A numerical solution of a Kawahara equation by using Multiquadric radial basis function
الموضوعات :
M. Zarebnia
1
(Department of Mathematics, University of mohaghegh Ardabili,
56199-11367, Ardabil, Iran.)
M. Takhti
2
(Department of Mathematics, University of mohaghegh Ardabili,
56199-11367, Ardabil, Iran.)
الکلمات المفتاحية: Kawahara equation, Traveling wave solution, Radial basis function(RBFs),
ملخص المقالة :
In this article, we apply the Multiquadric radial basis function (RBF) interpo-lation method for nding the numerical approximation of traveling wave solu-tions of the Kawahara equation. The scheme is based on the Crank-Nicolsonformulation for space derivative. The performance of the method is shown innumerical examples.
[1] J.K. Hunter, J. Scheurle, Existence of perturbed solitary wave
solutions to a model equation for water waves, Physica D 32 (1988)
253{268.
[2] T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys.
Soc. Japan 33(1) (1972) 260-264.
[3] B.I. Cohen, J.A. Krommes, W.M. Tang, M.N. Rosenbluth, Non-linear
saturation of the dissipative trapped ion mode by mode coupling, Nucl.
Fusion 16 (1976) 971{992.
[4] Y. Kuramoto, Di usion induced chaos in reactions systems, Progr.
Theoret. Phys. Suppl. 64 (1978) 346{367.
[5] G.I. Sivashinsky, Nonlinear analysis of hydrodynamic instability
in laminar ames, Part I. Derivation of basic equations, Acta
Astronautica 4 (1977) 1176{1206.
[6] G.I. Sivashinsky, On ame propagation under conditions of
stoichiometry, SIAM J. Appl. Math. 39 (1980) 67{82.
[7] D.J. Benney, Long Waves in Liquid lms, J. Math. Phys. 45 (1966)
150{155.
[8] A.P. Hooper, R. Grimshaw, Nonlinear instability at the interface
between two uids, Phys. Fluids 28 (1985) 37{45.
[9] A.V. Coward, D.T. Papageorgiou, Y.S. Smyrlis, Nonlinear stability
of oscillatory coreannular fow: A generalized Kuramoto-Sivashinsky
equation with time periodic coecients, Zeit. Angew. Math. Phys.
(ZAMP) 46 (1995) 1{39.
[10] S.G. Rubin, R.A. Graves, Cubic spline approximation for problems
in uid mechanics, NASA TR R-436, Washington, DC (1975).
[11] W.R. Madych, Miscellaneous error bounds for multiquadrics and
related interpolants, Comput. Math. Appl. 24(12) (1992) 121{38.
[12] W.R. Madych, SA. Nelson Multivariate interpolation and
conditionally positive de nite functions ii, Math. Comput. 54 (1990)
211-230.
[13] Z. Wu, R. Shaback, Local error estimates for radial basis function
interpolation of scaterred data, IMA J. Num. Anal. 13 (1993) 13{27.
[14] R.L. Hardy, Multiquadric equations of topography and other
irregular surfaces, Geophys Res 176 (1971) 1905{1915.
[15] R.L. Hardy, Theory and applications of the multiquadricbiharmonic
method: 20 years of discovery, Comput. Math. Applic. 19 (1990) 163{
208.
[16] E.M.E. Zayed, Kh. A. Gepreel, A generalized (G'/G)-expansion
method for nding traveling wave solutions of coupled nonlinear
evolution equations, Mathematics Scienti c Journal 6 (2010) 97{114.
