New RBF Collocation Methods and Kernel RBF with Applications

Title: New RBF Collocation Methods and Kernel RBF with Applications

Abstract: This research article introduces new radial basis function (RBF) discretization schemes for partial differential equations. It also presents five types of kernel RBFs. The main goal is to develop efficient and stable RBF solutions. The article discusses symmetric boundary knot methods, direct symmetric boundary knot methods, and symmetric boundary particle methods. It also introduces the modified Kansa's method and the finite knot method. The least squares RBF collocation method is proposed to improve solution accuracy. The article concludes with a discussion on kernel RBFs and their application in numerical PDEs.

Main Research Question: How can we develop efficient and stable RBF solutions for partial differential equations?

Methodology: The article introduces new RBF collocation methods for partial differential equations. It also presents five types of kernel RBFs. The methods include symmetric boundary knot methods, direct symmetric boundary knot methods, and symmetric boundary particle methods. The modified Kansa's method and the finite knot method are also discussed.

Results: The article presents the derivation of the symmetric Hermite BKM and direct BKM. It also discusses the symmetric boundary particle method. The modified Kansa's method and the finite knot method are introduced to improve solution accuracy.

Implications: The new RBF collocation methods and kernel RBFs can be applied to various problems in physics, engineering, and other fields. They can provide more accurate and efficient solutions for partial differential equations. The least squares RBF collocation method can help avoid the Gibbs phenomenon.

Link to Article: https://arxiv.org/abs/0111063v1 Authors: arXiv ID: 0111063v1