A Meshless Method for Solving Nonhomogeneous Cauchy Problems

In this paper the method of fundamental solutions (MFS) and the method of particular solution (MPS) are combined as a one-stage approach to solve the Cauchy problem for Poisson's equation. The main idea is to approximate the solution of Poisson's equation using a linear combination of fundamental solutions and radial basis functions. As a result, we provide a direct and effective meshless method for solving inverse problems with inhomogeneous terms. Numerical results in 2D and 3D show that our proposed method is effective for Cauchy problems.

[1]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[2]  D. L. Young,et al.  The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation , 2008, Comput. Math. Appl..

[3]  E. B. Saff,et al.  On separation of minimal Riesz energy points on spheres in Euclidean spaces , 2005 .

[4]  A robust data completion method for 2D Laplacian Cauchy problems , 2008 .

[5]  Carlos Alberto Brebbia,et al.  BOUNDARY ELEMENT METHODS IN ENGINEERING , 1982 .

[6]  Jin Cheng,et al.  Uniqueness and stability for an inverse problem of determining a part of boundary , 1998 .

[7]  Jin Cheng,et al.  Stability for an inverse boundary problem of determining a part of a boundary , 1999 .

[8]  D. L. Young,et al.  Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points , 2008, J. Comput. Phys..

[9]  Palghat A. Ramachandran,et al.  Osculatory interpolation in the method of fundamental solution for nonlinear Poisson problems , 2001 .

[10]  Y. Hon,et al.  A Meshless Computational Method For Solving Inverse Heat Conduction Problems , 2002 .

[11]  Xiang-Tuan Xiong,et al.  A regularization method for a Cauchy problem of the Helmholtz equation , 2010, J. Comput. Appl. Math..

[12]  Uno Hämarik,et al.  The use of monotonicity for choosing the regularization parameter in ill-posed problems , 1999 .

[13]  Y. Hon,et al.  A fundamental solution method for inverse heat conduction problem , 2004 .

[14]  A. Cheng,et al.  Heritage and early history of the boundary element method , 2005 .

[15]  M. Golberg.  Boundary integral methods : numerical and mathematical aspects , 1999 .

[16]  L. Marin.  The Method of Fundamental Solutions for Inverse Problems Associated with the Steady-State Heat Conduction in the Presence of Sources , 2008 .

[17]  Y. Hon,et al.  The method of fundamental solution for solving multidimensional inverse heat conduction problems , 2005 .

[18]  R. Mathon,et al.  The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions , 1977 .

[19]  A. Cheng,et al.  Direct solution of ill‐posed boundary value problems by radial basis function collocation method , 2005 .

[20]  Lloyd N. Trefethen,et al.  Reviving the Method of Particular Solutions , 2005, SIAM Rev..

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

[22]  L. Marin.  Stable MFS Solution to Singular Direct and Inverse Problems Associated with the Laplace Equation Subjected to Noisy Data , 2008 .

[23]  Q. Qin,et al.  A meshless method for generalized linear or nonlinear Poisson-type problems , 2006 .

[24]  T. Wei,et al.  Convergence analysis for the Cauchy problem of Laplace’s equation by a regularized method of fundamental solutions , 2010, Adv. Comput. Math..

[25]  D. L. Clements,et al.  Fundamental solutions for second order linear elliptic partial differential equations , 1998 .

[26]  Per Christian Hansen,et al.  Analysis of Discrete Ill-Posed Problems by Means of the L-Curve , 1992, SIAM Rev..

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

[28]  Rainer Kress,et al.  On the simulation source technique for exterior problems in acoustics , 1986 .

[29]  Per Christian Hansen,et al.  REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems , 1994, Numerical Algorithms.

[30]  Graeme Fairweather,et al.  The method of fundamental solutions for elliptic boundary value problems , 1998, Adv. Comput. Math..

[31]  V. D. Kupradze,et al.  The method of functional equations for the approximate solution of certain boundary value problems , 1964 .

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