×

zbMATH — the first resource for mathematics

Numerical solution of the generalized Zakharov equation by homotopy analysis method. (English) Zbl 1221.65269
Summary: The homotopy analysis method (HAM) is applied to obtain approximations to the analytic solution of the generalized Zakharov equation. The HAM contains the auxiliary parameter \(\hbar\), which provides us with a simple way to adjust and control the convergence region of the solution series.

MSC:
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35Q55 NLS equations (nonlinear Schrödinger equations)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Shang Y, Huang Y, Yuan W. The extended hyperbolic functions method and new exact solutions to the Zakharov equations. Appl Math Comput. doi:10.1016/j.amc.2007.10.059. · Zbl 1143.65083
[2] Malomed, B.; Anderson, D.; Lisak, M.; Quiroga-Teixeiro, M.L.; Stenflo, L., Dynamics of solitary waves in the Zakharov model equations, Phys rev E, 55, 962-968, (1997)
[3] Zakharov, V.E., Collapse of Langmuir waves, Zh eksp teor fiz, 62, 1745-1751, (1972)
[4] Golman, M.V., Langmuir wave solitons and collapse in plasma physics, Physica D, 18, 67-76, (1986) · Zbl 0613.76129
[5] Nicolson, D.R., Introduction to plasma theory, (1983), Wiley New York
[6] Wang, M.; Li, X., Extended F-expansion method and periodic wave solutions for the generalized Zakharov equations, Phys lett A, 343, 48-54, (2005) · Zbl 1181.35255
[7] Zhang, H., New exact travelling wave solutions of the generalized Zakharov equations, Reports math phys, 60, 97-106, (2007) · Zbl 1170.35524
[8] Wang, Y.Y.; Dai, C.Q.; Wu, L.; Zhang, J.F., Exact and numerical solitary wave solutions of generalized Zakharov equation by the Adomian decomposition method, Chaos solitons & fractals, 32, 1208-1214, (2007) · Zbl 1130.35120
[9] Zhang, J., Variational approach to solitary wave solution of the generalized Zakharov equation, Comput math appl, 54, 1043-1046, (2007) · Zbl 1141.65391
[10] Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis. Shanghai Jiao Tong University; 1992.
[11] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall CRC Press, Boca Raton
[12] 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
[13] Abbasbandy, S., Homotopy analysis method for heat radiation equations, Int commun heat mass transf, 34, 380-387, (2007)
[14] Abbasbandy, S.; Samadian Zakaria, F., Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear dynam, 51, 83-87, (2008) · Zbl 1170.76317
[15] Abbasbandy, S., Homotopy analysis method for generalized benjamin – bona – mahony equation, Z angew math phys (ZAMP), 59, 51-62, (2008) · Zbl 1139.35325
[16] Abbasbandy, S.; Tan, Y.; Liao, S.J., Newton-homotopy analysis method for nonlinear equations, Appl math comput, 188, 1794-1800, (2007) · Zbl 1119.65032
[17] Abbasbandy, S., Soliton solutions for the fitzhugh – nagumo equation with the homotopy analysis method, Appl math model, 32, 2706-2714, (2008) · Zbl 1167.35395
[18] 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
[19] Liao, S.J.; Magyari, E., Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Z angew math phys (ZAMP), 57, 777-792, (2006) · Zbl 1101.76056
[20] 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 transf, 49, 2437-2445, (2006) · Zbl 1189.76549
[21] Tan, Y.; Xu, H.; Liao, S.J., Explicit series solution of travelling waves with a front of Fisher equation, Chaos solitons & fractals, 31, 462-472, (2007) · Zbl 1143.35313
[22] Abbasbandy, S.; Liao, S.J., A new modification of false position method based on the homotopy analysis method, Appl math mech eng, 29, 223-228, (2008) · Zbl 1231.65084
[23] Sajid, M.; Siddiqui, A.M.; Hayat, T., Wire coating analysis using MHD Oldroyd 8-constant fluid, Int J eng sci, 45, 381-392, (2007)
[24] Hayat, T.; Sajid, M., Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid, Int J eng sci, 45, 393-401, (2007) · Zbl 1213.76137
[25] 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)
[26] Tan, Y.; Abbasbandy, S., Homotopy analysis method for quadratic Riccati differential equation, Commun nonlinear sci numer simul, 13, 539-546, (2008) · Zbl 1132.34305
[27] Abbasbandy, S., Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chem eng J, 136, 144-150, (2008)
[28] Abbasbandy, S.; Parkes, E.J., Solitary smooth-hump solutions of the camassa – holm equation by means of the homotopy analysis method, Chaos solitons & fractals, 36, 581-591, (2008) · Zbl 1139.76013
[29] Wang, J.; Chen, J.K.; Liao, S.J., An explicit solution of the large deformation of a cantilever beam under point load at the free tip, J comput appl math, 212, 320-330, (2008) · Zbl 1128.74026
[30] Hayat, T.; Abbas, Z.; Ali, N., MHD flow and mass transfer of an upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species, Phys lett A, 372, 4698-4704, (2008) · Zbl 1221.76031
[31] Sajid, M.; Hayat, T., The application of homotopy analysis method to the film flows of a third order fluid, Chaos solitons & fractals, 38, 506-515, (2008) · Zbl 1146.76588
[32] Hayat, T.; Ahmed, N.; Sajid, M., Analytic solution for MHD flow of third order fluid in a porous channel, J comp appl math, 214, 572-582, (2008) · Zbl 1144.76059
[33] Hayat, T.; Noreen, S.; Sajid, M., Heat transfer analysis of the steady flow of a fourth grade fluid, Int J thermal sci, 47, 591-599, (2008)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.