Haar Wavelet Method for Solving the Klein-Gordon and the Sine-Gordon Equations

Haar wavelet method for solving the Klein-Gordon and the sine-Gordon equations has been im- plemented. Application to partial differential equations is exemplified by solving the sine-Gordon equation. The efficiency of the method is demonstrated by five numerical examples. Computer simulation is carried out for problems the exact solution of which is known. This allows us to estimate the precision of the obtained numerical results. High accuracy of the results even in the case of a small number of collocation points is observed. The power of the manageable method is confirmed.

[1]  Qin Sheng,et al.  A predictor‐‐corrector scheme for the sine‐Gordon equation , 2000 .

[2]  Dogan Kaya,et al.  An implementation of the ADM for generalized one-dimensional Klein-Gordon equation , 2005, Appl. Math. Comput..

[3]  P. Clarkson,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering: References , 1991 .

[4]  Carlo Cattani.  Haar Wavelet Splines , 2001 .

[5]  Ülo Lepik,et al.  Numerical solution of evolution equations by the Haar wavelet method , 2007, Appl. Math. Comput..

[6]  G. Wei,et al.  Discrete singular convolution for the sine-Gordon equation , 2000 .

[7]  Petre P. Teodorescu,et al.  On the solitons and nonlinear wave equations , 2008 .

[8]  Gokul Hariharan,et al.  Haar wavelet method for solving Fisher's equation , 2009, Appl. Math. Comput..

[9]  Ulo Lepik,et al.  Haar wavelet method for solving higher order differential equations , 2008 .

[10]  Martin Eerme,et al.  Weak formulation based Haar wavelet method for solving differential equations , 2009, Appl. Math. Comput..

[11]  Mark J. Ablowitz,et al.  NUMERICAL HOMOCLINIC INSTABILITIES IN The SINE-GORDON EQUATION , 1992 .

[12]  Hongyan Zhi,et al.  A new Riccati equation expansion method with symbolic computation to construct new travelling wave solution of nonlinear differential equations , 2006, Appl. Math. Comput..

[13]  Salah M. El-Sayed,et al.  A numerical solution of the Klein-Gordon equation and convergence of the decomposition method , 2004, Appl. Math. Comput..

[14]  Mark A. M. Lynch.  Large amplitude instability in finite difference approximations to the Klein-Gordon equation , 1999 .

[15]  Abdul-Majid Wazwaz,et al.  The modified decomposition method for analytic treatment of differential equations , 2006, Appl. Math. Comput..

[16]  Ülo Lepik.  Solving integral and differential equations by the aid of non-uniform Haar wavelets , 2008, Appl. Math. Comput..

[17]  K. Kannan,et al.  Haar wavelet method for solving some nonlinear Parabolic equations , 2010, Forum for Interdisciplinary Mathematics.

[18]  K. Kannan,et al.  A Comparative Study of a Haar Wavelet Method and a Restrictive Taylor's Series Method for Solving Convection-diffusion Equations , 2010 .

[19]  Gokul Hariharan,et al.  Haar wavelet in estimating depth profile of soil temperature , 2009, Appl. Math. Comput..

[20]  A. Haar.  Zur Theorie der orthogonalen Funktionensysteme , 1910 .

[21]  Salah M. El-Sayed,et al.  The decomposition method for studying the Klein–Gordon equation , 2003 .

[22]  Abdul-Majid Wazwaz,et al.  A variable separated ODE method for solving the triple sine-Gordon and the triple sinh-Gordon equations , 2007 .

[23]  Mark J. Ablowitz,et al.  On the Numerical Solution of the Sine-Gordon Equation , 1996 .

[24]  Qing-Jiang Chen,et al.  Numerical solution of differential equations by using Haar wavelets , 2007, 2007 International Conference on Wavelet Analysis and Pattern Recognition.

[25]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[26]  Abdul-Majid Wazwaz,et al.  A new algorithm for calculating adomian polynomials for nonlinear operators , 2000, Appl. Math. Comput..

[27]  Silvia Bertoluzza,et al.  A wavelet collocation method for the static analysis of sandwich plates using a layerwise theory , 2010 .

[28]  Ülo Lepik.  Solving PDEs with the aid of two-dimensional Haar wavelets , 2011, Comput. Math. Appl..

[29]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[30]  C. F. Chen,et al.  Haar wavelet method for solving lumped and distributed-parameter systems , 1997 .

[31]  Abdul-Majid Wazwaz,et al.  A reliable modification of Adomian decomposition method , 1999, Appl. Math. Comput..