144
Views
2
CrossRef citations to date
0
Altmetric
Section B

On some modified variational iteration methods for solving the one-dimensional sine–Gordon equation

&
Pages 969-981 | Received 03 Aug 2009, Accepted 21 Apr 2010, Published online: 06 Jan 2011

References

  • Abbasbandy , S. 2007 . A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials . J. Comput. Appl. Math. , 207 : 59 – 63 .
  • Adomian , G. 1984 . Convergent series solution of nonlinear equations . J. Comput. Appl. Math. , 91 : 225 – 230 .
  • Argyris , J. , Haase , M. and Heinrich , J. C. 1991 . Finite element approximation to two-dimensional sine–Gordon solutions . Comput. Methods Appl. Mech. Eng. , 86 : 1 – 26 .
  • Barone , A. , Esposito , F. , Magee , C. J. and Scott , A. C. 1971 . Theory and applications of sine-Gordon equation . Rev. Nuovo Cimento , 1 : 227 – 267 .
  • Batiha , B. , Noorani , M. S.M. and Hashim , I. 2007 . Numerical solution of sine–Gordon equation by variational iteration method . Phys. Lett. A , 370 : 437 – 440 .
  • Bildik , N. and Konuralp , A. 2006 . Two-dimensional differential transform method, Adomian's decomposition method and variational iteration method for partial differential equations . Int. J. Comput. Math. , 83 : 973 – 987 .
  • Bratsos , A. G. 2008 . A numerical method for the one-dimensional sine-Gordon equation . Numer. Meth. Part. Diff. Equat. , 24 : 833 – 844 .
  • Cui , M. 2009 . Fourth-order compact scheme for the one-dimensional sine–Gordon equation . Numer. Methods Partial Differ. Equ. , 25 : 685 – 711 .
  • Deeba , E. Y. and Khuri , S. A. 1996 . A decomposition method for solving the nonlinear Klein–Gordon equation . J. Comput. Phys. , 124 : 442 – 448 .
  • Dehghan , M. 2004 . The use of Adomian decomposition method for solving the one-dimensional parabolic equation with non-local boundary specifications . Int. J. Comput. Math. , 81 : 25 – 34 .
  • Dehghan , M. and Shokri , Ali . 2007 . Numerical method for one-dimensional nonlinear sine–Gordon equation using collocation and radial basis functions . Numer. Methods Partial Differ. Equ. , 24 : 687 – 698 .
  • Dehghan , M. and Saadatmandi , A. 2009 . Variational iteration method for solving the wave equation subject to an integral conservation condition . Chaos Solitons Fractals , 41 : 1448 – 1453 .
  • Dehghan , M. and Shakeri , F. 2008 . The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics . Phys. Scr. , 78 : 1 – 11 .
  • Dehghan , M. and Shakeri , F. 2008 . Use of He's homotpy perturbation method for solving a partial differential equation arising in modeling of flow in porous media . J. Porous Media , 11 : 765 – 778 .
  • Dehghan , M. and Shakeri , F. 2009 . The numerical solution of the second Painleve equation . Numer. Methods Partial. Differ. Equ. , 25 : 1238 – 1259 .
  • Dehghan , M. , Shakourifar , M. and Hamidi , A. 2009 . The solution of linear and nonlinear systems of Volterra functional equations using Adomian-Pade technique . Chaos Solitons Fractals , 39 : 2509 – 2521 .
  • El-Sayed , S. M. 2003 . The decomposition method for studying the Klein–Gordon equation . Chaos Solitons Fractals , 18 : 1025 – 1030 .
  • Geng , F. and Cui , M. 2010 . Solving a system of third-order boundary value problem using variational iteration method . Int. J. Comput. Math. , 87 : 900 – 907 .
  • Ghorbani , A. 2009 . Beyond Adomian polynomials: He polynomials . Chaos Solitons Fractals , 39 : 1486 – 1492 .
  • Guo , B. Y. , Pascual , P. J. , Rodriguez , M. J. and Vazquez , L. 1986 . Numerical solution of the sine–Gordon equation . Appl. Math. Comput. , 18 : 1 – 14 .
  • He , J. H. 1997 . A new approach to nonlinear partial differential equations . Commun. Nonlinear Sci. Numer. Simul. , 2 : 230 – 235 .
  • He , J. H. 1999 . Homotopy perturbation technique . Comput. Methods Appl. Mech. Eng. , 178 : 257 – 262 .
  • He , J. H. 1999 . Variational iteration method: A kind of non-linear analytical technique, some examples . Int. J. Nonlinear Mech , 34 ( 4 ) : 699 – 708 .
  • He , J. H. 2000 . A coupling method of homotopy technique and perturbation technique for nonlinear problems . Int. J. Nonlinear Mech , 35 ( 1 ) : 115 – 123 .
  • He , J. H. 2000 . Variational iteration method for autonomous ordinary differential systems . Appl. Math. Comput. , 114 (2–3) : 115 – 123 .
  • He , J. H. 2005 . Homotopy perturbation method for bifurcation of nonlinear problems . Int. J. Nonlinear Sci. Numer. Simul. , 6 : 207 – 208 .
  • He , J. H. 2006 . Homotopy perturbation method for solving boundary value problems . Phys. Lett. A , 350 : 87 – 88 .
  • He , J. H. 2007 . Variational iteration method: Some recent results and new interpretations . J. Comput. Appl. Math. , 207 : 3 – 17 .
  • He , J. H. 2008 . Recent developments of the homotopy perturbation method . Topol. Methods Nonlinear Anal. , 31 : 205 – 209 .
  • He , J. H. and Wu , X. H. 2007 . Variational iteration method: New development and applications . Comput. Math. Appl. , 54 : 881 – 894 .
  • Herbst , B. M. and Ablowitz , M. J. 1992 . Numerical homoclinic instabilities in the sine-Gordon equation . Quaest. Math. , 15 : 345 – 363 .
  • Hirota , R. 1973 . Exact three-soliton solution of the two-dimensional sine-Gordon equation . J. Phys. Soc. Jpn. , 35 : 15 – 66 .
  • Kaya , D. 2003 . A numerical solution of the sine–Gordon equation using the modified decomposition method . Appl. Math. Comput. , 143 : 309 – 317 .
  • Leibbrandt , G. 1978 . New exact solutions of the classical sine–Gordon equation in 2+1 and 3+1 dimensions . Phys. Rev. Lett. , 41 : 435 – 438 .
  • Liu , S. , Fu , Z. and Liu , S. 2006 . Exact solutions to sine-Gordon-type equations . Phys. Lett. A , 351 : 59 – 63 .
  • Mohebbi , D. A. and Dehghan , M. 2010 . High-order solution of one-dimensional sine–Gordon equation using compact finite difference and DIRKN methods . Math. Comput. Model. , 51 : 537 – 549 .
  • Noor , M. A. and Mohyud-Din , S. T. 2008 . Homotopy perturbation method for nonlinear higher-order boundary value problems . Int. J. Nonlinear Sci. Numer. Simul. , 9 : 395 – 408 .
  • Ozis , T. and Yildirim , A. 2008 . Comparison between Adomian's method and He's homotopy perturbation method . Comput. Math. Appl. , 56 : 1216 – 1224 .
  • Ramos , J. J. 2001 . The sine–Gordon equation in the finite line . Appl. Math. Comput. , 124 : 45 – 93 .
  • Saadatmandi , A. , Dehghan , M. and Eftekhari , A. 2009 . Application of He's homotopy perturbation method for non-linear system of second-order boundary value problems . Nonlinear Anal. Real World Appl. , 10 : 1912 – 1922 .
  • Shakeri , F. and Dehghan , M. 2008 . Numerical solution of the Klein–Gordon equation via He's variational iteration method . Nonlinear Dyn. , 51 : 89 – 97 .
  • Shakeri , F. and Dehghan , M. 2008 . Solution of delay differential equations via a homotopy perturbation method . Math. Comput. Model. , 48 : 486 – 498 .
  • Tatari , M. and Dehghan , M. 2007 . On the convergence of He's variational iteration method . J. Comput. Appl. Math. , 207 : 121 – 128 .
  • Tatari , M. and Dehghan , M. 2009 . Improvement of He's variational iteration method for solving systems of differential equations . Comput. Math. Appl. , 58 : 2160 – 2166 .
  • Tourigny , Y. 1990 . Product approximation for nonlinear Klein–Gordon equations . IMA J. Numer. Anal. , 9 : 449 – 462 .
  • Wazwaz , A. M. 2005 . The tanh method: Exact solutions of the sine-Gordon and the sinh-Gordon equations . Appl. Math. Comput. , 167 : 1196 – 1210 .
  • Yucel , U. 2008 . Homotopy analysis method for the sine-Gordon equation with initial conditions . Appl. Math. Comput. , 203 : 387 – 395 .
  • Zagrodzinsky , J. 1979 . Particular solutions of the sine-Gordon equation in 2+1 dimensions . Phys. Lett. A , 72 : 284 – 286 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.