Solitary wave solutions to the Kuramoto-Sivashinsky equation by means of the homotopy analysis method. (English) Zbl 1173.35646

Summary: The homotopy analysis method is used to find a family of solitary solutions of the Kuramoto-Sivashinsky 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.


35Q53 KdV equations (Korteweg-de Vries equations)
35C10 Series solutions to PDEs
35A35 Theoretical approximation in context of PDEs
Full Text: DOI


[1] Lyapunov, A.M.: General Problem on Stability of Motion. Taylor and Francis, London (1992) · Zbl 0786.70001
[2] Karmishin, A.V., Zhukov, A.I., Kolosov, V.G.: Methods of Dynamics Calculation and Testing for Thin-Walled Structures. Mashinostroyenie, Moscow (1990)
[3] Abbasbandy, S.: Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method. Appl. Math. Comput. 172, 485–490 (2006) · Zbl 1088.65063 · doi:10.1016/j.amc.2005.02.014
[4] Abbasbandy, S.: A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method. Chaos Solitons Fractals 31, 257–260 (2007) · doi:10.1016/j.chaos.2005.10.071
[5] Ganji, D.D., Rajabi, A.: Assessment of homotopy–perturbation and perturbation methods in heat radiation equations. Int. Commun. Heat Mass 33, 391–400 (2006) · doi:10.1016/j.icheatmasstransfer.2005.11.001
[6] Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC, Boca Raton (2003)
[7] 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 Dyn. (2007, in press) doi: 10.1007/s11071-006-9140-y · Zbl 1181.76031
[8] Allan, F.M.: Derivation of the Adomian decomposition method using the homotopy analysis method. Appl. Math. Comput. (2007, in press) · Zbl 1125.65063
[9] Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006) · Zbl 1236.80010 · doi:10.1016/j.physleta.2006.07.065
[10] Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation. Phys. Lett. A 361, 478–483 (2007) · Zbl 1273.65156 · doi:10.1016/j.physleta.2006.09.105
[11] Abbasbandy, S.: Homotopy analysis method for heat radiation equations. Int. Commun. Heat Mass 34, 380–387 (2007) · doi:10.1016/j.icheatmasstransfer.2006.12.001
[12] Abbasbandy, S., Samadian Zakaria, F.: Soliton solutions for the fifth-order KdV equation with the homotopy analysis method. Nonlinear Dyn. (2007, in press) doi: 10.1007/s11071-006-9193-y · Zbl 1170.76317
[13] Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Transfer 48, 2529–39 (2005) · Zbl 1189.76142 · doi:10.1016/j.ijheatmasstransfer.2005.01.005
[14] Liao, S.J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate. Stud. Appl. Math. 117, 239–264 (2006) · Zbl 1145.76352 · doi:10.1111/j.1467-9590.2006.00354.x
[15] Liao, S.J., Magyari, E.: Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones. Z. Angew. Math. Phys. 57, 777–792 (2006) · Zbl 1101.76056 · doi:10.1007/s00033-006-0061-x
[16] Liao, S.J., Su, J., Chwang, A.T.: Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body. Int. J. Heat Mass Transfer 49, 2437–2445 (2006) · Zbl 1189.76549 · doi:10.1016/j.ijheatmasstransfer.2006.01.030
[17] Tan, Y., Xu, H., Liao, S.J.: Explicit series solution of traveling waves with a front of Fisher equation. Chaos Solitons Fractals 31, 462–472 (2007) · Zbl 1143.35313 · doi:10.1016/j.chaos.2005.10.001
[18] Wu, W., Liao, S.J.: Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos Solitons Fractals 26, 177–185 (2005) · Zbl 1071.76009 · doi:10.1016/j.chaos.2004.12.016
[19] Hayat, T., Sajid, M.: On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder. Phys. Lett. A 361, 316–322 (2007) · Zbl 1170.76307 · doi:10.1016/j.physleta.2006.09.060
[20] Hayat, T., Khan, M.: Homotopy solutions for a generalized second-grade fluid past a porous plate. Nonlinear Dyn. 42, 395–405 (2005) · Zbl 1094.76005 · doi:10.1007/s11071-005-7346-z
[21] Hayat, T., Khan, M., Ayub, M.: On non-linear flows with slip boundary condition. Z. Angew. Math. Phys. 56, 1012–1029 (2005) · Zbl 1097.76007 · doi:10.1007/s00033-005-4006-6
[22] Sajid, M., Hayat, T., Asghar, S.: On the analytic solution of the steady flow of a fourth grade fluid. Phys. Lett. A 355, 18–26 (2006) · doi:10.1016/j.physleta.2006.01.092
[23] Tan, Y., Abbasbandy, S.: Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simul. (2007, in press) · Zbl 1132.34305
[24] Wang, C.: Analytic solutions for a liquid film on an unsteady stretching surface. Heat Mass Transfer 42, 759–766 (2006) · doi:10.1007/s00231-005-0027-0
[25] Abbas, Z., Sajid, M., Hayat, T.: MHD boundary layer flow of an upper-convected Maxwell fluid in a porous channel. Theor. Comput. Fluid Dyn. 20, 229–238 (2006) · Zbl 1109.76065 · doi:10.1007/s00162-006-0025-y
[26] Hayat, T., Abbas, Z., Sajid, M.: Series solution for the upper-convected Maxwell fluid over a porous stretching plate. Phys. Lett. A 358, 396–403 (2006) · Zbl 1142.76511 · doi:10.1016/j.physleta.2006.04.117
[27] Hayat, T., Ellahi, R., Ariel, P.D., Asghar, S.: Homotopy solution for the channel flow of a third grade fluid. Nonlinear Dyn. 45, 55–64 (2006) · Zbl 1100.76005 · doi:10.1007/s11071-005-9015-7
[28] Hayat, T., Sajid, M.: Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet. Int. J. Heat Mass Transfer 50, 75–84 (2007) · Zbl 1104.80006 · doi:10.1016/j.ijheatmasstransfer.2006.06.045
[29] Hayata, T., Abbas, Z., Sajid, M., Asghar, S.: The influence of thermal radiation on MHD flow of a second grade fluid. Int. J. Heat Mass Transfer 50, 931–941 (2007) · Zbl 1124.80325 · doi:10.1016/j.ijheatmasstransfer.2006.08.014
[30] Wazwaz, A.M.: New solitary wave solutions to the Kuramoto–Sivashinsky and the Kawahara equations. Appl. Math. Comput. 182, 1642–1650 (2006) · Zbl 1107.65094 · doi:10.1016/j.amc.2006.06.002
[31] Conte, R.: Exact Solutions of Nonlinear Partial Differential Equations by Singularity Analysis. Lecture Notes in Physics. Springer, New York (2003) · Zbl 1060.35001
[32] Rademacher, J., Wattenberg, R.: Viscous shocks in the destabilized Kuramoto–Sivashinsky. J. Comput. Nonlinear Dyn. 1, 336–347 (2007) · doi:10.1115/1.2338656
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