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

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 (RBFs). 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.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

[1]  Mehdi Dehghan,et al.  Numerical solution to the unsteady two‐dimensional Schrödinger equation using meshless local boundary integral equation method , 2008 .

[2]  Mehdi Dehghan,et al.  A numerical method based on the boundary integral equation and dual reciprocity methods for one-dimensional Cahn–Hilliard equation , 2009 .

[3]  Mehdi Dehghan,et al.  Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions , 2006, Math. Comput. Model..

[4]  Idris Dag,et al.  Galerkin method for the numerical solution of the RLW equation using quadratic B-splines , 2004, Int. J. Comput. Math..

[5]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[6]  Bülent Saka,et al.  Galerkin method for the numerical solution of the RLW equation using quintic B-splines , 2006 .

[7]  Bülent Saka.  A COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF THE RLW EQUATION USING CUBIC B-SPLINE BASIS , 2005 .

[8]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .

[9]  L. Gardner,et al.  Solitary waves of the regularized long-wave equation , 1990 .

[10]  M. Dehghan.  A computational study of the one‐dimensional parabolic equation subject to nonclassical boundary specifications , 2006 .

[11]  Abdulkadir Dogan,et al.  Numerical solution of regularized long wave equation using Petrov–Galerkin method , 2001 .

[12]  Bülent Saka,et al.  A numerical solution of the RLW equation by Galerkin method using quartic B‐splines , 2007 .

[13]  Edward H. Twizell,et al.  A linearized implicit pseudo‐spectral method for some model equations: the regularized long wave equations , 2003 .

[14]  Hossein Jafari,et al.  New solitary wave solutions for generalized regularized long-wave equation , 2010, Int. J. Comput. Math..

[15]  A. A. Soliman,et al.  Collocation solution for RLW equation with septic spline , 2005, Appl. Math. Comput..

[16]  Mehdi Dehghan,et al.  Parameter determination in a partial differential equation from the overspecified data , 2005, Math. Comput. Model..

[17]  Idris Dag,et al.  Application of cubic B-splines for numerical solution of the RLW equation , 2004, Appl. Math. Comput..

[18]  Satya N. Atluri,et al.  New concepts in meshless methods , 2000 .

[19]  I. Dag,et al.  Approximation of the RLW equation by the least square cubic B-spline finite element method , 2001 .

[20]  Mehdi Dehghan,et al.  Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices , 2006, Math. Comput. Simul..

[21]  Mehdi Dehghan,et al.  A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions , 2008, Math. Comput. Simul..

[22]  Zongmin Wu,et al.  Additive Schwarz Domain Decomposition with Radial Basis Approximation , 2002 .

[23]  Haiming Gu,et al.  Least‐squares mixed finite element methods for the RLW equations , 2008 .

[24]  D. Peregrine.  Calculations of the development of an undular bore , 1966, Journal of Fluid Mechanics.

[25]  Dheeraj Bhardwaj,et al.  A computational method for regularized long wave equation , 2000 .

[26]  Yun-Xin Zhang,et al.  Solve partial differential equations by meshless subdomains method combined with RBFs , 2006, Appl. Math. Comput..

[27]  Mehdi Dehghan,et al.  On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation , 2005 .

[28]  Zhihua Xie,et al.  High‐order compact difference scheme for the regularized long wave equation , 2006 .

[29]  Ling Guo,et al.  H1-Galerkin Mixed Finite Element Method for the Regularized Long Wave Equation , 2006, Computing.

[30]  Ted Belytschko,et al.  Overview and applications of the reproducing Kernel Particle methods , 1996 .

[31]  K. R. Raslan,et al.  A computational method for the regularized long wave (RLW) equation , 2005, Appl. Math. Comput..

[32]  A. Rashid.  Numerical Solution of Korteweg-de Vries Equation by the Fourier Pseudospectral Method , 2007 .

[33]  E. Kansa,et al.  Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations , 2000 .

[34]  Mehdi Dehghan,et al.  A numerical method for KdV equation using collocation and radial basis functions , 2007 .

[35]  Weiming Cao,et al.  The Fourier pseudospectral method with a restrain operator for the RLW equation , 1988 .

[36]  P. Jain,et al.  Numerical solution of regularized long-wave equation , 1993 .

[37]  F. J. Seabra-Santos,et al.  A Petrov–Galerkin finite element scheme for the regularized long wave equation , 2004 .

[38]  B. Saka,et al.  Quartic B‐spline collocation algorithms for numerical solution of the RLW equation , 2007 .

[39]  R. Schreiber.  Numerical Methods for Partial Differential Equations , 1999 .

[40]  Mehdi Dehghan,et al.  The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data , 2007 .

[41]  A. Santarelli,et al.  Numerical analysis of the regularized long-wave equation: Anelastic collision of solitary waves , 1978 .

[42]  J. C. Eilbeck,et al.  Numerical study of the regularized long-wave equation I: Numerical methods , 1975 .

[43]  M. A. Helal,et al.  Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics , 2002 .

[44]  Alaattin Esen,et al.  A finite difference solution of the regularized long-wave equation , 2006 .

[45]  Carsten Franke,et al.  Solving partial differential equations by collocation using radial basis functions , 1998, Appl. Math. Comput..

[46]  M. Dehghan,et al.  Meshless Local Petrov--Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity , 2009 .

[47]  Mehdi Dehghan,et al.  USE OF HES HOMOTOPY PERTURBATION METHOD FOR SOLVING A PARTIAL DIFFERENTIAL EQUATION ARISING IN MODELING OF FLOW IN POROUS MEDIA , 2008 .

[48]  Peter J. Olver,et al.  Euler operators and conservation laws of the BBM equation , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.

[49]  Benny Y. C. Hon,et al.  An efficient numerical scheme for Burgers' equation , 1998, Appl. Math. Comput..

[50]  Yunxin Zhang.  Convergence of meshless Petrov-Galerkin method using radial basis functions , 2006, Appl. Math. Comput..

[51]  Dogan Kaya,et al.  A Numerical Simulation of Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation , 2002, Appl. Math. Comput..

[52]  Ching-Shyang Chen,et al.  A numerical method for heat transfer problems using collocation and radial basis functions , 1998 .

[53]  L. Gardner,et al.  A B-spline finite element method for the regularized long wave equation , 1995 .

[54]  Abdulkadir Dogan,et al.  A least‐squares finite element scheme for the RLW equation , 1996 .

[55]  Modelling an undular bore with B-splines , 1997 .

[56]  Mehdi Dehghan,et al.  Identification of a time‐dependent coefficient in a partial differential equation subject to an extra measurement , 2005 .

[57]  Jichun Li,et al.  Radial basis function method for 1-D and 2-D groundwater contaminant transport modeling , 2003 .

[58]  S. Zaki,et al.  Solitary waves of the splitted RLW equation , 2001 .

[59]  I Dağ.  Corrigendum to: “Least-squares quadratic B-spline finite element method for the regularized long wave equation” [Comput. Methods Appl. Mech. Engrg. 182 (2000) 205–215] , 2001 .

[60]  Juan M. Restrepo,et al.  Stable and Unstable Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation , 2000, J. Nonlinear Sci..

[61]  J. Bona,et al.  Model equations for long waves in nonlinear dispersive systems , 1972, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[62]  J. A. Tjon,et al.  Resonant production of solitons in the RLW equation , 1979 .

[63]  Solitary waves of the EW and RLW equations , 2007 .

[64]  G. R. McGuire,et al.  Numerical Study of the Regularized Long-Wave Equation. II: Interaction of Solitary Waves , 1977 .

[65]  Y. C. Hon,et al.  Numerical comparisons of two meshless methods using radial basis functions , 2002 .

[66]  Khaled Omrani,et al.  A fully Galerkin method for the damped generalized regularized long‐wave (DGRLW) equation , 2009 .

[67]  Yunxin Zhang,et al.  Solving partial differential equations by meshless methods using radial basis functions , 2007, Appl. Math. Comput..

[68]  Idris Dag,et al.  B-Spline Collocation Methods For Numerical Solutions Of The Rlw Equation , 2003, Int. J. Comput. Math..

[69]  Mehdi Dehghan,et al.  On the solution of the non-local parabolic partial differential equations via radial basis functions , 2009 .