## Abstract and Applied Analysis

### A Fourth Order Finite Difference Method for the Good Boussinesq Equation

#### Abstract

The “good” Boussinesq equation is transformed into a first order differential system. A fourth order finite difference scheme is derived for this system. The resulting scheme is analyzed for accuracy and stability. Newton’s method and linearization techniques are used to solve the resulting nonlinear system. The exact solution and the conserved quantity are used to assess the accuracy and the efficiency of the derived method. Head-on and overtaking interactions of two solitons are also considered. The numerical results reveal the good performance of the derived method.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 323260, 10 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412273197

Digital Object Identifier
doi:10.1155/2014/323260

Mathematical Reviews number (MathSciNet)
MR3176737

Zentralblatt MATH identifier
07022169

#### Citation

Ismail, M. S.; Mosally, Farida. A Fourth Order Finite Difference Method for the Good Boussinesq Equation. Abstr. Appl. Anal. 2014 (2014), Article ID 323260, 10 pages. doi:10.1155/2014/323260. https://projecteuclid.org/euclid.aaa/1412273197

#### References

• M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, vol. 4 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1981.
• A. Biswas and K. R. Raslan, “Numerical simulation of the modified Korteweg-de Vries equation,” Physics of Wave Phenomena, vol. 19, no. 2, pp. 142–147, 2011.
• G. Domairry, A. G. Davodi, and A. G. Davodi, “Solutions for the double sine-Gordon equation by exp-function, tanh, and extended tanh methods,” Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 384–398, 2010.
• R. Hirota, “Exact $N$-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices,” Journal of Mathematical Physics, vol. 14, pp. 810–814, 1973.
• H. Leblond, H. Triki, and D. Mihalache, “Derivation of a generalized double sine-Gordon equation describing ultrashort soliton propagation in optical media composed of multilevel atoms,” Physical Review A, vol. 86, Article ID 063825, 9 pages, 2012.
• H. Liu and J. Li, “Symmetry reductions, dynamical behavior and exact explicit solutions to the Gordon types of equations,” Journal of Computational and Applied Mathematics, vol. 257, pp. 144–156, 2014.
• R. Mokhtari and M. Mohseni, “A meshless method for solving mKdV equation,” Computer Physics Communications, vol. 183, no. 6, pp. 1259–1268, 2012.
• G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, NY, USA, 1974.
• A. Ayd\in and B. Karasözen, “Lobatto IIIA-IIIB discretization of the strongly coupled nonlinear Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 235, no. 16, pp. 4770–4779, 2011.
• D. Bai and J. Wang, “The time-splitting Fourier spectral method for the coupled Schrödinger-Boussinesq equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1201–1210, 2012.
• M. S. Ismail and T. R. Taha, “Parallel methods and higher dimensional NLS equations,” Abstract and Applied Analysis, vol. 2013, Article ID 497439, 9 pages, 2013.
• M. S. Ismail and A. Biswas, “1-soliton solution of the Klein-Gordon-Zakharov equation with power law nonlinearity,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 4186–4196, 2010.
• M. S. Ismail, “Numerical solution of complex modified Korteweg-de Vries equation by collocation method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 749–759, 2009.
• M. S. Ismail, “Numerical solution of a coupled Korteweg-de Vries equations by collocation method,” Numerical Methods for Partial Differential Equations, vol. 25, no. 2, pp. 275–291, 2009.
• M. S. Ismail, “A fourth-order explicit schemes for the coupled nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 196, no. 1, pp. 273–284, 2008.
• M. S. Ismail, “Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method,” Mathematics and Computers in Simulation, vol. 78, no. 4, pp. 532–547, 2008.
• M. S. Ismail and T. R. Taha, “A linearly implicit conservative scheme for the coupled nonlinear Schrödinger equation,” Mathematics and Computers in Simulation, vol. 74, no. 4-5, pp. 302–311, 2007.
• S. Kutluay and Y. Ucar, “Numerical solution of coupled modied Korteweg-devries equation by Galerkin method using quadratic spline,” International Journal of Computer Mathematics, vol. 90, no. 11, pp. 2353–2371, 2013.
• V. S. Manoranjan, A. R. Mitchell, and J. Ll. Morris, “Numerical solutions of the good Boussinesq equation,” Society for Industrial and Applied Mathematics, vol. 5, no. 4, pp. 946–957, 1984.
• S. S. Ray, “Soliton solutions for time fractional coupled modified KdV equations using new coupled fractional reduced differential transform method,” Journal of Mathematical Chemistry, vol. 51, no. 8, pp. 2214–2229, 2013.
• S. Zhou and X. Cheng, “Numerical solution to coupled nonlinear Schrödinger equations on unbounded domains,” Mathematics and Computers in Simulation, vol. 80, no. 12, pp. 2362–2373, 2010.
• H. Hu, L. Liu, and L. Zhang, “New analytical positon, negaton and complexiton solutions of a coupled KdV-mKdV system,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 5743–5749, 2013.
• A. G. Bratsos, “A second order numerical scheme for the solution of the one-dimensional Boussinesq equation,” Numerical Algorithms, vol. 46, no. 1, pp. 45–58, 2007.
• A. G. Bratsos, Ch. Tsitouras, and D. G. Natsis, “Linearized numerical schemes for the Boussinesq equation,” Applied Numerical Analysis and Computational Mathematics, vol. 2, no. 1, pp. 34–53, 2005.
• M. S. Ismail and A. G. Bratsos, “A predictor-corrector scheme for the numerical solution of the Boussinesq equation,” Journal of Applied Mathematics & Computing, vol. 13, no. 1-2, pp. 11–27, 2003.
• A. Mohebbi and Z. Asgari, “Efficient numerical algorithms for the solution of “good” Boussinesq equation in water wave propagation,” Computer Physics Communications, vol. 182, no. 12, pp. 2464–2470, 2011.
• A.-M. Wazwaz, “Multiple-soliton solutions for the Boussinesq equation,” Applied Mathematics and Computation, vol. 192, no. 2, pp. 479–486, 2007.
• B. S. Attili, “The Adomian decomposition method for solving the Boussinesq equation arising in water wave propagation,” Numerical Methods for Partial Differential Equations, vol. 22, no. 6, pp. 1337–1347, 2006.
• A.-M. Wazwaz, “New travelling wave solutions to the Boussinesq and the Klein-Gordon equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 5, pp. 889–901, 2008.
• A. M. Wazwaz, “Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method,” Chaos, Solitons and Fractals, vol. 12, no. 8, pp. 1549–1556, 2001.
• A. Yildirim and S. T. Mohyud-Din, “A variational approach for soliton solutions of good Boussinesq equation,” Journal of King Saud University, vol. 22, no. 4, pp. 205–208, 2010.
• A. Biswas, D. Milovic, and A. Ranasinghe, “Solitary waves of Boussinesq equation in a power law media,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 11, pp. 3738–3742, 2009.
• A. G. Bratsos, “A parametric scheme for the numerical solution of the Boussinesq equation,” The Korean Journal of Computational & Applied Mathematics, vol. 8, no. 1, pp. 45–57, 2001.
• H. El-Zoheiry, “Numerical investigation for the solitary waves interaction of the “good” Boussinesq equation,” Applied Numerical Mathematics, vol. 45, no. 2-3, pp. 161–173, 2003.
• A. Aydin and B. Karasözen, “Symplectic and multisymplectic Lobatto methods for the “good” Boussinesq equation,” Journal of Mathematical Physics, vol. 49, no. 8, Article ID 083509, 2008.
• P. Daripa and W. Hua, “A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: filtering and regularization techniques,” Applied Mathematics and Computation, vol. 101, no. 2-3, pp. 159–207, 1999.
• T. Matsuo, “New conservative schemes with discrete variational derivatives for nonlinear wave equations,” Journal of Computational and Applied Mathematics, vol. 203, no. 1, pp. 32–56, 2007.
• M. Dehghan and R. Salehi, “A meshless based numerical technique for traveling solitary wave solution of Boussinesq equation,” Applied Mathematical Modelling, vol. 36, no. 5, pp. 1939–1956, 2012.
• A. R. Mitchell and D. Gritthes, Finite Difference Method, Wiley, New York, NY, USA, 1980. \endinput