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Homotopy analysis method for the Kawahara equation. (English) Zbl 1181.35224

Summary: The homotopy analysis method (HAM) is used to find a family of travelling-wave solutions of the Kawahara equation. This approximate solution, which is obtained as a series of exponentials, has a reasonable residual error. The homotopy analysis method contains the auxiliary parameter \(\hbar\), which provides us with a simple way to adjust and control the convergence region of series solution. This method is reliable and manageable.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C07 Traveling wave solutions
35C10 Series solutions to PDEs
35A20 Analyticity in context of PDEs
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[1] Liao, S. J., Beyond Perturbation: Introduction to the Homotopy Analysis Method (2003), Chapman & Hall/CRC Press: Chapman & Hall/CRC Press Boca Raton
[2] Abbasbandy, S., Phys. Lett. A, 360, 109 (2006) · Zbl 1236.80010
[3] Abbasbandy, S., Phys. Lett. A, 361, 478 (2007) · Zbl 1273.65156
[4] Allan, F. M., Appl. Math. Comput., 190, 6 (2007) · Zbl 1125.65063
[5] Sajid, M.; Hayat, T.; Asghar, S., Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt, Nonlinear Dynam., 50, 27 (2007) · Zbl 1181.76031
[6] Abbasbandy, S., Int. Commun. Heat Mass, 34, 380 (2007)
[7] Abbasbandy, S.; Samadian Zakaria, F., Nonlinear Dynam., 51, 83 (2008) · Zbl 1170.76317
[8] Abbasbandy, S.; Angew, Z., Math. Phys. (ZAMP), 59, 51 (2008) · Zbl 1139.35325
[9] Liao, S. J., Int. J. Heat Mass Transfer, 48, 2529 (2005) · Zbl 1189.76142
[10] Liao, S. J., Stud. Appl. Math., 117, 239 (2006) · Zbl 1145.76352
[11] Liao, S. J.; Magyari, E.; Angew, Z., Math. Phys. (ZAMP), 57, 777 (2006)
[12] Liao, S. J.; Su, J.; Chwang, A. T., Int. J. Heat Mass Transfer, 49, 2437 (2006)
[13] Tan, Y.; Xu, H.; Liao, S. J., Chaos, Solitons Fractals, 31, 462 (2007) · Zbl 1143.35313
[14] Wu, W.; Liao, S. J., Chaos, Solitons Fractals, 26, 177 (2005) · Zbl 1071.76009
[15] Hayat, T.; Sajid, M., Phys. Lett. A, 361, 316 (2007) · Zbl 1170.76307
[16] Hayat, T.; Khan, M., Nonlinear Dynam., 42, 395 (2005) · Zbl 1094.76005
[17] Hayat, T.; Khan, M.; Ayub, M.; Angew, Z., Math. Phys. (ZAMP), 56, 1012 (2005) · Zbl 1097.76007
[18] Sajid, M.; Hayat, T.; Asghar, S., Phys. Lett. A, 355, 18 (2006)
[19] Tan, Y.; Abbasbandy, S., Commun. Nonlinear Sci. Numer. Simul., 13, 539 (2008) · Zbl 1132.34305
[20] Wang, C., Heat Mass. Transfer., 42, 759 (2006)
[21] Abbasbandy, S., Appl. Math. Model., 32, 2706 (2008) · Zbl 1167.35395
[22] Hayat, T.; Sajid, M.; Pop, I., Three-dimensional flow over a stretching surface in a viscoelastic fluid, Nonlinear Anal. RWA, 9, 1811 (2008) · Zbl 1154.76315
[23] Wazwaz, A. M., Appl. Math. Comput., 182, 1642 (2006) · Zbl 1107.65094
[24] Kawahara, T., J. Phys. Soc. Japan, 33, 260 (1972)
[25] Sirendaoreji, S. J., Chaos, Solitons Fractals, 19, 147 (2004) · Zbl 1068.35141
[26] Kaya, D.; Al-Khaled, K., Phys. Lett. A, 363, 433 (2007) · Zbl 1197.65201
[27] Yusufoglu, E.; Bekir, A.; Alp, M., Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using Sine-Cosine method, Chaos, Solitons Fractals, 37, 1193 (2008) · Zbl 1148.35351
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