Numerical solutions of the MRLW equation by cubic B-spline Galerkin finite element method
Keywords:
Cubic B-splines, finite element method, Galerkin, MRLW equation, solitary waves.Abstract
In this paper, a numerical solution of the modified regularized long wave (MRLW) equationhas been obtained by a numerical technique based on a lumped Galerkin method using cubicB-spline finite elements. Solitary wave motion, interaction of two and three solitary waveshave been studied to validate the proposed method. The three invariants ( 1 2 3 I , I , I ) of themotion have been calculated to determine the conservation properties of the scheme. Errornorms 2 L and ∞ L have been used to measure the differences between the exact and numericalsolutions. Also, a linear stability analysis of the scheme is proposed.References
Ali, I.A. 2009. Mesh free collocation method for numerical solution of initial-boundary value problems using radial basis functions, Ph.D. Thesis.
Alzubaidi, H.M.O. 2006. Numerical Methods For Solving The Modified Regularized Long Wave Equation, MSc Thesis, King Saud University.
Benjamin T.B., Bona, J.L. & Mahoney, J.L. 1972. Model equations for long waves in nonlinear
dispersive media, Philosophical Transactions of the Royal Society A 272, 47-78.
Benjamin, T. B., F.R.S., Bona, J. L. & Mahony, J. J. 1971. Model Equations for Long Waves in
Nonlinear Dispersive Systems, Fluid Mechanics Research Institute, University of Esex, Colchester,
Esex.
Bona, J.L. & Pryant, P.J. 1973. A mathematical model for long wave generated by wave makers in
nonlinear dispersive systems, Mathematical Proceedings of the Cambridge Philosophical Society,
: 391-405.
Eilbeck J.C. & McGuire, G.R. 1977. Numerical study of the regularized long wave equation,
II:Interaction of solitary wave, Journal of Computational Physics, 23: 63-73.
Esen, A. & Kutluay, S. 2005. Application of lumped Galerkin method to the regularized long wave
equation, Applied Mathematics and Computation.
Gardner, L.R.T. & Gardner, G.A. 1990. Solitary waves of the regularized long wave equation, Journal
of Computational Physics, 91: 441-459.
Gardner, L.R.T., Gardner, G.A. & Geyikli, T. 1994. The boundary forced MKdV equation, Journal of
Computational Physics, 11: 5-12.
Gardner, L.R.T., Gardner, G.A., Ayoup, F.A. & Amein, N.K. 1997. Simulations of solitary waves of
the MRLW equation by B-spline finite element, The Arabian Journal for Science and Engineering,
: 183-193.
Gou, B.Y. & Cao, W.M. 1988. The Fourier pseudo-spectral method with a restrain operator for the RLW
equation, Journal of Computational Physics, 74: 110-126.
Haq, F., Islam, S. & Tirmizi, I.A. 2010. A numerical technique for solution of the MRLW equation using
quartic B-splines, Applied Mathematical Modelling, 34: 4151-4160.
Jain, P.C., Shankar, R. & Singh, T.V. 1993. Numerical solution of regularized long wave equation,
Communications in Numerical Methods in Engineering, 9: 579-586.
Kaya, D. & El-Sayed, S.M. 2003. An application of the decomposition method for the generalized KdV
and RLW equations, Chaos, Solitons and Fractals, 17: 869-877.
Karakoc, S.B.G. 2011. Numerical solutions of modified equal width wave equation with finite elements
method, PhD. Thesis.
Karakoc, S.B.G., Yagmurlu N.M. & Ucar, Y. 2013. Numerical approximation to a solution of the
modified regularized long wave equation using quintic B-splines, Boundary Value Problems,
:27.
Karakoc, S.B.G. & Geyikli, T. 2013. Petrov-Galerkin finite element method for solving the MRLW
equation, Mathematical Sciences, 7:25.
Khalifa, A.K., Raslan, K.R. & Alzubaidi, H.M. 2008. A collocation method with cubic B- splines for
solving the MRLW equation, The Journal of Computational and Applied Mathematics, 212: 406-
Khalifa, H.M., Raslan, K.R. & Alzubaidi, H.M. 2007. A finite difference scheme for the MRLW and
solitary wave interactions, Applied Mathematics and Computation, 189: 346-354.
Olver, P.J. 1979. Euler operators and conservation laws of the BBM equation, Mathematical Proceedings
of the Cambridge Philosophical Society, 85: 143-159.
Peregrine, D.H. 1966. Calculations of the development of an undular bore, Journal of Fluid Mechanics,
: 321-330.
Ramos, J.I. 2007. Solitary wave interactions of the GRLW equation, Chaos, Solitons and Fractals, 33:
-491.
Raslan, K.R. 2009. Numerical study of the modified regularized long wave equation, Chaos, Solitons and
Fractals, 42: 1845-1853.
Raslan, K.R. & Hassan, S.M. 2009. Solitary waves for the MRLW equation, Applied Mathematics
Letters, 22: 984-989.
Roshan, T. 2012. A Petrov-Galerkin method for solving the generalized regularized long wave (GRLW)
equation, Computers & Mathematics with Applications, 63: 943-956.
Zhang, L. 2005. A finite difference scheme for generalized long wave equation, Applied Mathematics and
Computation, 168:2: 962-972.