zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Soliton solutions for the fifth-order KdV equation with the homotopy analysis method. (English) Zbl 1170.76317
Summary: An analytic technique, the homotopy analysis method (HAM), is applied to obtain the soliton solution of the fifth-order KdV equation. The homotopy analysis method (HAM) provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter $\hbar$, which provides us with a simple way to adjust and control the convergence region of series solution.

76B25Solitary waves (inviscid fluids)
76M45Asymptotic methods, singular perturbations (fluid mechanics)
Full Text: DOI
[1] 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
[2] 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
[3] Abbasbandy, S.: Homotopy analysis method for heat radiation equations. Int. Commun. Heat Mass (in Press) · Zbl 1236.80010
[4] Ayub, M., Rasheed, A., Hayat, T.: Exact flow of a third grade fluid past a porous plate using homotopy analysis method. Int. J. Eng. Sci. 41, 2091--2103 (2003) · Zbl 1211.76076 · doi:10.1016/S0020-7225(03)00207-6
[5] 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
[6] 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
[7] Lax, P.D.: Periodic solutions of the Korteweg--de Vries equation. Commun. Pure Appl. Math. 28, 141--188 (1975) · Zbl 0302.35008 · doi:10.1002/cpa.3160280105
[8] Liao, S.J.: The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University (1992)
[9] Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC, Boca Raton, FL (2003)
[10] Liao, S.J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech. 488, 189--212 (2003) · Zbl 1063.76671
[11] Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Transfer 48, 2529--2539 (2005) · Zbl 1189.76142 · doi:10.1016/j.ijheatmasstransfer.2005.01.005
[12] 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
[13] 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
[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] 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
[16] Tan, Y., Abbasbandy, S.: Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simul. (in press) · Zbl 1132.34305
[17] Wazwaz, A.: Solitons and periodic solutions for the fifth-order KdV equations. Appl. Math. Lett. 19, 1162--1167 (2006) · Zbl 1179.35296 · doi:10.1016/j.aml.2005.07.014
[18] Wang, C., Wu, Y., Wu, W.: Solving the nonlinear periodic wave problems with the homotopy analysis method. Wave Motion 41, 329--337 (2005) · Zbl 1189.35293 · doi:10.1016/j.wavemoti.2004.08.002