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Using the homotopy analysis method (HAM), the nonlinear equation of the jamming transition problem (JTP) in traffic flow is discussed, which is based on the Lorentz system. Through choosing different initial approximation solutions and different linear operators, approximation solutions of the JTP and the corresponding residual errors are obtained respectively. By comparing the present results with the previous related studies, the following conclusions can be drawn that the HAM is superior to the differential transform method; however, a linear operator should be chosen as best you can to approach the linear part of the original operator in using the HAM. A new method to choose the initial approximation solution (named double HAM) is given. The correctness of the theoretical analysis is verified by numerical simulation.
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Keywords:
- homotopy analysis method /
- traffic jam /
- approximate solution /
- residual error
[1] Orosz G, Wilson R E, Stépán G 2010 Phil. Trans. Math. Phys. Eng. Sci. 368 4455
[2] Helbing D, Tilch B 1998 Phys. Rev. E 58 133
[3] Wagner P 2011 Eur. Phys. J. B 84 713
[4] Wang T, Gao Z Y, Zhao X M 2006 Acta Phys. Sin. 55 634 (in Chinese) [王涛, 高自友, 赵小梅 2006 55 634]
[5] Ge H X, Han X L 2006 Physica A 371 667
[6] Han X L, Jiang C Y, Ge H X, Dai S Q 2007 Acta Phys. Sin. 56 4383 (in Chinese) [韩祥临, 姜长元, 葛红霞, 戴世强 2007 56 4383]
[7] Jiang R, Wu Q S 20015 Eur. Phys. J. B 46 581
[8] Tang C F, Jiang R, Wu Q S 2007 Phys. A 377 641
[9] Olemskoi A I, Khomenko A V 1996 Am. Inst. Phys. 83 1180
[10] Ganji S S, Barari A, Ibsen L B, Domairry D 2012 Cent. Eur. J. Oper. Res. 20 87
[11] Ganji S S, Barari A, Najafi M, Domairry D 2011 Can. J. Phys. 89 729
[12] Chowdhury D, Santen L, Schadschneider A 2000 Phys. Rep. 329 199
[13] Helbing D R 2001 Mod. Phys. 73 1067
[14] Li X L, Song T, Kuang H, Dai S Q 2008 Chin. Phys. B 17 3014
[15] Nagatani T 1998 Phys. Rev. E 58 4271
[16] Olemskoi A I, Khomenko A V 1996 J. Exp. Theor. Phys. 83 1180
[17] Olemskoi A I, Khomenko A V 2001 Phys. Rev. E 63 036116
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[1] Orosz G, Wilson R E, Stépán G 2010 Phil. Trans. Math. Phys. Eng. Sci. 368 4455
[2] Helbing D, Tilch B 1998 Phys. Rev. E 58 133
[3] Wagner P 2011 Eur. Phys. J. B 84 713
[4] Wang T, Gao Z Y, Zhao X M 2006 Acta Phys. Sin. 55 634 (in Chinese) [王涛, 高自友, 赵小梅 2006 55 634]
[5] Ge H X, Han X L 2006 Physica A 371 667
[6] Han X L, Jiang C Y, Ge H X, Dai S Q 2007 Acta Phys. Sin. 56 4383 (in Chinese) [韩祥临, 姜长元, 葛红霞, 戴世强 2007 56 4383]
[7] Jiang R, Wu Q S 20015 Eur. Phys. J. B 46 581
[8] Tang C F, Jiang R, Wu Q S 2007 Phys. A 377 641
[9] Olemskoi A I, Khomenko A V 1996 Am. Inst. Phys. 83 1180
[10] Ganji S S, Barari A, Ibsen L B, Domairry D 2012 Cent. Eur. J. Oper. Res. 20 87
[11] Ganji S S, Barari A, Najafi M, Domairry D 2011 Can. J. Phys. 89 729
[12] Chowdhury D, Santen L, Schadschneider A 2000 Phys. Rep. 329 199
[13] Helbing D R 2001 Mod. Phys. 73 1067
[14] Li X L, Song T, Kuang H, Dai S Q 2008 Chin. Phys. B 17 3014
[15] Nagatani T 1998 Phys. Rev. E 58 4271
[16] Olemskoi A I, Khomenko A V 1996 J. Exp. Theor. Phys. 83 1180
[17] Olemskoi A I, Khomenko A V 2001 Phys. Rev. E 63 036116
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