New Radial Basis Function (RBF) Numerical Schemes for Partial Differential Equations

Title: New Radial Basis Function (RBF) Numerical Schemes for Partial Differential Equations

Research Question: How can we improve the accuracy and efficiency of radial basis function (RBF) numerical schemes for solving partial differential equations?

Methodology: The authors propose several new RBF-based numerical schemes, including indirect and direct symmetric boundary knot methods (BKM), boundary particle methods (BPM), modified Kansa method (MKM), and finite knot method (FKM). These methods are designed to be symmetric, meshless, and integration-free, making them efficient for solving complex problems.

Results: The authors demonstrate that their new schemes, particularly the direct BKM and MKM, produce accurate and efficient solutions for a variety of problems. They also show that these methods reduce calculation errors at nodes adjacent to boundaries, further improving their performance.

Implications: The new RBF numerical schemes presented in this paper have several important implications. First, they offer a more accurate and efficient way to solve partial differential equations using RBF methods. Second, they provide a practical alternative to traditional mesh-based methods, as they do not require the time-consuming process of generating and managing meshes. Finally, the new schemes can be applied to a wide range of problems in fields such as engineering, physics, and mathematics, making them a valuable tool for researchers and practitioners alike.

Link to Article: https://arxiv.org/abs/0104018v1 Authors: arXiv ID: 0104018v1