Generalized finite difference method for solving two-dimensional inverse Cauchy problems

Abstract

In this paper, a meshless numerical scheme is adopted for solving two-dimensional inverse Cauchy problems which are governed by second-order linear partial differential equations. In Cauchy problems, over-specified boundary conditions are imposed on portions of the boundary while on parts of boundary no boundary conditions are imposed. The application of conventional numerical methods to Cauchy problems yields highly ill-conditioned matrices. Hence, small noise added in the boundary conditions will tremendously enlarge the computational errors. The generalized finite difference method (GFDM), which is a newly developed domain-type meshless method, is adopted to solve in a stable manner the two-dimensional Cauchy problems. The GFDM can overcome time-consuming mesh generation and numerical quadrature. Besides, Cauchy problems can be solved stably and accurately by the GFDM. We present three numerical examples to validate the accuracy and the simplicity of the meshless scheme. In addition, different levels of noise are added into the boundary conditions to verify the stability of the proposed method.