Quasi-reversibility and truncation methods to solve a Cauchy problem for the modified Helmholtz equation

In this paper, the Cauchy problem for the modified Helmholtz equation in a rectangular domain is investigated. We use a quasi-reversibility method and a truncation method to solve it and present convergence estimates under two different a priori boundedness assumptions for the exact solution. The numerical results show that our proposed numerical methods work effectively.

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

[2]  Pham Hoang Quan,et al.  Determination of a two-dimensional heat source: uniqueness, regularization and error estimate , 2006 .

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

[4]  Laurent Bourgeois,et al.  A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation , 2005 .

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

[6]  V. Isakov.  Appendix -- Function Spaces , 2017 .

[7]  Michael V. Klibanov,et al.  Carleman estimates for coefficient inverse problems and numerical applications , 2004 .

[8]  Xin Li,et al.  On solving boundary value problems of modified Helmholtz equations by plane wave functions , 2006 .

[9]  Toshihisa Honma,et al.  An analysis of axisymmetric modified Helmholtz equation by using boundary element method , 1990 .

[10]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[11]  Chu-Li Fu,et al.  Two regularization methods for a Cauchy problem for the Laplace equation , 2008 .

[12]  H. Berendsen,et al.  The electric potential of a macromolecule in a solvent: A fundamental approach , 1991 .

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

[14]  Dang Duc Trong,et al.  Nonhomogeneous heat equation: Identification and regularization for the inhomogeneous term , 2005 .

[15]  Fredrik Berntsson,et al.  Wavelet and Fourier Methods for Solving the Sideways Heat Equation , 1999, SIAM J. Sci. Comput..

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

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

[18]  L. Eldén,et al.  Solving an inverse heat conduction problem by a `method of lines' , 1997 .

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

[20]  Lars Eldén,et al.  Approximations for a Cauchy problem for the heat equation , 1987 .

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

[22]  Jingfang Huang,et al.  An adaptive fast solver for the modified Helmholtz equation in two dimensions , 2006 .

[23]  Jacques-Louis Lions,et al.  The method of quasi-reversibility : applications to partial differential equations , 1969 .

[24]  Charles F. Weber,et al.  Analysis and solution of the ill-posed inverse heat conduction problem , 1981 .

[25]  Y. C. Hon,et al.  Solving Cauchy Problems Of Elliptic EquationsBy The Method Of Fundamental Solutions , 2005 .

[26]  Michael V. Klibanov,et al.  A computational quasi-reversiblility method for Cauchy problems for Laplace's equation , 1991 .

[27]  J. Hadamard,et al.  Lectures on Cauchy's Problem in Linear Partial Differential Equations , 1924 .