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Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM. (English) Zbl 1244.65153
Summary: A generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation is solved by using two recent semi-analytic methods, the differential transform method (DTM) and a reduced form of the differential transformation method (so-called RDTM). The concepts of DTM and RDTM are briefly introduced, and their application for the generalized Hirota-Satsuma coupled KdV equation is studied. The results obtained employing DTM and RDTM are compared with together and the exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by classic DTM. The numerical results reveal that the RDTM is very effective, convenient and quite accurate to systems of nonlinear equations. It is predicted that the RDTM can be found widely applicable in engineering.
MSC:
65M99Numerical methods for IVP of PDE
35Q53KdV-like (Korteweg-de Vries) equations
References:
[1]Ablowitz, M. J.; Segur, H.: Solitons and inverse scattering transform, (1981)
[2]Tam, H. W.; Hu, X. B.: Soliton solutions and Bäcklund transformation for the kupershmidt five-field lattice: a bilinear approach, Appl math lett 15, 987-993 (2002) · Zbl 1009.37047 · doi:10.1016/S0893-9659(02)00074-5
[3]Hirota, R.: The direct method in soliton theory, (2004)
[4]Borhanifar A, Reza Abazari, An unconditionally stable parallel difference scheme for telegraph equation, vol. 2009, Article ID 969610, 2009, p. 17. doi:10.1155/2009/969610. · Zbl 1181.78018 · doi:10.1155/2009/969610
[5]Freeman, N. C.; Nimmo, J. J. C.: Soliton solitons of the KdV and KP equations: the wroanskian technique, Proc roy soc lond A 389, 319-329 (1983)
[6]Shahrbabaki, A. Shabani; Abazari, Reza: Perturbation method for heat exchange between a gas and solid particles, J appl mech tech phys 50, No. 6, 959-964 (2009)
[7]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994)
[8]He, J. H.: Variational iteration method – a kind of nonlinear analytical technique: some examples, Int J non-linear mech 34, 708-799 (1999)
[9]Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[10]Zhou, J. K.: Differential transformation and its application for electrical circuits, (1986)
[11]Kurnaz, A.; Oturanç, G.: The differential transform approximation for the system of ordinary differential equations, Int J comput math 82, 709-719 (2005) · Zbl 1072.65101 · doi:10.1080/00207160512331329050
[12]Arikoglu, A.; Ozkol, I.: Solution of differential-difference equations by using differential transform method, Appl math comput 181, 153-162 (2006) · Zbl 1148.65310 · doi:10.1016/j.amc.2006.01.022
[13]Borhanifar, A.; Abazari, Reza: Exact solutions for non-linear Schrödinger equations by differential transformation method, J appl math comput 35, 37-51 (2011) · Zbl 1211.35250 · doi:10.1007/s12190-009-0338-2
[14]Abazari, Reza; Borhanifar, A.: Numerical study of Burgers and coupled Burgers equations by differential transformation method, Comput math appl 59, 2711-2722 (2010) · Zbl 1193.65178 · doi:10.1016/j.camwa.2010.01.039
[15]Borhanifar, A.; Abazari, Reza: Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method, Opt commun 283, 2026-2031 (2010)
[16]Abazari, Reza: Solution of Riccati types matrix differential equations using matrix differential transform method, J appl math inform 27, 1133-1143 (2009)
[17]Karakoç, F.; Bereketouglu, H.: Solutions of delay differential equations by using differential transform method, Int J comput math 86, No. 5, 914-923 (2009) · Zbl 1167.65038 · doi:10.1080/00207160701750575 · doi:http://www.informaworld.com/smpp/./content~db=all~content=a794818469
[18]Arikoglu, A.; Ozkol, I.: Solution of fractional differential equations by using differential transform method, Chaos soliton fract 34, 1473-1481 (2007) · Zbl 1152.34306 · doi:10.1016/j.chaos.2006.09.004
[19]Tari, A.; Rahimi, M. Y.; Shahmorad, S.; Talati, F.: Solving a class of twodimensional linear and nonlinear Volterra integral equations by the differential transform method, J comput appl math 228, 70-76 (2009) · Zbl 1176.65164 · doi:10.1016/j.cam.2008.08.038
[20]Wu, Y. T.; Geng, X. G.; Hu, X. B.; Zhu, S. M.: A generalized hirotasatsuma coupled Korteweg-de Vries equation and miura transformations, Phys lett A 255, 259-264 (1999) · Zbl 0935.37029 · doi:10.1016/S0375-9601(99)00163-2
[21]Hirota, R.; Satsuma, J.: Soliton solutions of a coupled Korteweg-de Vries equation, Phys lett A 85, 407-408 (1981)
[22]Fan, E.: Soliton solutions for a generalized Hirota – satsuma coupled KdV equation and a coupled mkdv equation, Phys lett A 282, 18-22 (2001) · Zbl 0984.37092 · doi:10.1016/S0375-9601(01)00161-X
[23]He, J. H.; Wu, X. H.: Construction of solitary solution and compaction-like solution by variational iteration method, Chaos soliton fract 29, 108-113 (2006) · Zbl 1147.35338 · doi:10.1016/j.chaos.2005.10.100
[24]Kaya, D.: Solitary wave solutions for a generalized Hirota – satsuma coupled KdV equations, Appl math comput 147, 69-78 (2004) · Zbl 1037.35069 · doi:10.1016/S0096-3003(02)00651-3
[25]Ganji, D. D.; Rafei, M.: Solitary wave solutions for a generalized Hirota – satsuma coupled KdV equations by homotopy perturbation method, Phys lett A 356, 131-137 (2006) · Zbl 1160.35517 · doi:10.1016/j.physleta.2006.03.039
[26]Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota – satsuma coupled KdV equation, Phys lett A 361, 478-483 (2007)
[27]Keskin, Y.; Oturanç, G.: Reduced differential transform method for partial differential equations, Int J nonlinear sci numer simul 10, No. 6, 741-749 (2009)
[28]Keskin, Y.; Oturanç, G.: The reduced differential transform method: a new approach to factional partial differential equations, Nonlinear sci lett A 1, 207-217 (2010)
[29]Abazari, Reza; Ganji, Masoud: Extended two-dimensional DTM and its a pplication on nonlinear pdes with proportional delay, Int J comput math 88, No. 8, 1749-1762 (2011) · Zbl 1232.35012 · doi:10.1080/00207160.2010.526704