Numerical simulation of the generalized regularized long wave equation by He’s variational iteration method. (English) Zbl 1152.65467

Soliman, A. A.

Math. Comput. Simul. 70, No. 2, 119-124 (2005).

Summary: The solution for the generalized regularized long wave equation based on variational iteratiom method, is exactly obtained. In this method, the solution is calculated in the form of convergent power series with easily computable componentes. This approach does need linearization, weak nonlinearity assumptions or perturbation theory. The results reveal that the method is very effective and convenient.

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L75	Higher-order nonlinear hyperbolic equations

Keywords:

variational iteration method; generalized RLW equation; solitary wave equation; nonlinear PDE

References:

[1]	Peregrine, D.H., Calculations of the development of an undular bore, J. fluid mech., 25, 321-330, (1966)
[2]	Abdullove, Kh.O.; Bogalubsky, H.; Makhankov, V.G., One more example of inelastic soliton interaction, Phys. lett. A, 56, 427-428, (1976)
[3]	Eilbek, J.C.; MeGuire, G.R., Numerical study of the regularized long wave equation. II. interaction of solitary wave, J. comput. phys., 23, 63-73, (1977) · Zbl 0361.65100
[4]	Bona, J.L.; Pritchard, W.G.; Scott, L.R., Numerical scheme for a model of nonlinear dispersive waves, J. comput. phys., 60, 167-176, (1985) · Zbl 0578.65120
[5]	Alexander, M.E.; Morris, J.H., Galerkin method for some model equation for nonlinear dispersive waves, J. comput. phys., 30, 428-451, (1979) · Zbl 0407.76014
[6]	Gardner, L.R.; Gardner, G.A., Solitary waves of the regularized long wave equation, J. comput. phys., 91, 441-459, (1990) · Zbl 0717.65072
[7]	Gardner, L.R.; Gardner, G.A.; Dogan, A., A least-squares finite element scheme for RLW equation, Commun. numer. meth. eng., 12, 795-804, (1996) · Zbl 0867.76040
[8]	Soliman, A.A.; Raslan, K.R., Collocation method using quadratic B-spline for the RLW equation, Int. J. comput. math., 78, 399-412, (2001) · Zbl 0990.65116
[9]	Soliman, A.A.; Hussien, M.H., Appl. math. comput., 161, 623-636, (2005)
[10]	Soliman, A.A., Int. J. comput. math., 81, 1281-1288, (2004)
[11]	He, J.H., Comput. meth. appl. mech. eng., 167, 57-68, (1998)
[12]	He, J.H., Comput. meth. appl. mech. eng., 167, 69-73, (1998)
[13]	He, J.H., Int. J. nonlinear mech., 34, 699-708, (1999)
[14]	He, J.H., Commun. nonlinear sci. numer. simul., 2, 4, 230-235, (1997)
[15]	He, J.H., Appl. math. comput., 114, 2-3, 115-123, (2000)
[16]	Marinca, V., Int. J. nonlinear sci. numer. simul., 3, 107-110, (2002)
[17]	Abdou, M.A.; Soliman, A.A., Variational iteration method for solving burger’s and coupled burger’s equations’, J. comput. appl. math., 181, 2, 245-251, (2005) · Zbl 1072.65127
[18]	Draganescu, Gh.G.; Capalnasan, V., Int. J. nonlinear sci. numer. simul., 4, 219-226, (2004)
[19]	He, J.H., Approximate analytical methods in science and engineering, (2002), Henan Science and Technical Press Zhengzhou, (in Chinese)
[20]	He, J.H., Generalized variational principles in fluids, (2003), Science and Culture Publishing House of China Hongkong, (in Chinese) · Zbl 1054.76001