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Chin. Phys. B, 2015, Vol. 24(5): 050206    DOI: 10.1088/1674-1056/24/5/050206
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Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation

Reza Mohammadi
Department of Mathematics, University of Neyshabur, Neyshabur 91136-899, Iran
Abstract  The aim of the present paper is to present a numerical algorithm for the time-dependent generalized regularized long wave equation with boundary conditions. We semi-discretize the continuous problem by means of the Crank–Nicolson finite difference method in the temporal direction and exponential B-spline collocation method in the spatial direction. The method is shown to be unconditionally stable. It is shown that the method is convergent with an order of O(k2+h2). Our scheme leads to a tri-diagonal nonlinear system. This new method has lower computational cost in comparison to the Sinc-collocation method. Finally, numerical examples demonstrate the stability and accuracy of this method.
Keywords:  solitary waves      GRLW equation      exponential B-spline      collocation  
Received:  02 November 2014      Revised:  16 December 2014      Accepted manuscript online: 
PACS:  02.60.Lj (Ordinary and partial differential equations; boundary value problems)  
  03.65.Ge (Solutions of wave equations: bound states)  
Corresponding Authors:  Reza Mohammadi     E-mail:  rez.mohammadi@gmail.com, mohammadi@neyshabur.ac.ir
About author:  02.60.Lj; 03.65.Ge

Cite this article: 

Reza Mohammadi Exponential B-spline collocation method for numerical solution of the generalized regularized long wave equation 2015 Chin. Phys. B 24 050206

[1] Feng Z, Wang X D and Ouyang J 2012 Acta Phys. Sin. 61 230204 (in Chinese)
[2] Cai W J, Wang Y S and Song Y Z 2014 Chin. Phys. Lett. 31 040201
[3] Wang X, Ma T B and Ning J G 2014 Chin. Phys. Lett. 31 030201
[4] Ge H X and Cheng R J 2014 Chin. Phys. B 23 040203
[5] Yu X, Ren Z G and Xu C 2014 Chin. Phys. B 23 040201
[6] Peregrine D H 1966 J. Fluid Mech. 25 321
[7] Lin J, Xie Z and Zhou J 2007 Commun. Numer. Meth. Eng. 23 135
[8] Saka B and Dag I 2008 Commun. Numer. Meth. Eng. 24 1339
[9] Abdulloev K H O, Bogalubsky H and Markhankov V G 1976 Phys. Lett. A 56 427
[10] Peregrine D H 1967 J. Fluid Mech. 27 815
[11] Bona J L, McKinney W R and Restrepo J M 2000 J. Nonlinear Sci. 10 603
[12] Durán A and López-Marcos M A 2002 Appl. Numer. Math. 42 95
[13] Djidjeli K, Price W G, Twizell E H and Cao Q 2003 Commun. Numer. Meth. Eng. 19 847
[14] Kaya D and Syed E 2003 Chaos Soliton. Fract. 17 869
[15] Kaya D 2004 Appl. Math. Comput. 149 833
[16] Zhang L 2005 Appl. Math. Comput. 168 962
[17] Xie S, Kim S, Woo G and Yi S 2008 SIAM J. Sci. Comput. 30 2263
[18] Jafari H, Borhanifar A and Karimi S A 2010 Int. J. Comput. Math. 87 509
[19] Mokhtari R and Mohammadi M 2010 Comput. Phys. Commun. 181 1266
[20] Mohammadi M and Mokhtari R 2011 J. Comput. Appl. Math. 235 4003
[21] Wang J F, Bai F N and Cheng Y M 2011 Chin. Phys. B 20 030206
[22] Roshan T 2012 Comput. Math. Appl. 63 943
[23] Lin B A 2014 Stud. Appl. Math. 132 160
[24] Benjamin T B, Bona J L and Mahony J J 1972 Trans. R. Soc. Lond. Ser. A 272 47
[25] Dağ I 2000 Comput. Method. Appl. Mech. Eng. 182 205
[26] Dağ I and Özer M N 2001 Appl. Math. Model. 25 221
[27] Gardner L R T, Gardner G A and Doğan A 1996 Commun. Numer. Method. Eng. 12 795
[28] Dağ I, Dogan A and Saka B 2003 Int. J. Comput. Math. 80 743
[29] Esen A and Kutluay S 2006 Appl. Math. Comput. 174 833
[30] Dağ I, Saka B and Irk D 2006 J. Comput. Appl. Math. 190 532
[31] Raslan K R 2005 Appl. Math. Comput. 167 1101
[32] Saka B, Dağ I and Dogan A 2004 Int. J. Comput. Math. 81 727
[33] Saka B and Dağ I 2005 Arab J. Sci. Eng. 30 39
[34] Dogan A 2002 Appl. Math. Model. 26 771
[35] Gardner L R T, Gardner G A and Dağ I 1995 Commun. Numer. Meth. Eng. 11 59
[36] Mei L and Chen Y 2012 Comput. Phys. Commun. 183 1609
[37] Dogan A 2001 Commun. Numer. Meth. Eng. 17 485
[38] Islam S, Haq S and Ali A 2009 J. Comput. Appl. Math. 223 997
[39] Mokhtari R and Torabi Z S 2011 Int. J. Appl. Comput. Math. 10 428
[40] Korkmaz A 2010 Numerical Solutions of Some One-dimensional Partial Differential Equations Using B-spline Differential Quadrature Methods (Ph. D. thesis) (Eskisehir: Eskisehir Osmangazi University of Turkey)
[41] Shokri A and Dehghan M 2010 Numer. Meth. Partial Differential Eq. 26 807
[42] Alexander M E and Morris J L 1979 J. Comput. Phys. 30 428
[43] Dağ I, Saka B and Irk D 2006 J. Comput. Appl. Math. 190 532
[44] Dağ I, Saka B and Irk D 2004 Appl. Math. Comput. 159 373
[45] Soliman A A and Hussein M H 2005 J. Appl. Math. Comput. (Science) 161 623
[46] Ali A H A 2008 Nonlinear Dyn. 51 59
[47] Chegini N G, Salaripanah A, Mokhtari R and Isvand D 2012 Nonlinear Dyn. 69 459
[48] Gardner L R T, Gardner G A, Ayoub F A and Amein N K 1997 Arab J. Sci. Eng. 22 183
[49] Khalifa A K, Raslan K R and Alzubaidi H M 2007 Appl. Math. Comput. 189 346
[50] Khalifa A K, Raslan K R and Alzubaidi H M 2008 J. Comput. Appl. Math. 212 406
[51] Khalifa A K, Raslan K R and Alzubaidi H M 2008 Appl. Math. Model. 32 2962
[52] Ali A 2009 Mesh Free Collocation Method for Numerical Solution of Initial-boundary Value Problems using Radial Basis Functions (Ph. D. thesis) (Ghulam Ishaq Khan Institute of Engineering Sciences and Technology)
[53] Raslan K R 2009 Chaos Soliton. Fract. 42 1845
[54] Raslan K R and Hassan S M 2009 Appl. Math. Lett. 22 984
[55] Raslan K R and EL-Danaf T S 2010 J. King Saud University 22 161
[56] Haq F, Islam S and Tirmizi I A 2010 Appl. Math. Model. 34 4151
[57] Achouri T and Omrani K 2010 Numer. Meth. Partial Differential Eq. 26 399
[58] Labidi M and Omrani K 2011 Numer. Meth. Partial Differential Eq. 27 478
[59] Dereli Y 2012 Int. J. Nonlinear Sci. 13 28
[60] Dereli Y 2012 Numer. Meth. Partial Differential Eq. 28 235
[61] Khan Y, Taghipour R, Falahian M and Nikkar A 2013 Neural Comput. Appl. 23 1335
[62] Karakoc S B G, Yagmurlu N M and Ucar Y 2013 Boundary Value Problems 27 1
[63] Rashidinia J and Mohammadi R 2010 Comput. Phys. Commun. 181 78
[64] Rashidinia J and Mohammadi R 2011 Numer. Algor. 56 129
[65] Mohammadi R 2012 Appl. Anal. 91 2189
[66] Mohammadi R 2013 Appl. Math. 4 933
[67] Mohammadi R 2014 Comput. Phys. Commun. 185 917
[68] Mohammadi R 2014 Appl. Math. Comput. 241 151
[69] Bian X B 2014 Phys. Rev. A 90 033403
[70] Chandra S R S and Kumar M 2008 Appl. Numer. Math. 58 1572
[71] Chandra S R S and Kumar M 2009 Nonlinear Anal. 71 1579
[72] Mc Cartin B J 1991 J. Approx. Theory 66 1
[73] Pruess S 1976 J. Approx. Theory 17 86
[74] Mc Cartin B J 1981 Theory, Computation, and Application of Exponential Splines (Courant Mathematics and Computing Laboratory Research and Development Report DOE/ER/03077-171)
[75] Prenter P M 1975 Splines and Variational Methods (New York: John Wiley and Sons)
[76] Clavero C and Jorge J C and Lisbona F 1993 Uniformly Convergent Schemes for Singular Perturbation Problems Combining Alternating Directions and Exponential Fitting Techniques, in: J. J. H. Miller (Ed.) (Boole, Dublin: Applications of Advanced Computational Methods for Boundary and Interior Layers) pp. 33-52
[77] Rashidinia J, Ghasemi M and Jalilian R 2010 Math. Sci. 4 87
[78] Mohammadi R 2013 Numer. Meth. Partial Differential Eq. 29 1173
[79] Khalifa A K 1979 Theory and Applications of the Collocation Method with Splines for Ordinary and Partial Differential Equations (Ph.D. thesis) (Heriot-Watt University)
[80] Gardner L R T and Gardner G A 1990 J. Comput. Phys. 91 441
[81] Soliman A A and Raslan K R 2001 Int. J. Comput. Math. 78 399
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