IJMNTA  Vol.4 No.1 , March 2015
The k = 1 Finite Element Numerical Solution for the Improved Boussinesq Equation
Abstract: The improved Boussinesq equation is solved with classical finite element method using the most basic Lagrange element k = 1, which leads us to a second order nonlinear ordinary differential equations system in time; this can be solved by any standard accurate numerical method for example Runge-Kutta-Fehlberg. The technique is validated with a typical example and a fourth order convergence in space is confirmed; the 1- and 2-soliton solutions are used to simulate wave travel, wave splitting and interaction; solution blow up is described graphically. The computer symbolic system MathLab is quite used for numerical simulation in this paper; the known results in the bibliography are confirmed.
Cite this paper: López, F. , Tapia, E. , Ongay, F. and Aguero, M. (2015) The k = 1 Finite Element Numerical Solution for the Improved Boussinesq Equation. International Journal of Modern Nonlinear Theory and Application, 4, 88-99. doi: 10.4236/ijmnta.2015.41006.

[1]   Bogolyubskii, I.L. (1976) Modified Equation of a Nonlinear String and Inelastic Interaction of Solitons. Journal of Experimental and Theoretical Physics, 24, 160.

[2]   Khan, K. and Ali Akbar, M. (2013) Traveling Wave Solutions of Some Coupled Nonlinear Evolution Equations. ISRN Mathematical Physics, 2013, 1-8.

[3]   Debnath, L. (1997) Nonlinear PDEs for Scientists and Engineers. 3rd Edition, Birkhauser, Springer New York Dordrecht Heidelberg London.

[4]   Ciarlet, P.G. (2002) The Finite Element Method for Elliptic Problems (Classics in Applied Mathematics). SIAM.

[5]   Johnson, C. (2009) Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Books on Mathematics. 1987 Edition, Reprint of the Cambridge University Press, New York.

[6]   Bogolyubskii, I.L. (1977) Some Examples of Inelastic Soliton Interaction. Computer Physics Communications, 13, 149-155.

[7]   Smith, G.D. (1987) Numerical Solution of PDEs Finite Difference Methods. 3rd Edition, Oxford University Press, New York.

[8]   Iskandar, L. and Jain, P.C. (1980) Numerical Solution of the Improved Boussinesq Equation. Proceedings of the Indian Academy of Science (Mathematical Sciences), 89, 171-181.

[9]   Bratsos, A.G. (1998) The Solution of the Boussinesq Equation Using the Method of Lines. Computer Methods in Applied Mechanics and Engineering, 157, 33-44.

[10]   El-Zoheiry, H. (2002) Numerical Study of Improved Boussinesq Equation. Chaos, Solitons and Fractals, 14, 377-384.

[11]   Bratsos, A.G. (2007) A Second Order Numerical Shame for the Solution of the One-Dimensional Boussinesq Equation. Numerical Algorithms, 46, 45-58.

[12]   Bratsos, A.G. (2007) A Second Order Numerical Scheme for the Improved Boussinesq Equation. Physics Letters A, 370, 145-147.

[13]   Ismail, M.S. and Mosally, F. (2014) A Fourth Order Finite Difference Method for the Good Boussinesq Equation. Hindawi Publishing Corporation, Abstract and Applied Analysis, 2014, Article ID: 323260.

[14]   Lin, Q., Wu, Y.H., Loxton, R. and Lai, S.Y. (2009) Linear B-Spline Finite Elemnt Method for the Improved Boussinesq Equation. Journal of Computational and Applied Mathematics, 224, 658-667.

[15]   Adams, R.A. and Fournier, J.J.F. (2003) Sobolev Spaces. Vol. 140, Pure and Applied Mathematics. 2nd Edition, Elsevier, Amsterdam.

[16]   Atkinson, K. and Han, W.M. (2009) Numerical Solution of Ordinary Differential Equations. Wiley, Hoboken.

[17]   Contreras, F., Cervantes, H., Aguero, M. and de Lourdes Najera, Ma. (2013) Classic and Non-Classic Soliton Like Structures for Traveling Nerves Pulses. International Journal of Modern Nonlinear Theory and Application, 2, 7-13.

[18]   Whitham, G.B. (1999) Linear and Nonlinear Waves. John Wiley and Son Inc., Hoboken.

[19]   Yang, Z. (1998) Existence and Non-Existence of Global Solutions to a Generalized Modification of the Improved Boussinesq Equation. Mathematical Methods in the Applied Sciences, 21, 1467-1477.

[20]   Yang, Z. and Wang, X. (2007) Blowup of Solutions for Improved Boussinesq-Type Equation. Journal of Mathematical Analysis and Applications, 278, 335-353.