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Chin. Phys. B, 2013, Vol. 22(6): 060209    DOI: 10.1088/1674-1056/22/6/060209
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Analysis of the generalized Camassa and Holm equation with the improved element-free Galerkin method

Cheng Rong-Jun, Wei Qi
Ningbo Institute of Technology, Zhejiang University, Ningbo 315100, China
Abstract  In this paper, we analyze the generalized Camassa and Holm (CH) equation by the improved element-free Galerkin (IEFG) method. By employing the improved moving least-square (IMLS) approximation, we derive the formulas for the generalized CH equation with the IEFG method. A variational method is used to obtain the discrete equations, and the essential boundary conditions are enforced by the penalty method. Because there are less coefficients in the IMLS approximation than in the MLS approximation, and in the IEFG method, less nodes are selected in the entire domain than in the conventional EFG method, the IEFG method should result in a higher computing speed. The effectiveness of the IEFG method for the generalized CH equation is investigated by numerical examples in this paper.
Keywords:  meshless method      improved moving least-square (IMLS) approximation      improved element-free Galerkin (IEFG) method      generalized Camassa and Holm (CH) equation  
Received:  04 October 2012      Revised:  29 October 2012      Accepted manuscript online: 
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 Natural Science Foundation of Ningbo City, Zhejiang Province, China (Grant Nos. 2012A610038 and 2012A610023) and the Natural Science Foundation of Zhejiang Province, China (Grant No. Y6110007).
Corresponding Authors:  Wei Qi     E-mail:  weiqi@nit.zju.edu.cn

Cite this article: 

Cheng Rong-Jun, Wei Qi Analysis of the generalized Camassa and Holm equation with the improved element-free Galerkin method 2013 Chin. Phys. B 22 060209

[1] Camassa R and Holm D D 1993 Phys. Rev. Lett. 71 1661
[2] Dai H H 1998 Acta Mech. 127 193
[3] Boyd J P 1997 Appl. Math. Comput. 81 173
[4] Liu Z R and Ouyang Z Y 2007 Phys. Lett. A 366 377
[5] Constantin A and Ivanov R 2006 Lett. Math. Phys. 76 93
[6] Wazwaz A M 2005 Appl. Math. Comput. 165 485
[7] Parker A 2006 Inverse Problems 22 599
[8] Artebrant R and Schroll H J 2006 Appl. Numer. Math. 56 695
[9] Mustafa O G 2006 Nonlinear Anal. 64 1382
[10] Yomba 2005 J. Math. Phys. 46 12
[11] Liu Z R, Li Q X and Lin Q M 2004 Int. J. Bifur. Chaos 14 3541
[12] Boyd J P 2005 Phys. Lett. A 336 342
[13] Belytschko T, Krongauz Y and Organ D 1996 Comput. Meth. Appl. Mech. Engng. 139 3
[14] Cheng Y M and Peng M J 2005 Sci. China G: Phys. Mech. & Astron. 48 641
[15] Cheng Y M and Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese)
[16] Qin Y X and Cheng Y M 2006 Acta Phys. Sin. 55 3215 (in Chinese)
[17] Cheng R J and Cheng Y M 2007 Acta Phys. Sin. 56 5569 (in Chinese)
[18] Dai B D and Cheng Y M 2007 Acta Phys. Sin. 56 597 (in Chinese)
[19] Cheng R J and Cheng Y M 2008 Acta Phys. Sin. 57 6037 (in Chinese)
[20] Cheng R J 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] Cheng R J and Ge H X 2010 Chin. Phys. B 19 090201
[23] Cheng Y M and Li J H 2006 Sci. China G: Phys. Mech. & Astron. 49 46
[24] Wang J F and Cheng Y M 2011 Chin. Phys. B 20 030206
[25] Cheng R J and Cheng Y M 2011 Chin. Phys. B 20 070206
[26] Cheng R J and Cheng Y M 2011 Acta Phys. Sin. 60 070206 (in Chinese)
[27] Cheng R J and Liew K M 2012 Engineering Analysis with Boundary Elements 36 203
[28] Chen L and Cheng Y M 2008 Acta Phys. Sin. 57 1 (in Chinese)
[29] Chen L and Cheng Y M 2008 Acta Phys. Sin. 57 6047 (in Chinese)
[30] Bai F N, Li D M, Wang J F and Cheng Y M 2012 Chin. Phys. B 21 020204
[31] Cheng Y M, Liew K M and Kitipornchai S 2009 Int. J. Numer. Meth. Eng. 78 1258
[32] Li S C and Cheng Y M 2004 Chin. J. Theor. Appl. Mech. 36 496
[33] Cheng R J and Cheng Y M 2007 Chin. J. Theor. Appl. Mech. 39 843
[34] Cheng R J and Ge H X 2012 Chin. Phys. B 21 040203
[35] Gao H F and Cheng Y M 2010 Int. J. Comput. Meth. 7 55
[36] Gao H F and Cheng Y M 2009 Chin. J. Theor. Appl. Mech. 41 480
[37] Monaghan J J 1988 Comput. Phys. Commun. 48 89
[38] Chen W 2000 "New RBF Collocation Method and Kernel RBF with Application" in Meshfree Methods for Partial Differential Equations (Berline/Heidelberg: Springer) Vol. 26, pp 75-86
[39] Belytschko T, Lu Y Y and Gu L 1994 Int. J. Numer. Meth. Eng. 37 229
[40] Liu W K, Jun S and Zhang Y F 1995 Int. J. Numer. Meth. Fluids 20 1081
[41] Cheng R J and Liew K M 2009 Comput. Mech. 45 1
[42] Atluri S N and Zhu T 1998 Comput. Mech. 22 117
[43] Cheng R J and Cheng Y M 2008 Appl. Numer. Math. 58 884
[44] Cheng R J and Liew K M 2012 Engineering Analysis with Boundary Elements 36 1322
[45] Cheng R J and Ge H X 2012 Chin. Phys. B 21 100209
[46] Cheng R J and Liew K M 2012 Comput. Meth. Appl. Mech. Eng. 245-246 132
[47] Liew K M, Cheng Y M and Kitipornchai S 2005 Int. J. Numer. Meth. Eng. 64 1610
[48] Sun Y Z, Zhang Z, Kitipornchai S and Liew K M 2006 Int. J. Eng. Sci. 44 37
[49] Liew K M, Cheng Y M and Kitipornchai S 2006 Int. J. Numer. Meth. Eng. 65 1310
[50] Ren H P and Cheng Y M 2011 Int. J. Appl. Mech. 3 735
[51] Bai F N, Li D M, Wang J F and Cheng Y M 2012 Chin. Phys. B 21 020204
[52] Ren H P and Cheng Y M 2012 Engineering Analysis with Boundary Elements 36 873
[53] Ren H P and Cheng Y M 2012 Int. J. Comput. Mater. Sci. Eng. 1 1250011
[54] Zhang Z, Li D M, Cheng Y M and Liew K M 2012 Acta Mech. Sin. 28 808
[55] Cheng Y M, Wang J F and Bai F N 2012 Chin. Phys. B 21 090203
[56] Wang J F, Sun F X and Cheng Y M 2012 Chin. Phys. B 21 090204
[57] Cheng Y M, Li R X and Peng M J 2012 Chin. Phys. B 21 090205
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