A meshless method using the radial basis functions for numerical solution of the regularized long wave equation.

*(English)*Zbl 1195.65142Summary: This article discusses on the solution of the regularized long wave (RLW) equation, which is introduced to describe the development of the undular bore, has been used for modeling in many branches of science and engineering. A numerical method is presented to solve the RLW equation. The main idea behind this numerical simulation is to use the collocation and approximating the solution by radial basis functions. To avoid solving the nonlinear system, a predictor-corrector scheme is proposed.

Several test problems are given to validate the new technique. The numerical simulation, includes the propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves and the development of an undular bore. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the presented scheme.

Several test problems are given to validate the new technique. The numerical simulation, includes the propagation of a solitary wave, interaction of two positive solitary waves, interaction of a positive and a negative solitary wave, the evaluation of Maxwellian pulse into stable solitary waves and the development of an undular bore. The three invariants of the motion are calculated to determine the conservation properties of the algorithm. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the presented scheme.

##### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

35L75 | Higher-order nonlinear hyperbolic equations |

##### Keywords:

collocation; predictor-corrector; thin plate splines radial basis functions; regularized long wave equation; numerical examples; algorithm
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\textit{A. Shokri} and \textit{M. Dehghan}, Numer. Methods Partial Differ. Equations 26, No. 4, 807--825 (2010; Zbl 1195.65142)

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