Local radial basis function–finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher–Kolmogorov equations

  • Mehdi DehghanEmail author
  • Nasim Shafieeabyaneh
Original Article


In this investigation, we concentrate on solving the regularized long-wave (RLW) and extended Fisher–Kolmogorov (EFK) equations in one-, two-, and three-dimensional cases by a local meshless method called radial basis function (RBF)–finite-difference (FD) method. This method at each stencil approximates differential operators such as finite-difference method. In each stencil, it is necessary to solve a small-sized linear system with conditionally positive definite coefficient matrix. This method is relatively efficient and has low computational cost. How to choose the shape parameter is a fundamental subject in this method, since it has a palpable effect on coefficient matrix. We will employ the optimal shape parameter which results from algorithm of Sarra (Appl Math Comput 218:9853–9865, 2012). Then, we compare the approximate solutions acquired by RBF–FD method with theoretical solution and also with results obtained from other methods. The numerical results show that the RBF–FD method is suitable and robust for solving the RLW and EFK equations.


Nonlinear regularized long-wave (RLW) equation Extended Fisher–Kolmogorov (EFK) equation Radial basis functions (RBFs) Local meshless method RBF–FD method Shape parameter 

Mathematics Subject Classification

65M06 65M70 35K55 



The authors would like to acknowledge one of the referee for his (her) valuable comment.


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© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematics and Computer SciencesAmirkabir University of TechnologyTehranIran

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