small perturbation in the data can cause a dramatically large error in the solution for 0

[1]  Y. Y. Belov,et al.  Inverse Problems for Partial Differential Equations , 2002 .

[2]  Derek B. Ingham,et al.  Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation , 2004 .

[3]  Hans-Jürgen Reinhardt,et al.  Stability and Regularization of a Discrete Approximation to the Cauchy Problem for Laplace's Equation , 1999 .

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

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

[6]  Y. Meyer Wavelets and Operators , 1993 .

[7]  Masahiro Yamamoto,et al.  Unique continuation on a line for harmonic functions , 1998 .

[8]  Ting Wei,et al.  Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation , 2008 .

[9]  Xiang-Tuan Xiong,et al.  Fourth-order modified method for the Cauchy problem for the Laplace equation , 2006 .

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

[11]  Daniel Lesnic,et al.  The Cauchy problem for Laplace’s equation via the conjugate gradient method , 2000 .

[12]  C. Fu,et al.  Wavelets and regularization of the Cauchy problem for the Laplace equation , 2008 .

[13]  Ting Wei,et al.  Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation , 2001 .

[14]  L. Marin An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation , 2010 .

[15]  Jianxin Zhu,et al.  An operator marching method for inverse problems in range-dependent waveguides , 2008 .

[16]  Chu-Li Fu,et al.  The Fourier regularization for solving the Cauchy problem for the Helmholtz equation , 2009 .

[17]  Derek B. Ingham,et al.  BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method , 2004 .

[18]  S Subramaniam,et al.  Computation of molecular electrostatics with boundary element methods. , 1997, Biophysical journal.

[19]  U. Tautenhahn,et al.  Optimal stable approximations for the sideways heat equation , 1997 .

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

[21]  Liviu Marin,et al.  A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations , 2005, Appl. Math. Comput..

[22]  Xiang-Tuan Xiong,et al.  Two approximate methods of a Cauchy problem for the Helmholtz equation , 2007 .

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

[24]  Hao Cheng,et al.  A regularization method for solving the Cauchy problem for the Helmholtz equation , 2011 .

[25]  Bangti Jin,et al.  The plane wave method for inverse problems associated with Helmholtz-type equations , 2008 .

[26]  Teresa Regińska,et al.  Approximate solution of a Cauchy problem for the Helmholtz equation , 2006 .

[27]  Ting Wei,et al.  Modified regularization method for the Cauchy problem of the Helmholtz equation , 2009 .

[28]  H. Reinhardt,et al.  Regularization of a non-characteristic Cauchy problem for a parabolic equation , 1995 .

[29]  I. Daubechies Ten Lectures on Wavelets , 1992 .

[30]  T. Wei,et al.  Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation , 2009 .

[31]  Bangti Jin,et al.  Boundary knot method for some inverse problems associated with the Helmholtz equation , 2005 .

[32]  Xiang-Tuan Xiong,et al.  Central difference regularization method for the Cauchy problem of the Laplace's equation , 2006, Appl. Math. Comput..

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