Indirect RBFN method with thin plate splines for numerical solution of differential equations

Abstract

This paper reports a mesh-free Indirect Radial Basis Function Network method (IRBFN) using Thin Plate Splines (TPSs) for numerical solution of Differential Equations (DEs) in rectangular and curvilinear coordinates. The adjustable parameters required by the method are the number of centres, their positions and possibly the order of the TPS. The first and second order TPSs which are widely applied in numerical schemes for numerical solution of DEs are employed in this study. The advantage of the TPS over the multiquadric basis function is that the former, with a given order, does not contain the adjustable shape parameter (i.e. the RBF's width) and hence TPS-based RBFN methods require less parametric study. The direct TPS-RBFN method is also considered in some cases for the purpose of comparison with the indirect TPS-RBFN method. The TPS-IRBFN method is verified successfully with a series of problems including linear elliptic PDEs, nonlinear elliptic PDEs, parabolic PDEs and Navier-Stokes equations in rectangular and curvilinear coordinates. Numerical results obtained show that the method achieves the norm of the relative error of the solution of O(10 -6) for the case of 1D second order DEs using a density of 51, of O(10 -7) for the case of 2D elliptic PDEs using a density of 20 × 20 and a Reynolds number Re = 200 for the case of Jeffery-Hamel flow with a density of 43 × 12.