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 variational iteration method for studying the Klein-Gordon equation. (English) Zbl 1152.65475
Summary: We use He’s variational iteration method for solving linear and nonlinear Klein-Gordon equations. Also, the results are compared with those obtained by Adomian’s decomposition method (ADM). The results reveal that the method is very effective and simple.

MSC:
65M99Numerical methods for IVP of PDE
WorldCat.org
Full Text: DOI
References:
[1] Ablowitz, M. J.; Clarkson, P. A.: Solitons, nonlinear evolution equations and inverse scattering transform. (1990) · Zbl 0762.35001
[2] He, J. H.: A new approach to nonlinear partial differential equations. Comm. nonlinear sci. Numer. simul. 2, 230-235 (1997)
[3] He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. methods appl. Mech. engrg. 167, 57-68 (1998) · Zbl 0942.76077
[4] He, J. H.: Approximate solution of nonlinear differential equations with convolution product nonlinearities. Comput. methods appl. Mech. engrg. 167, 69-73 (1998) · Zbl 0932.65143
[5] He, J. H.: Variational iteration method--A kind of non-linear analytical technique: some examples. Internat. J. Non-linear mech. 34, 699-708 (1999) · Zbl 05137891
[6] He, J. H.: Variational iteration method for autonomous ordinary differential systems. Appl. math. Comput. 114, 115-123 (2000) · Zbl 1027.34009
[7] He, J. H.: Variational principle for some nonlinear partial differential equations with variable coefficients. Chaos solitons fractals 19, 847-851 (2004) · Zbl 1135.35303
[8] Abdou, M. A.; Soliman, A. A.: New applications of variational iteration method. Physica D 211, 1-8 (2005) · Zbl 1084.35539
[9] Abdou, M. A.; Soliman, A. A.: Variational iteration method for solving Burgers and coupled Burgers equations. J. comput. Appl. math. 181, 245-251 (2005) · Zbl 1072.65127
[10] Momani, S.; Abuasad, S.: Application of he’s variational iteration method to Helmholtz equation. Chaos solitons fractals 27, 1119-1123 (2006) · Zbl 1086.65113
[11] Odibat, Z. M.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Internat. J. Nonlinear sci. Numer. simul 7, 27-34 (2006) · Zbl 05675858
[12] He, J. H.: Some asymptotic methods for strongly nonlinear equations. Internat. J. Modern phys. B 20, 1141-1199 (2006) · Zbl 1102.34039
[13] Abulwafa, E. M.; Abdou, M. A.; Mahmoud, A. A.: The solution of nonlinear coagulation problem with mass loss. Chaos solitons fractals 26, 313-330 (2006) · Zbl 1101.82018
[14] Bildik, N.; Konuralp, A.: The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Internat. J. Nonlinear sci. Numer. simul. 7, 65-70 (2006) · Zbl 1115.65365
[15] Dodd, R. K.; Eilbeck, I. C.; Gibbon, J. D.; Morris, H. C.: Solitons and nonlinear wave equations. (1982) · Zbl 0496.35001
[16] Khaliq, A. Q.; Abukhodair, B.; Sheng, Q.; Ismail, M. S.: A predictor--corrector scheme for sine--Gordon equation. Numer. methods partial differential equations 16, 133-146 (2000) · Zbl 0951.65089
[17] Lynch, M. A. M.: Large amplitude instability in finite difference approximations to the Klein--Gordon equation. Appl. numer. Math. 31, 173-182 (1999) · Zbl 0937.65098
[18] Lu, X.; Schmid, R.: Symplectic integration of sine--Gordon type systems. Math. comput. Simulation 50, 255-263 (1999)
[19] Kaya, D.: An implementation of the ADM for generalized one-dimensional Klein--Gordon equation. Appl. math. Comput. 166, 426-433 (2005) · Zbl 1074.65118
[20] Kaya, D.; El-Sayed, S. M.: A numerical solution of the Klein--Gordon equation and convergence of the decomposition method. Appl. math. Comput. 156, 341-353 (2004) · Zbl 1084.65101
[21] El-Sayed, S. M.: The decomposition method for studying the Klein--Gordon equation. Chaos solitons fractals 18, 1025-1030 (2003) · Zbl 1068.35069
[22] Wazwaz, A. M.: The modified decomposition method for analytic treatment of differential equations. Appl. math. Comput. 173, 165-176 (2006) · Zbl 1089.65112
[23] Adomian, G.: Solving frontier problems of physics: the decomposition methods. (1994) · Zbl 0802.65122
[24] Wazwaz, A. M.: A reliable modification of Adomian decomposition method. Appl. math. Comput. 102, 77-86 (1997) · Zbl 0928.65083
[25] Wazwaz, A. M.: A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl. math. Comput. 111, 53-69 (2000) · Zbl 1023.65108
[26] Zhao, Xueqin; Zhi, Hongyan; Yu, Yaxuan; Zhang, Hongqing: A new Riccati equation expansion method with symbolic computation to construct new travelling wave solution of nonlinear differential equations. Appl. math. Comput. 172, 24-39 (2006) · Zbl 1088.65095