The Local Radial Basis Function Collocation Method for Solving Two-Dimensional Inverse Cauchy Problems

Abstract

In this study, inverse Cauchy problems, which are governed by the Poisson equation, inhomogeneous Helmholtz equation, and inhomogeneous convection-diffusion-reaction equation, are analyzed by the local radial basis function collocation method (LRBFCM). In the inverse Cauchy problem, overspecified boundary conditions are given along part of the boundary and no boundary condition is imposed on the rest of the boundary. The inverse problems are generally very unstable and ill-posed, so the inverse Cauchy problem is very difficult to solve stably using any numerical scheme. The LRBFCM is one kind of domain-type meshless method and can get rid of mesh generation and numerical quadrature. In addition, the localization in LRBFCM can reduce the ill-conditioning problem and full matrix. Therefore, in this study the LRBFCM is adopted to analyze two-dimensional inverse Cauchy problems. Five numerical examples are provided to verify the proposed meshless scheme. In addition, the stability of the proposed scheme is validated by adding noise into boundary conditions.