zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
The homotopy analysis method for solving the fornberg-whitham equation and comparison with Adomian’s decomposition method. (English) Zbl 1193.65179
Summary: An analytical technique, namely the homotopy analysis method (HAM), is applied to obtain an approximate analytical solution of the Fornberg-Whitham equation. A comparison is made between the HAM results and the Adomian’s decomposition method (ADM) and the homotopy perturbation method (HPM). The results reveal that HAM is very simple and effective. The HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of solution series.
MSC:
65M99Numerical methods for IVP of PDE
References:
[1]Zhou, J.; Tian, L.: A type of bounded traveling wave solutions for the fornberg–Whitham equation, J. math. Anal. appl. 346, 255-261 (2008) · Zbl 1146.35025 · doi:10.1016/j.jmaa.2008.05.055
[2]Whitham, G. B.: Variational methods and applications to water wave, Proc. R. Soc. lond. A 299, 6-25 (1967) · Zbl 0163.21104 · doi:10.1098/rspa.1967.0119
[3]Fornberg, B.; Whitham, G. B.: A numerical and theoretical study of certain nonlinear wave phenomena, Phil. trans. R. soc. Lond. A 289, 373-404 (1978) · Zbl 0384.65049 · doi:10.1098/rsta.1978.0064
[4]Adomian, G.: A review of the decomposition method in applied mathematics, J. math. Anal. appl. 135, 44-501 (1988) · Zbl 0671.34053 · doi:10.1016/0022-247X(88)90170-9
[5]Adomian, G.: Solving frontier problems of physics: the decomposition method, (1999)
[6]Cherruault, Y.: Convergence of Adomian’s method, Kybernetes 18, No. 2, 8-31 (1989) · Zbl 0697.65051 · doi:10.1108/eb005812
[7]Cherruault, Y.; Adomian, G.: Decomposition methods: A new proof of convergence, Math. comput. Modelling 18, 103-106 (1993) · Zbl 0805.65057 · doi:10.1016/0895-7177(93)90233-O
[8]Abbaoui, K.; Cherruault, Y.: Convergence of Adomian’s method applied to differential equations, Comput. math. Appl. 28, 8-31 (1994) · Zbl 0809.65073 · doi:10.1016/0898-1221(94)00144-8
[9]Mavoungou, T.; Cherruault, Y.: Convergence of Adomian’s method and applications to non-linear partial differential equation, Kybernetes 21, No. 6, 13-25 (1992) · Zbl 0801.35007 · doi:10.1108/eb005942
[10]Wazwaz, A. M.: A reliable modification of Adomian decomposition method, Appl. math. Comput. 102, 77-86 (1999) · Zbl 0928.65083 · doi:10.1016/S0096-3003(98)10024-3
[11]A.M. Wazwaz, The existence of noise terms for systems of inhomogeneous differential and integral equations. 146 (1) (2003) pp. 81–92. · Zbl 1032.65114 · doi:10.1016/S0096-3003(02)00527-1
[12]Wazwaz, A. M.: A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. math. Comput. 111, 53-69 (2000) · Zbl 1023.65108 · doi:10.1016/S0096-3003(99)00047-8
[13]Achouri, T.; Omrani, K.: Numerical solutions for the damped generalized regularized long-wave equation with a variable coefficient by Adomian decomposition method, Commun. nonlinear sci. Numer. simul. 14, 869-877 (2009) · Zbl 1221.65270 · doi:10.1016/j.cnsns.2008.07.011
[14]Kaya, D.; El-Sayed, S. M.: An application of the decomposition method for the generalized KdV and RLW equations, Chaos solitons fractals 17, 869-877 (2003) · Zbl 1030.35139 · doi:10.1016/S0960-0779(02)00569-6
[15]Ray, S. Saha; Chaudhuri, K. S.; Bera, R. K.: Application of modified decomposition method for the analytical solution of space fractional diffusion equation, Appl. math. Comput. 196, 2025-2033 (2008) · Zbl 1133.65119 · doi:10.1016/j.amc.2007.05.048
[16]S.J. Liao, The proposed homotopy analysis method techniques for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.
[17]Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[18]Liao, S. J.: An approximate solution technique which does not depend upon small parameters: A special example, Internat. J. Non-linear mech. 30, 371-380 (1995) · Zbl 0837.76073 · doi:10.1016/0020-7462(94)00054-E
[19]Liao, S. J.: On the homotopy analysis method for nonlinear problems, Appl. math. Comput. 147, 499-513 (2004) · Zbl 1086.35005 · doi:10.1016/S0096-3003(02)00790-7
[20]Liao, S. J.: Comparison between the homotopy analysis method and the homotopy perturbation method, Appl. math. Comput. 169, 1186-1194 (2005) · Zbl 1082.65534 · doi:10.1016/j.amc.2004.10.058
[21]Tari, H.; Ganji, D. D.; Rostamian, M.: Approximate solutions of K(2,2), KdV and the modified KdV equations by hes methods and liao’s method, Int. J. Nonlinear sci. Numer. simul. 8, No. 2 (2007)
[22]Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. lett. A 360, 109-113 (2006)
[23]Abbasbandy, S.: The application of homotopy analysis method to solve a generalized Hirota satsuma coupled KdV equation, Phys. lett. A 36, 478-483 (2007)
[24]Fakhari, A.; Domairry, G.; Ebrahimpour: Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with exact solution, Phys. lett. A 368, 64-68 (2007) · Zbl 1209.65109 · doi:10.1016/j.physleta.2007.03.062
[25]Bataineh, A. Sami; Nourani, M. S. M.; Hashim, I.: Solving systems of odes by homotopy analysis method, Commun. nonlinear sci. Numer. simul., 5-26 (2007)
[26]Ayab, M.; Rasheed, A.; Hayat, T.: Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Internat. J. Engrg. sci. 41, 2091-2103 (2003) · Zbl 1211.76076 · doi:10.1016/S0020-7225(03)00207-6
[27]Hayat, T.; Khan, M.: Homotopy solutions for a generalized second-grade fluid past a porous plate, Nonlinear dynam. 42, 395-405 (2005) · Zbl 1094.76005 · doi:10.1007/s11071-005-7346-z
[28]Rashidi, M. M.; Domairry, G.; Dinarvand, S.: The homotopy analysis method for explicit analytical solutions of Jaulent–Miodek equations, Numer. methods partial differential equations 25, No. 2, 430-439 (2009) · Zbl 1168.35428 · doi:10.1002/num.20358
[29]Yildirim, A.; Koçak, H.: Homotopy perturbation method for solving the space–time fractional advection–dispersion equation, Adv. water resour. 32, 1711-1716 (2009)
[30]Berberler, M. E.; Yildirim, A.: He’s homotopy perturbation method for solving the shock wave equation, Appl. anal. 88, 997-1004 (2009) · Zbl 1172.76042 · doi:10.1080/00036810903114767
[31]Koçak, H.; Yildirim, A.: Series solution for a delay differential equation arising in electrodynamics, Comm. numer. Methods engrg. 25, 1084-1096 (2009) · Zbl 1177.78059 · doi:10.1002/cnm.1288
[32]He, J. H.: Homotopy perturbation technique, Comput. methods appl. Mech. engrg. 178, 257-262 (1999)