In this paper, we present a meshless hybrid method combining the Generalized Finite Difference (GFD) and Finite Difference based Radial Basis Function (RBF-FD) approaches to solve non-homogeneous partial differential equations (PDEs) involving both lower and higher order derivatives. The proposed method eliminates the need for mesh generation by leveraging the strengths of both GFD and RBF-FD techniques. The GFD method is robust and stable, effectively handling ill-conditioned systems, while the RBF-FD method excels in extending to higher-order derivatives and higher-dimensional problems. Despite their individual advantages, each method has its limitations. To address these, we developed a hybrid GFD-RBF approach that combines their strengths. Specifically, the GFD method is employed to approximate lower order terms (convective terms), and the RBF method is used for higher order terms (diffusive terms). The performance of the proposed hybrid method is tested on both linear and nonlinear PDEs, considering uniform and non-uniform distributions of nodes within the domain. This approach demonstrates the versatility and effectiveness of the hybrid GFD-RBF method in solving second and higher order convection-diffusion problems.

翻译：本文提出了一种结合广义有限差分(GFD)方法与基于有限差分的径向基函数(RBF-FD)方法的无网格混合方法，用于求解包含低阶和高阶导数的非齐次偏微分方程(PDEs)。该方法通过整合GFD和RBF-FD技术的优势，消除了网格生成的需求。GFD方法具有鲁棒性和稳定性，能有效处理病态系统；而RBF-FD方法在扩展到高阶导数和更高维问题方面表现优异。尽管各自具有优势，但两种方法均存在局限性。为此，我们开发了一种结合两者优势的混合GFD-RBF方法。具体而言，采用GFD方法逼近低阶项（对流项），而RBF方法用于处理高阶项（扩散项）。通过在均匀与非均匀节点分布条件下对线性和非线性PDEs进行测试，验证了所提混合方法的性能。该研究证明了混合GFD-RBF方法在求解二阶及更高阶对流扩散问题中的通用性和有效性。