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Sarra, S.A. and Kansa, E.J. (2009) Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations. Advances in Computational Mechanics, 2, 1-206.

has been cited by the following article:

• TITLE: A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

  AUTHORS: Scott A. Sarra, Derek Musgrave, Marcus Stone, Joseph I. Powell

  KEYWORDS: Numerical Partial Differential Equations, Boundary Value Problems, Radial Basis Function Methods, Ghost Points, Variable Shape Parameter, Least Squares

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.12 No.7, July 24, 2024

  ABSTRACT: Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.