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QUINTIC B-SPLINE COLLOCATION METHOD FOR NUMERICAL SOLUTION OF THE RLW EQUATION

Part of:Partial differential equations, boundary value problemsIncompressible inviscid fluidsNumerical approximation and computational geometry

Published online by Cambridge University Press:  01 January 2008

BÜLENT SAKA,
İDRIS DAĞ and
DURSUN IRK
BÜLENT SAKA*
Affiliation:
Mathematics Department, Eskişehir Osmangazi University, 26480 Eskişehir, Turkey (email: bsaka@ogu.edu.tr)
İDRIS DAĞ
Affiliation:
Mathematics Department, Eskişehir Osmangazi University, 26480 Eskişehir, Turkey (email: bsaka@ogu.edu.tr)
DURSUN IRK
Affiliation:
Mathematics Department, Eskişehir Osmangazi University, 26480 Eskişehir, Turkey (email: bsaka@ogu.edu.tr)
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Abstract

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Quintic B-spline collocation schemes for numerical solution of the regularized long wave (RLW) equation have been proposed. The schemes are based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The quintic B-spline collocation method over finite intervals is also applied to the time-split RLW equation and space-split RLW equation. After stability analysis is applied to all the schemes, the results of the three algorithms are compared by studying the propagation of the solitary wave, interaction of two solitary waves and wave undulation.

Keywords

RLW equationcollocationsolitary wavesquintic B-splinesundular bore

MSC classification

Secondary:65D07: Splines65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods65N35: Spectral, collocation and related methods76B25: Solitary waves

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 49 , Issue 3 , January 2008, pp. 389 - 410
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Abdulloev, Kh. O., Bogolubsky, I. L. and Makhankov, V. G., “One more example of inelastic soliton interaction”, Phys. Lett. A 56 (1976) 427–428.CrossRefGoogle Scholar
[2]Avilez-Valente, P. and Seabra-Santos, F. J., “A Petrov–Galerkin finite element scheme for the regularized long wave equation”, Comput. Mech. 34 (2004) 256–270.CrossRefGoogle Scholar
[3]Benjamin, T. B., Bona, J. L. and Mahony, J. J., “Model equations for long waves in nonlinear dispersive systems”, Philos. Trans. R. Soc., Lond., Ser. A 272 (1972) 47–78.Google Scholar
[4]Bhardwaj, D. and Shankar, R., “A computational method for regularized long wave equation”, Comput. Math. Appl. 40 (2000) 1397–1404.CrossRefGoogle Scholar
[5]Bona, J. L. and Bryant, P. J., “A mathematical model for long waves generated by wave makers in nonlinear dispersive systems”, Proc. Cambridge Philos. Soc. 73 (1973) 391–405.CrossRefGoogle Scholar
[6]Dağ, İ., Doğan, A. and Saka, B., “B-spline collocation methods for numerical solutions of the RLW equation”, Int. J. Comput. Math. 80 (2003) 743–757.Google Scholar
[7]Dağ, İ., Saka, B. and Boz, A., “Quintic B-spline Galerkin methods for numerical solutions of the Burgers’ equation”, in Proc. Int. Conf. Dynamical Systems and Applications, Antalya, Turkey, 5–10 July 2004 (Altas Conferences Inc.), 295–309.Google Scholar
[8]Esen, A. and Kutluay, S., “Application of a lumped Galerkin method to the regularized long wave equation”, Appl. Math. Comput. 174 (2006) 833–845.Google Scholar
[9]Gardner, L. R. T. and Gardner, G. A., “Solitary wave of the regularized long wave equation”, J. Comput. Phys. 91 (1990) 441–459.CrossRefGoogle Scholar
[10]Gardner, G. A., Gardner, L. R. T. and Ali, A. H. A., “Modelling solitons of the Korteweg–de Vries equation with quintic B-splines”, U.C.N.W. Math., Preprint, 1990.Google Scholar
[11]Gardner, L. R. T., Gardner, G. A., Ayoub, F. A. and Amein, N. K., “Modelling an undular bore with B-splines”, Comput. Methods Appl. Mech. Engrg. 147 (1997) 147–152.CrossRefGoogle Scholar
[12]Gardner, L. R. T., Gardner, G. A. and Daǧ, İ., “A B-spline finite element method for the regularized long wave equation”, Comm. Numer. Methods Engrg. 11 (1995) 59–68.CrossRefGoogle Scholar
[13]Irk, D., Dağ, İ. and Doğan, A., “Numerical integration of the RLW equation using cubic splines”, ANZIAM J. 47 (2005) 131–142.CrossRefGoogle Scholar
[14]Olver, P. J., “Euler operators and conservation laws of the BBM equation”, Math. Proc. Cambridge Philos. Soc. 85 (1979) 143–159.CrossRefGoogle Scholar
[15]Peregrine, D. H., “Calculations of the development of an undular bore”, J. Fluid. Mech. 25 (1966) 321–330.CrossRefGoogle Scholar
[16]Prenter, P. M., Splines and variational methods (John Wiley & Sons, New York, 1975).Google Scholar
[17]Raslan, K. R., “A computational method for the regularized long wave (RLW) equation”, Appl. Math. Comput. 167 (2005) 1101–1118.Google Scholar
[18]Rosenberg, V., Methods for solution of partial differential equations, Vol. 113 (Elsevier, New York, 1969).Google Scholar
[19]Saka, B., “A finite element method for equal width equation”, Appl. Math. Comput. 175 (2006) 730–747.Google Scholar
[20]Saka, B. and Dağ, İ., “A collocation method for the numerical solution of the RLW equation using cubic B-spline basis”, Arab. J. Sci. Eng. 30 (2005) 39–50.Google Scholar
[21]Saka, B., Dağ, İ. and Doğan, A., “Galerkin method for the numerical solution of the RLW equation using quadratic B-splines”, Int. J. Comput. Math. 81 (2004) 727–739.CrossRefGoogle Scholar
[22]Shashkov, M., Conservative finite-difference methods on general grids (CRC Press, Boca Raton, FL, 1996).Google Scholar
[23]Zaki, S. I., “A quintic B-spline finite elements scheme for the KdVB equation”, Comput. Methods Appl. Mech. Engrg. 188 (2000) 121–134.CrossRefGoogle Scholar