An accurate meshless generalized finite difference framework with fictitious nodes for high-order boundary value problems

Abstract

An accurate meshless numerical framework is presented for solving two-dimensional boundary value problems governed by high-order partial differential equations. The proposed method combines the generalized finite difference method (GFDM) with a fictitious-nodes technique to efficiently handle high-order derivatives and multiple boundary conditions. Owing to its Taylor-series-based formulation, the GFDM allows partial derivatives of arbitrary order to be directly approximated using scattered nodes without mesh generation or numerical integration. However, high-order boundary value problems typically lead to overdetermined algebraic systems. This difficulty is resolved by introducing fictitious nodes outside the computational domain, which transforms the overdetermined system into a square system. The method is applied to polyharmonic equations of orders ranging from 6th to 12th on simply connected and multiply connected domains with complex geometries. Four benchmark problems with analytical solutions are examined to assess the accuracy and robustness of the proposed method. Numerical results show excellent agreement with analytical solutions and demonstrate that high accuracy can be achieved with relatively few computational nodes when sufficiently high-order Taylor expansions are employed. The proposed approach provides a robust and flexible meshless tool for high-order boundary value problems in engineering analysis.