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The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics. (English) Zbl 1255.37022

Summary: By introducing a new general ansatz, the improved fractional sub-equation method is proposed to construct analytical solutions of nonlinear evolution equations involving Jumarie’s modified Riemann-Liouville derivative. By means of this method, the space-time fractional Whitham-Broer-Kaup and generalized Hirota-Satsuma coupled KdV equations are successfully solved. The obtained results show that the proposed method is quite effective, promising and convenient for solving nonlinear fractional differential equations.

MSC:

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
35R11 Fractional partial differential equations
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References:

[1] Diethelm, K., The Analysis of Fractional Differential Equations (2010), Springer-Verlag: Springer-Verlag Berlin
[2] Li, C.; Chen, A.; Ye, J., J. Comput. Phys., 230, 3352 (2011)
[3] Odibat, Z.; Momani, S., Comput. Math. Appl., 58, 2199 (2009)
[4] He, J. H., Bull. Sci. Technol., 15, 86 (1999)
[5] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press California · Zbl 0918.34010
[6] Yang, X., Prog. Nonlinear Sci., 4, 1 (2011)
[7] Cui, M., J. Comput. Phys., 228, 7792 (2009)
[8] Huang, Q.; Huang, G.; Zhan, H., Adv. Water Resour., 31, 1578 (2008)
[9] El-Sayed, A. M.A.; Gaber, M., Phys. Lett. A, 359, 175 (2006)
[10] El-Sayed, A. M.A.; Behiry, S. H.; Raslan, W. E., Comput. Math. Appl., 59, 1795 (2010)
[11] Odibat, Z.; Momani, S., Appl. Math. Lett., 21, 194 (2008)
[12] He, J. H., Commun. Nonlinear Sci. Numer. Simul., 2, 235 (1997)
[13] Wu, G.; Lee, E. W.M., Phys. Lett. A, 374, 2506 (2010)
[14] Guo, S.; Mei, L., Phys. Lett. A, 375, 309 (2011)
[15] He, J. H., Comput. Methods Appl. Mech. Engrg., 178, 257 (1999)
[16] He, J. H., Internat. J. Non-Linear Mech., 35, 37 (2000)
[17] Mophou, G. M., Nonlinear Anal., 72, 1604 (2010)
[18] Huang, Q.; Huang, G.; Zhan, H., Adv. Water Resour., 31, 1578 (2008)
[19] Jiang, W.; Lin, Y., Comput. Phys. Commun., 181, 557 (2010)
[20] Pandey, R. K.; Singh, O. P.; Baranwal, V. K., Comput. Phys. Commun., 182, 1134 (2011)
[21] Xue, C.; Nie, J.; Tan, W., Nonlinear Anal., 69, 2086 (2008)
[22] Molliq, R. Y.; Noorani, M. S.M.; Hashim, I., Nonlinear Anal., 10, 1854 (2009)
[23] He, J. H.; Wu, X., Chaos Solitons Fractals, 30, 700 (2006)
[24] Zhang, S.; Zong, Q. A.; Liu, D.; Gao, Q., Commun. Fract. Calc., 1, 48 (2010)
[25] Zhang, S., Phys. Lett. A, 371, 65 (2007)
[26] Bekir, A.; Boz, A., Phys. Lett. A, 372, 1619 (2008)
[27] Wu, X. H.; He, J. H., Comput. Math. Appl., 54, 966 (2007)
[28] Khani, F.; Hamedi-Nezhad, S., Comput. Math. Appl., 58, 2325 (2009)
[29] Zhang, S., Phys. Lett. A, 372, 1873 (2008)
[30] Zhang, S., Appl. Math. Comput., 199, 242 (2008)
[31] Ganji, D. D.; Kachapi, S., Prog. Nonlinear Sci., 3, 1 (2011)
[32] Zhang, S.; Zhang, H., Phys. Lett. A, 375, 1069 (2011)
[33] Wang, M., Phys. Lett. A, 199, 169 (1995)
[34] Jumarie, G., Comput. Math. Appl., 51, 1367 (2006)
[35] Jumarie, G., J. Appl. Math. Comput., 24, 31 (2007)
[36] Jumarie, G., Appl. Math. Lett., 23, 1444 (2010)
[37] Xu, T.; Li, J.; Zhang, H.; Zhang, Y.; Yao, Z.; Tian, B., Phys. Lett. A, 369, 458 (2007)
[38] Ping, Z., Appl. Math. Comput., 217, 1688 (2010)
[39] Kupershmidt, B. A., Commun. Math. Phys., 99, 51 (1985)
[40] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0762.35001
[41] Wu, Y.; Geng, X., Phys. Lett. A, 255, 259 (1999)
[42] Abazari, R.; Abazari, M., Commun. Nonlinear Sci. Numer. Simul., 17, 619 (2012)
[43] Ganji, Z. Z.; Ganji, D. D.; Rostamiyan, Y., Appl. Math. Model, 33, 3107 (2009)
[44] Zhou, Y.; Wang, M.; Wang, Y., Phys. Lett. A, 308, 31 (2003)
[45] He, J. H., Phys. Lett. A, 375, 3362 (2011)
[46] Abdou, M. A., Chaos Solitons Fractals, 31, 95 (2007)
[47] El-Wakil, S. A.; Abdou, M. A., Chaos Solitons Fractals, 31, 840 (2007)
[48] El-Wakil, S. A.; Abdou, M. A., Phys. Lett. A, 358, 275 (2006)
[49] Rosenau, P.; Hyman, J. M., Phys. Rev. Lett., 70, 564 (1993)
[50] Ma, W., Phys. Lett. A, 301, 35 (2002)
[51] Odibat, Z., Phys. Lett. A, 372, 1219 (2008)
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