×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[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)
[3] Abbasbandy, S., Phys. Lett. A, 361, 478 (2007) · Zbl 1273.65156
[4] Allan, F. M., Appl. Math. Comput., 190, 6 (2007)
[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)
[8] Abbasbandy, S.; Angew, Z., Math. Phys. (ZAMP), 59, 51 (2008)
[9] Liao, S. J., Int. J. Heat Mass Transfer, 48, 2529 (2005)
[10] Liao, S. J., Stud. Appl. Math., 117, 239 (2006)
[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)
[14] Wu, W.; Liao, S. J., Chaos, Solitons Fractals, 26, 177 (2005)
[15] Hayat, T.; Sajid, M., Phys. Lett. A, 361, 316 (2007) · Zbl 1170.76307
[16] Hayat, T.; Khan, M., Nonlinear Dynam., 42, 395 (2005)
[17] Hayat, T.; Khan, M.; Ayub, M.; Angew, Z., Math. Phys. (ZAMP), 56, 1012 (2005)
[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)
[20] Wang, C., Heat Mass. Transfer., 42, 759 (2006)
[21] Abbasbandy, S., Appl. Math. Model., 32, 2706 (2008)
[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)
[24] Kawahara, T., J. Phys. Soc. Japan, 33, 260 (1972)
[25] Sirendaoreji, S. J., Chaos, Solitons Fractals, 19, 147 (2004)
[26] Kaya, D.; Al-Khaled, K., Phys. Lett. A, 363, 433 (2007)
[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
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.