A Numerical Investigation on a Hybrid-Parameterless Radial Basis Function Applied with a Meshless Method


  • T Moonsan
  • S Kaennakham
  • N Chuathong




A Hybrid RBF has recently been proposed and tested with some scattered data interpolation problems and the results have appeared promising whereas the appearance of the shape parameter remains a difficulty when deploying. This work, therefore, focuses on three objectives; firstly, it is aimed to extend the use of the newly proposed-RBF to application of RBF-collocation method. Secondly, realizing the burden attributed to the lack of information on choosing an optimum shape parameter, the hybrid RBF is then modified where the shape parameter is no longer included. Thirdly, it is to investigate its application/implementation towards solving PDEs particularly those in both linear and non-linear form. It has been found in this work that the new RBF of this HyBrid form with no parameter can well be a good candidate and truly deserves further study with more complex problems.


Kansa E. J., Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, Computers Math. Applic, 1990, vol. 19, p. 147-161. DOI: https://doi.org/10.1016/0898-1221(90)90271-K

Sarra S. A. and Kansa E. J., Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations, Adv. Comput. Math, 2009.

Yaghouti M. and Azarboni H. R., Determining optimal value of the shape parameter c in RBF for unequal distances topographical points by Cross-Validation algorithm, Journal of Mathematical Modeling, 2017, vol. 5, p. 53-60. DOI: https://dx.doi.org/10.22124/jmm.2017.2225

Chanthawara K. and Kaennakham S., A Numerical Experiment on Optimal Inverse Multiquadric RBF Shape Parameter in the Dual Reciprocity Boundary Element Method for Convective-Dominated Problems, The 11th IMT-GT International Conference on Mathematics, Statistics and Its Applications 2015(ICMSA 2015), 2016, p. 119-131

Kaennakham S., Chanthawara K. and Toutip W., Optimal Radial Basis Function (RBF) for Dual Reciprocity Boundary Element Method (DRBEM) applied to Coupled Burgers’ Equations with Increasing Reynolds Number, Australian Journal of Basic and Applied Sciences, 2014, p. 462-476.

Wang J. G. and Liu G. R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Computer Methods in Applied Mechanics and Engineering, Boca Raton, 2002, vol. 191, p. 2611-2630. DOI: https://doi.org/10.1016/S0045-7825(01)00419-4

Chuathong N., Kaennakham S. and Toutip W., Numerical solutions of 2D nonlinear PDEs using Kansa’s meshless method and the search for optimal radial basis function, Proceeding of 19th International Annual Symposium on Computational Science and Engineering, Ubon Ratchthani university, Ubon Ratchathani, Thailand, 2015, p. 13-15

Mishra P. K., Nath S. K., Sen M. K. and Fasshauer G. E., Hybrid Gaussian-cubic radial basis functions for scattered data interpolation, Computational Geosciences, 2018, p. 1203-1218. DOI: https://doi.org/10.1007/s10596-018-9747-3

Kaennakham S. and Chuathong N., Solution to a Convection-Diffusion Problem Using a New Variable Inverse-Multiquadric Parameter in a Collocation Meshfree Scheme, International Journal of Multiphysics, 2017, vol. 11, p. 359-374.

Kaennakham S. and Chuathong N., Numerical simulation of convection-diffusion phenomena by four inverse-quadratic-RBF domain-meshfree schemes, International Journal of Multiphysics, 2019, vol. 13, p. 1-30. DOI: http://dx.doi.org/10.21152/1750-9548.13.1.1


Chen C. S., Kuhn G., Li J. and Mishuris G., Radial basis functions for solving near singular Poisson problems, Communications in Numerical Methods in Engineering, 2003, p. 333-347. DOI: https://doi.org/10.1002/cnm.593

Dehghan M. and Tatari M., The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data, Numerical Methods for Partial Differential Equations, 2007, vol. 23, p. 984 - 997. DOI: https://doi.org/10.1002/num.20204

Fasshauer G. E., Newton iteration with multiquadrics for the solution of nonlinear PDEs, Computers & Mathematics with Applications, 2002, vol. 43, p. 423-438. DOI: https://doi.org/10.1016/S0898-1221(01)00296-6

Linesawat K., The dual reciprocity boundary element method for nonlinear problems using compactly supported radial basis function, Khon Kaen University, Thailand, 2010.

Pao C. V., Block monotone iterative methods for numerical solutions of nonlinear elliptic equations, Numer. Math., 1995, vol. 72, p. 239-262. DOI: https://doi.org/10.1007/s002110050168



How to Cite

Moonsan, T., Kaennakham, S. and Chuathong, N. (2020) “A Numerical Investigation on a Hybrid-Parameterless Radial Basis Function Applied with a Meshless Method”, The International Journal of Multiphysics, 14(4), pp. 315-330. doi: 10.21152/1750-9548.14.4.315.