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# Indirect RBFN Method with Thin Plate Splines for Numerical Solution of Differential Equations

N. Mai-Duy, T. Tran-Cong1
Corresponding author: Telephone +61 7 46312539, Fax +61 7 46312526, E-mail trancong@usq.edu.au, Faculty of Engineering and Surveying, University of Southern Queensland, Toowoomba, QLD 4350, Australia

Computer Modeling in Engineering & Sciences 2003, 4(1), 85-102. https://doi.org/10.3970/cmes.2003.004.085

### 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$20\times 20$ and a Reynolds number$Re=200$ for the case of Jeffery-Hamel flow with a density of$43\times 12$.

### Keywords

mesh-free indirect RBFN method, thin plate splines, rectangular coordinates, curvilinear coordinates, Jeffery-Hamel flow, numerical solution, differential equation.

Mai-Duy, N., Tran-Cong, T. (2003). Indirect RBFN Method with Thin Plate Splines for Numerical Solution of Differential Equations. CMES-Computer Modeling in Engineering & Sciences, 4(1), 85–102.

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