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Abstract
In this paper, numerical solutions for nonlinear coupled Korteweg-de Vries(abbreviated as KdV) equations are calculated by the Sinc-collocation method. This approach is based on a global collocation method using Sinc basis functions. The first step is to discretize time derivative of the KdV equations by a classic finite difference formula, while the space derivatives are approximated by a weighted scheme. Sinc functions are used to solve these two equations. Soliton solutions are constructed to show the nature of the solution. The numerical results are shown to demonstrate the efficiency of the newly proposed method.
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References
- Alquran, M. and Al-Khaled, K. Sinc and solitary wave solutions to the generalized Benjamin-Bona-Mahony-Burgers equations. Phys. Scr, 2011, 83, 6.
- Alquran, M. and Al-Khaled, K. The tanh and sine–cosine methods for higher order equations of Korteweg-de Vries type. Phys. Scr, 2011, 84.
- Wazwaz, A.M., The tanh and the sine-cosine methods for the complex modified KdV and the generalized KdV equations. Comput. and Math. with Appl., 2005, 49, 1101-1112.
- Wazwaz, A.M. Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Appl. Math. and Comput., 2005, 169, 639-656.
- Triki, H., Taha, T. and Wazwaz, A.M., Solitary wave solutions for a generalized KdV-mKdV equation with variable coefficients. Math. and Comput. Simul., 2010, 80(9), 1867-1873.
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- Kaya, D. and Inan, I.E., Exact and numerical travelling wave solutions for nonlinear coupled equations using symbolic computation. Appl. Math. Comput., 2004, 151, 775-787.
- Khare, A.H., Temsah, R.S. and Callebaut, D.K. Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method. Math. Computer Model., 2008, 48, 1237-1253.
- Fan, E. Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation. Phys. Lett. A, 2001, 282, 18-22.
- Siraj-ul-Islam, Haq, S. and Uddin, M. A meshfree interpolation method for the numerical solution of KdV-Burgers equation. Appl. Math. Model. 2009, 33, 3442-3449.
- Siraj-ul-Islam, Haq, S. and Uddin, M. A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations. Engineering Analysis with Boundary Elements, 2009, 33, 399-409.
- Arshed, A., Siraj-ul-Islam and Haq, S. A computational meshfree technique for the numerical solution of the two-dimensional coupled Burgers' equations. Intern. J. of Comput. Eng. Sci. Mech., 2009, 10(5), 406-422.
- Mokhtari, R. and Mohammad, M. Numerical solution of GRLW equation using Sinc-collocation method. Computer Phys. Commun., 2010, 181, 1266-1274.
- Stenger, F. Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, New York, (1993).
- Lund. J. and Bowers, K.L.Sinc Methods for Quadrature and Differential Equations,SIAM, Philadelphia, (1992).
- Al-Khaled, K. Numerical study of Fisher's reaction-diffusion equation by the Sinc collocation method. J. Comput. Appl. Math., 2001, 137, 245-255.
- Al-Khaled, K. Sinc numerical solution for solitons and solitary waves. J. Comput. Appl. Math., 2001, 130(1-2), 283-292.
References
Alquran, M. and Al-Khaled, K. Sinc and solitary wave solutions to the generalized Benjamin-Bona-Mahony-Burgers equations. Phys. Scr, 2011, 83, 6.
Alquran, M. and Al-Khaled, K. The tanh and sine–cosine methods for higher order equations of Korteweg-de Vries type. Phys. Scr, 2011, 84.
Wazwaz, A.M., The tanh and the sine-cosine methods for the complex modified KdV and the generalized KdV equations. Comput. and Math. with Appl., 2005, 49, 1101-1112.
Wazwaz, A.M. Travelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations, Appl. Math. and Comput., 2005, 169, 639-656.
Triki, H., Taha, T. and Wazwaz, A.M., Solitary wave solutions for a generalized KdV-mKdV equation with variable coefficients. Math. and Comput. Simul., 2010, 80(9), 1867-1873.
Hirota, R. and Satsuma, J. Soliton solutions of a coupled Korteweg-de Vries equations. Phys. Lett. A, 1981, 85, 407-408.
Kaya, D. and Inan, I.E., Exact and numerical travelling wave solutions for nonlinear coupled equations using symbolic computation. Appl. Math. Comput., 2004, 151, 775-787.
Khare, A.H., Temsah, R.S. and Callebaut, D.K. Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method. Math. Computer Model., 2008, 48, 1237-1253.
Fan, E. Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a coupled MKdV equation. Phys. Lett. A, 2001, 282, 18-22.
Siraj-ul-Islam, Haq, S. and Uddin, M. A meshfree interpolation method for the numerical solution of KdV-Burgers equation. Appl. Math. Model. 2009, 33, 3442-3449.
Siraj-ul-Islam, Haq, S. and Uddin, M. A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations. Engineering Analysis with Boundary Elements, 2009, 33, 399-409.
Arshed, A., Siraj-ul-Islam and Haq, S. A computational meshfree technique for the numerical solution of the two-dimensional coupled Burgers' equations. Intern. J. of Comput. Eng. Sci. Mech., 2009, 10(5), 406-422.
Mokhtari, R. and Mohammad, M. Numerical solution of GRLW equation using Sinc-collocation method. Computer Phys. Commun., 2010, 181, 1266-1274.
Stenger, F. Numerical Methods Based on Sinc and Analytic Functions. Springer-Verlag, New York, (1993).
Lund. J. and Bowers, K.L.Sinc Methods for Quadrature and Differential Equations,SIAM, Philadelphia, (1992).
Al-Khaled, K. Numerical study of Fisher's reaction-diffusion equation by the Sinc collocation method. J. Comput. Appl. Math., 2001, 137, 245-255.
Al-Khaled, K. Sinc numerical solution for solitons and solitary waves. J. Comput. Appl. Math., 2001, 130(1-2), 283-292.