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Section B

New modification of the HPM for numerical solutions of the sine-Gordon and coupled sine-Gordon equations

, , &
Pages 908-919 | Received 03 Jun 2007, Accepted 25 May 2008, Published online: 14 Oct 2008

References

  • Abbasbandy , S. 2006 . Application of He's homotopy perturbation method for Laplace transform . Chaos, Solitons & Fractals , 30 : 1206 – 1212 .
  • Abdelwahid , F. 2000 . A mathematical model of Adomian polynomials . Appl. Math. Comput. , 141 : 447 – 453 .
  • Ablowitz , M. J. , Herbst , B. M. and Schober , C. 1996 . Constance on the numerical solution of the sine-Gordon equation, I: integrable discretizations and homoclinic manifolds . J. Comput. Phys. , 126 : 299 – 314 .
  • Adomian , G. 1994 . Solving Frontier Problems of Physics: The Decomposition Method , Boston, MA : Kluwer Academic Publisher .
  • Adomian , G. 1998 . A review of the decomposition method in applied mathematics . J. Math. Anal. Appl. , 135 : 501 – 544 .
  • Chen , W. and Lu , Z. 2004 . An algorithm for Adomian decomposition method . Appl. Math. Comput. , 159 : 221 – 235 .
  • Chowdhury , M. S.H. and Hashim , I. Application of homotopy perturbation method to Klein–Gordon and sine-Gordon equations , Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2007.06.091, in press
  • Darvishi , M. T. and Khani , F. 2008 . Application of He's homotopy perturbation method to stiff systems of ordinary differential equations . Zeitschrift fuer Naturforschung A , 63a ( 1–2 ) : 19 – 23 .
  • Drazin , P. G. and Johnson , R. S. 1989 . Solutions: An Introduction , Cambridge : Cambridge University Press .
  • El-Shahed , M. 2005 . Application of He's homotopy perturbation method to Voltera's integro-differential equation . Int. J. Nonlinear Sci. Numer. Simul. , 6 ( 2 ) : 163 – 168 .
  • Ganji , D. D. and Sadighi , A. 2006 . Application of He's homotopy perturbation method to nonlinear coupled systems of reaction-diffusion equations . Int. J. Nonlinear Sci. Numer. Simul. , 7 ( 4 ) : 411 – 418 .
  • Ghorbani , A. Beyond Adomian polynomials: He polynomials , Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2007. 06.034, in press
  • Ghorbani , A. and Saberi-Nadjafi , J. 2007 . He's homotopy perturbation method for calculating Adomian polynomials . Int. J. Nonlinear Sci. Numer. Simul. , 8 ( 2 ) : 229 – 232 .
  • He , J. H. 1998 . Approximate analytical solution for seepage flow with fractional derivatives in porous media . Comput. Methods Appl. Mech. Eng. , 167 ( 1–2 ) : 57 – 68 .
  • He , J. H. 1999 . Homotopy perturbation technique . Comput. Methods Appl. Mech. Eng. , 178 ( 3/4 ) : 230 – 235 .
  • He , J. H. 2000 . A coupling method of homotopy technique and perturbation technique for nonlinear problems . Int. J. Nonlinear Mech. , 35 ( 1 ) : 37 – 43 .
  • He , J. H. 2003 . Homotopy perturbation method: a new nonlinear analytical technique . Appl. Math. Comput. , 135 : 73 – 79 .
  • He , J. H. 2004 . Comparison of homotopy perturabation method and homotopy analysis method . Appl. Math. Comput. , 156 : 527 – 539 .
  • He , J. H. 2004 . Asymptopology by homotopy perturbation method . Appl. Math. Comput. , 156 ( 3 ) : 591 – 596 .
  • He , J. H. 2004 . The homotopy perturbation method for nonlinear oscillators with discontinuities . Appl. Math. Comput. , 151 : 287 – 292 .
  • He , J. H. 2005 . Limit cycle and bifurcation of nonlinear problems . Chaos, Solitons & Fractals , 26 ( 3 ) : 827 – 833 .
  • He , J. H. 2005 . Homotopy perturbation method for bifurcation of nonlinear problems . Int. J. Nonlinear Sci. Numer. Simul. , 6 ( 2 ) : 207 – 218 .
  • He , J. H. 2005 . Application of homotopy perturbation method to nonlinear wave equations . Chaos, Solitons & Fractals , 26 ( 3 ) : 695 – 700 .
  • He , J. H. 2006 . Homotopy perturbation method for solving boundary value problems . Phys. Lett. A , 50 ( 1–2 ) : 87 – 98 .
  • Khusnutdinova , K. R. and Pelinovsky , D. E. 2003 . On the exchange of energy in coupled Klein–Gordon equations . Wave Motion , 38 : 1 – 10 .
  • Molabahrami , A. and Khani , F. 2008 . Numerical solutions of highly oscillatory integrals . Appl. Math. Comput. , 198 ( 2 ) : 657 – 664 . no. 2
  • Molabahrami , A. , Khani , F. and Hamedi-Nezad , S. 2007 . Soliton solutions of the two-dimensional KdV–Burgers equation by homotopy perturbation method . Phys. Lett. A , 370 : 433 – 436 .
  • Nayfeh , A. H. 1985 . Problems in Purturbation , New York : John Wiley .
  • Odibat , Z. M. 2006 . A reliable modification of the rectangular decomposition method . Appl. Math. Comput. , 183 : 1226 – 1234 .
  • Odibat , Z. M. 2007 . A new modification of the homotopy perturbation method for linear and nonlinear operators . Appl. Math. Comput. , 189 ( 1 ) : 746 – 753 .
  • Ozis , T. and Yildirim , A. 2007 . Traveling wave solution of Korteweg–de Vries equation using He's homotopy perturbation method . Int. J. Nonlinear Sci. Numer. Simul. , 8 ( 2 ) : 239 – 242 .
  • Rafei , M. and Ganji , D. D. 2006 . Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method . Int. J. Nonlinear Sci. Numer. Simul. , 7 ( 3 ) : 321 – 328 .
  • Tari , H. , Ganji , D. D. and Rostamian , M. 2007 . Approximate solutions of K(2,2), KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method . Int. J. Nonlinear Sci. Numer. Simul. , 8 ( 2 ) : 203 – 210 .
  • Wazwaz , A. M. 2000 . A new algorithm for calculating Adomian polynomials for nonlinear operators . Appl. Math. Comput. , 111 : 53 – 59 .
  • Wei , G. W. 2000 . Discrete singular convolution for the sine-Gordon equation . Phys. D , 137 : 247 – 259 .

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