Numerical solution of the Sturm–Liouville problem with local RBF-based differential quadrature collocation method

Abstract

In this article, we present a local RBF-based differential quadrature (LRBFDQ) collocation method for the Sturm–Liouville problem with Dirichlet, Neumann, mixed, periodic, and semi-periodic boundary conditions. Compared with the globally supported RBF (GSRBF) collocation method, this novel method approximates the derivatives by RBF interpolation using a small set of nodes in the neighbourhood of any collocation node. Less computational time is needed than the GSRBF collocation method. Compared with the GSRBF collocation method and the finite difference method (FDM), numerical results demonstrate the accuracy and easy implementation of the LRBFDQ collocation method.

Keywords:

• local RBF-based differential quadrature
• radial basis function
• globally supported radial basis function
• Sturm–Liouville problem
• collocation method
• multiquadric
• finite difference method
• meshless

2010 AMS Subject Classifications :

• 65L15
• 74S30

Acknowledgements

The author is grateful to the reviewers for their valuable comments and constructive suggestions which improved the article.