Application of the Generalized Finite-Difference Method to Inverse Biharmonic Boundary-Value Problems

Abstract

In this article, the generalized finite-difference method (GFDM), one kind of domain-type meshless method, is adopted for analyzing inverse biharmonic boundary-value problems. In inverse problems governed by fourth-order partial differential equations, overspecified boundary conditions are imposed at part of the boundary, and, on the other hand, part of the boundary segment lacks enough boundary conditions. The ill-conditioning problems will appear when conventional numerical simulations are used for solving the inverse problems. Thus, small perturbations added in the boundary conditions will result in problems of instability and large numerical errors. In this article, we adopt the GFDM to stably and accurately analyze the inverse problems governed by fourth-order partial differential equations. The GFDM is truly free from time-consuming mesh generation and numerical quadrature. Six numerical examples are provided to validate the accuracy and the simplicity of the GFDM. Furthermore, different levels of noise are added into the boundary conditions to verify the satisfying stability of the GFDM.