×

Variational iteration method for solving Burgers and coupled Burgers equations. (English) Zbl 1072.65127

Summary: By means of variational iteration method the solutions of Burgers equation and coupled Burgers equations are exactly obtained, a comparison with the Adomian decomposition method is made, showing that the former is more effective than the later. In this paper, J. H. He’s variational iteration method [Appl. Math. Comput. 114, No. 2–3, 115–123 (2000; Zbl 1027.34009)] is introduced to overcome the difficulty arising in calculating Adomian polynomials.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)

Citations:

Zbl 1027.34009
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001
[2] Adomian, G., Math. Comput. Modelling, 22, 103 (1995) · Zbl 0833.60062
[3] Ali, A. H.A.; Gardner, G. A.; Gardner, L. R.T., Comput. Methods Appl. Mech. Eng., 100, 325-337 (1992) · Zbl 0762.65072
[4] Burgers, J., (Advances in Applied Mechanics (1948), Academic Press: Academic Press New York), 171-199 · Zbl 0038.36807
[5] J. Caldwell, E. Hinton, et al. (Eds.), Numerical Methods for Nonlinear Problems, Pineridge, Swansea, vol. 3, 1987, pp. 253-261.; J. Caldwell, E. Hinton, et al. (Eds.), Numerical Methods for Nonlinear Problems, Pineridge, Swansea, vol. 3, 1987, pp. 253-261.
[6] Caldwell, J.; Wanless, P.; Cook, A. E., Appl. Math. Modelling, 5, 189-193 (1981) · Zbl 0476.76054
[7] A. Coely, et al. (Eds.), Backlund and Darboux Transformations, American Mathematical Society, Providence, RI, 2001.; A. Coely, et al. (Eds.), Backlund and Darboux Transformations, American Mathematical Society, Providence, RI, 2001.
[8] Cole, J. D., Quart. Appl. Math., 9, 225-236 (1951) · Zbl 0043.09902
[9] Draganescu, Gh. E.; Capalnasan, V., Internat. J. Nonlinear Sci. Numer. Simulation, 4, 219-226 (2004)
[10] Esipov, S. E., Phys. Rev. E, 52, 3711-3718 (1995)
[11] Fan, E., Phys. Lett. A, 282, 18 (2001) · Zbl 0984.37092
[12] Fan, E. G.; Zhang, H. Q., Phys. Lett. A, 246, 403 (1998) · Zbl 1125.35308
[13] Gardner, C. S.; Green, J. M.; Kruskal, M. D.; Miura, R. M., Phys. Rev. Lett., 19, 1095 (1967) · Zbl 1061.35520
[14] He, J. H., Comm. Nonlinear Sci. Numer. Simulation, 2, 4, 230-235 (1997)
[15] He, J. H., Comput. Methods Appl. Mech. Eng., 167, 57-68 (1998) · Zbl 0942.76077
[16] He, J. H., Comput. Methods Appl. Mech. Eng., 167, 69-73 (1998) · Zbl 0932.65143
[17] He, J. H., Internat. J. Non-linear Mech., 34, 699-708 (1999) · Zbl 1342.34005
[18] He, J. H., Appl. Math. Comput., 114, 2,3, 115-123 (2000) · Zbl 1027.34009
[19] He, J. H., Approximate Analytical Methods in Science and Engineering (2002), Henan Sci. & Tech. Press: Henan Sci. & Tech. Press Zhengzhou, (in Chinese) · Zbl 1021.34001
[20] He, J. H., Generalized Variational Principles in Fluids (2003), Science & Culture Publishing House of China: Science & Culture Publishing House of China Hong Kong, (in Chinese) · Zbl 1054.76001
[21] Herbst, B. M.; Schoombie, S. W.; Mitchell, A. R., Internat. J. Numer. Methods Eng., 18, 1321-1336 (1982) · Zbl 0485.65093
[22] Hirota, R., Phys. Rev. Lett., 27, 1192 (1971) · Zbl 1168.35423
[23] Hirota, R.; Satsuma, J., Phys. Lett. A, 85, 407 (1981)
[24] Hopf, E., The partial differential equation, Comm. Pure Appl. Math., 3, 201-230 (1950) · Zbl 0039.10403
[25] Kaya, D., Internat. J. Math. Math. Sci., 27, 675 (2001) · Zbl 0997.35077
[26] Kaya, D., Appl. Math. Comput., 144, 353-363 (2003) · Zbl 1024.65096
[27] Malfeit, W., Amer. J. Phys., 60, 650 (1992)
[28] Malfliet, W., Amer. J. Phys., 60, 650 (1992) · Zbl 1219.35246
[29] Marinca, V., Internat. J. Nonlinear Sci. Numer. Simulation, 3, 107-120 (2002) · Zbl 1079.34028
[30] Nee, J.; Duan, J., Appl. Math. Lett., 11, 1, 57-61 (1998) · Zbl 1076.35537
[31] S.G. Rubin, R.A. Graves, Computers and Fluids, vol. 3, Pergamon Press, Oxford, 1975, p. 136.; S.G. Rubin, R.A. Graves, Computers and Fluids, vol. 3, Pergamon Press, Oxford, 1975, p. 136.
[32] Satsuma, J.; Hirota, R., J. Phys. Soc. Japan, 51, 332 (1982)
[33] A.A. Soliman, International Conference on Computational Fluid Dynamics, Beijing, China, October 17-20, 2000, pp. 559-566.; A.A. Soliman, International Conference on Computational Fluid Dynamics, Beijing, China, October 17-20, 2000, pp. 559-566.
[34] Wadati, M.; Sanuki, H.; Konno, K., Prog. Theor. Phys., 53, 419 (1975) · Zbl 1079.35506
[35] Wang, M. L., Phys. Lett. A, 215, 279 (1996)
[36] Wazwaz, A. M., Appl. Math. Comput., 111, 53 (2000) · Zbl 1023.65108
[37] Wazwaz, A. M., Comput. Math. Appl., 4, 1237-1244 (2001) · Zbl 0983.65090
[38] Wazwaz, A. M., Chaos Solitons Fractical, 12, 2283 (2001) · Zbl 0992.35092
[39] Wu, Y. T.; Geng, X. G.; Hu, X. B.; Zhu, S. M., Phys. Lett. A, 255, 259 (1999) · Zbl 0935.37029
[40] Yan, C. T., Phys. Lett. A, 224, 77 (1996) · Zbl 1037.35504
[41] Yan, Z. Y.; Zhang, H. Q., Appl. Math. Mech., 21, 382 (2000)
[42] Yan, Z. Y.; Zhang, H. Q., J. Phys. A, 34, 1785 (2001) · Zbl 0970.35147
[43] Yan, Z. Y.; Zhang, H. Q., Phys. Lett. A, 285, 355 (2001) · Zbl 0969.76518
[44] Yan, Z. Y., Phys. Lett. A, 292, 100 (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.