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Chin. Phys. B, 2011, Vol. 20(3): 030206    DOI: 10.1088/1674-1056/20/3/030206
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A meshless method for the nonlinear generalized regularized long wave equation

Bai Fu-Nonga, Cheng Yu-Mina, Wang Ju-Fengb
a Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China; b Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China;Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China
Abstract  This paper presents a meshless method for the nonlinear generalized regularized long wave (GRLW) equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method. A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm. Compared with numerical methods based on mesh, the meshless method for the GRLW equation only requires the scattered nodes instead of meshing the domain of the problem. Some examples, such as the propagation of single soliton and the interaction of two solitary waves, are given to show the effectiveness of the meshless method.
Keywords:  generalized regularized long wave equation      meshless method      moving least-squares approximation      convergence  
Received:  18 June 2010      Revised:  10 October 2010      Published:  15 March 2011
PACS:  02.60.Lj (Ordinary and partial differential equations; boundary value problems)  
  03.65.Ge (Solutions of wave equations: bound states)  
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 10871124) and the Innovation Program of the Shanghai Municipal Education Commission, China (Grant No. 09ZZ99).

Cite this article: 

Wang Ju-Feng, Bai Fu-Nong, Cheng Yu-Min A meshless method for the nonlinear generalized regularized long wave equation 2011 Chin. Phys. B 20 030206

[1] Peregrine D H 1966 J. Flui. Mech. 25 321
[2] Peregrine D H 1967 J. Flui. Mech. 27 815
[3] Zaki S I 2001 Comput. Phys. Comm. 138 80
[4] El-Danaf T S, Ramadan M A and Abd Alaal FEI 2005 Chaos, Solitons & Fractals 26 747
[5] Kaya D and El-Sayed S M 2003 Chaos, Solitons & Fractals 17 869
[6] Davg .I 2000 Comput. Methods Appl. Mech. Engng. 182 205
[7] Saka B and Davg .I 2005 Arab. J. Sci. Eng. 30 39
[8] Islam S U, Sirajul H and Arshed A 2009 J. Comput. Appl. Math. 223 997
[9] Bona J L and Soyeur A 1994 J. Nonlinear Sci. 4 449
[10] Hamdi S and Enright W H 2004 Math. Comput. Simulat. 65 535
[11] Soliman A A 2005 Math. Comput. Simulat. 70 119
[12] Zhang L 2005 Appl. Math. Comput. 168 962
[13] Ramos J I 2007 Solitons & Fractals 33 479
[14] Kaya D 2004 Appl. Math. Comput. 149 833
[15] Belytschko T, Krongauz Y and Organ D 1996 Comput. Meth. Appl. Mech. Engng. 139 3
[16] Monaghan J J 1988 Comput. Phys. Comm. 48 89
[17] Kansa E J 1990 Comput. Math. Appl. 19 127
[18] Kansa, E J 1990 Comput. Math. Appl. 19 147
[19] Belytschko T, Lu Y Y and Gu L 1994 Int. J. Numer. Meth. Engng. 37 229
[20] Cheng R J, Cheng Y M and Ge H X 2009 Chin. Phys. B 18 4059
[21] Wang J F, Sun F X and Cheng R J 2010 Chin. Phys. B 19 060201
[22] Xiong Y B and Wang H 2006 Chin. Phys. B 15 2352
[23] Atluri S N and Zhu T 1998 Comput. Mech. 22 117
[24] Cheng R J, Cheng Y M and Ge H X 2010 Chin. Phys. B 19 090201
[25] Onate E, Idelsohn S and Zienkiewicz O C 1996 Int. J. Num. Meth. Engng. 39 3839
[26] Cheng R J and Cheng Y M 2007 Acta Phys. Sin. 56 5569 (in Chinese)
[27] Dai B D and Cheng Y M 2007 Acta Phys. Sin. 56 597 (in Chinese)
[28] Liu W K, Jun S and Zhang Y F 1995 Int. J. Numer. Meth. Engng. 20 1081
[29] Cheng Y M and Chen M J 2003 Acta Mech. Sin. 35 181 (in Chinese)
[30] Cheng Y M and Peng M J 2005 Sci. China G 48 641
[31] Ren H P, Cheng Y M and Zhang W 2009 Chin. Phys. B 18 4065
[32] Cheng Y M and Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese)
[33] Chen L and Cheng Y M 2008 Acta Phys. Sin. 57 1 (in Chinese)
[34] Chen L and Cheng Y M 2008 Acta Phys. Sin. 57 6047 (in Chinese)
[35] Peng M J and Cheng Y M 2009 Eng. Anal. Bound. Elem. 33 77
[36] Chen L and Cheng Y M 2010 Chin. Phys. B 19 090204
[37] Kim Y, Kim D W, Jun S and Lee J H 2007 Comput. Meth. Appl. Mech. Engng. 196 3095
[38] Khattak A J, Tirmizi S I A and Islam S U 2009 Eng. Anal. Bound. Elem. 33 661
[39] Lancaster P and Salkauskas K 1981 Math. Comput. 37 141 endfootnotesize
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