Year 2013,
Volume 3,
Issue 2,
231 - 244,
01.12.2013
S. Battal Gazi KARAKOC
Turabi GEYİKLİ
Ali BASHAN
References
- D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. V.25,1966, pp.321-330.
- T. B. Benjamin, J. L. Bona and J. L. Mahoney, Model equations for long waves in nonlinear dispersive media, Phil. Trans. Roy. Soc. Lond. A 272, 1972, pp.47-78.
- J. L. Bona and P. J. Pryant, A mathematical model for long wave generated by wave makers in nonlinear dispersive systems, Proc. Cambridge Phil. Soc. V.73, 1973, pp.391-405.
- J. C. Eilbeck, G. R. McGuire, Numerical study of the regularized long wave equation, II:Interaction of solitary wave, J. Comput. Phys. V.19, N.1, 1975, pp. 43-57.
- P. C. Jain, R. Shankar, T. V. Singh, Numerical solution of regularized long wave equation, Commun. Numer. Meth. Eng. V.9, N.7, 1993, pp. 579-586.
- D. Bhardwaj, R. Shankar, A computational method for regularized long wave equation, Comput. Math. Appl., V.40, N.12, 2000, pp. 1397-1404.
- Q. Chang, G. Wang, B. Guo, Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion, J. Comput. Phys. V.93, N.2, 1995, pp. 360-375.
- L. R. T. Gardner, G. A. Gardner, Solitary waves of the regularized long wave equation, J. Comput. Phys. V.91, 1990, pp.441-459.
- L. R. T. Gardner, G. A. Gardner, A. Dogan, A least-squares Şnite element scheme for the RLW equation, Commun. Numer. Meth. Eng. V.12, N.11, 1996, pp.795-804.
- L. R. T. Gardner, G. A. Gardner, I. Dag, A B-spline Şnite element method for the regularized long wave equation, Commun. Numer. Meth. Eng. V.11, N.1,1995, pp.59-68.
- M. E. Alexander, J. L. Morris, Galerkin method applied to some model equations for nonlinear dispersive waves, J. Comput. Phys. V.30, N.3, 1979, pp. 428-451.
- J. M. Sanz Serna, I. Christie, Petrov Galerkin methods for nonlinear dispersive wave, J. Comput. Phys. V.39 ,1981, pp.94-102.
- A.Dogan, Numerical solution of RLW equation using linear Şnite elements within Galerkin’s method, Appl. Math. Model., V.26, N.7, 2002, pp.771-783.
- A. Esen, S. Kutluay, Application of lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput. V.174,N.2,2006, pp.833-845
- A. A. Soliman, K. R. Raslan, Collocation method using quadratic B-spline for the RLW equation, Int. J. Comput. Math. V.78, N.3, 2001, pp.399-412.
- A. A. Soliman, M. H. Hussien, Collocation solution for RLW equation with septic spline, Appl. Math. Comput. V.161, N.2, 2005, pp.623-636.
- K. R. Raslan, A computational method for the regularized long wave (RLW) equation, Appl. Math. Comput. V.167, N.2, 2005, pp.1101-1118.
- B. Saka, I. Dag and A. Dogan, Galerkin method for the numerical solution of the RLW equation using quadratic B-splines, Int. J. Comput. Math. V.81, N.6, 2004, pp.727-739.
- I. Dag, B. Saka, D. Irk, Application of cubic B-splines for numerical solution of the RLW equation, Appl. Math. Comput. V.159, N.2, 2004, pp.373-389.
- I. Dag, M. N. Ozer, Approximation of RLW equation by least-square cubic B-spline Şnite element method, Appl. Math. Model. V.25, N.3, 2001, pp.221-231.
- S. I .Zaki, Solitary waves of the splitted RLW equation, Comput. Phys. Commun. V.138, N.1, 2001, pp. 80-91.
- B. Y. Gou, W. M. Cao, The Fourier pseudo-spectral method with a restrain operator for the RLW equation, J. Comput. Phys. V.74, N.1, 1988, pp.110-126.
- L. Zhang, A Şnite diﬀerence scheme for generalized long wave equation, Appl. Math. Comput. V.168,N.2 2005,pp. 962-972.
- D. Kaya, S. M. El-Sayed, An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons and Fractals, V.17, N.5, 2003, pp.869-877.
- J. I. Ramos, Solitary wave interactions of the GRLW equation, Chaos, Solitons and Fractals, V.33, N.2, 2007, pp.479-491.
- T. Roshan, A Petrov-Galerkin method for solving the generalized regularized long wave (GRLW) equation, Comput. Math. Appl. V.63, N.5, 2012, pp.943-956.
- L. R. T. Gardner, G. A. Gardner, F. A. Ayoup, N. K. Amein, Simulations of solitary waves of the MRLW equation by B-spline Şnite element, Arab. J. Sci. Eng. V.22, 1997, pp. 183-193.
- A. K. Khalifa, K. R. Raslan, H. M. Alzubaidi, A collocation method with cubic B- splines for solving the MRLW equation, J. Comput. Appl. Math. V.212, N.2, 2008, pp.406-418.
- A. K. Khalifa, K. R. Raslan, H. M. Alzubaidi, A Şnite diﬀerence scheme for the MRLW and solitary wave interactions, Appl. Math. Comput. V.189, N.1, 2007, pp.346-354.
- K. R. Raslan, Numerical study of the modiŞed regularized long wave equation, Chaos, Solitons and Fractals, V.42, N.3, 2009, pp.1845-1853.
- K. R. Raslan and S. M. Hassan, Solitary waves for the MRLW equation, Appl. Math. Lett. V.22, N.7, , pp.984-989. S. B. Gazi Karakoc and T. Geyikli, Petrov-Galerkin Şnite element method for solving the MRLW equation, Mathematical Sciences, 7:25, 2013.
- F. Haq, S. Islam and I. A. Tirmizi, A numerical technique for solution of the MRLW equation usibng quartic B-splines, Appl. Math. Model. V.34, N.12, 2010, pp.4151-4160.
- P. J. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Cambridge Philos.Soc. V.85, 1979, pp.143-159.
- P. M. Prenter, Splines and Variational Methods, (New York:John Wiley), (1975).
A NUMERICAL SOLUTION OF THE MODIFIED REGULARIZED LONG WAVE MRLW EQUATION USING QUARTIC B-SPLINES
Year 2013,
Volume 3,
Issue 2,
231 - 244,
01.12.2013
S. Battal Gazi KARAKOC
Turabi GEYİKLİ
Ali BASHAN
Abstract
In this paper, a numerical solution of the modified regularized long wave MRLW equation is obtained by subdomain finite element method using quartic B-spline functions. Solitary wave motion, interaction of two and three solitary waves and the development of the Maxwellian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the proposed method are tested by calculating the numerical conserved laws and error norms L2 and L∞ . The obtained results show that the method is an effective numerical scheme to solve the MRLW equation. In addition, a linear stability analysis of the scheme is found to be unconditionally stable.
References
- D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. V.25,1966, pp.321-330.
- T. B. Benjamin, J. L. Bona and J. L. Mahoney, Model equations for long waves in nonlinear dispersive media, Phil. Trans. Roy. Soc. Lond. A 272, 1972, pp.47-78.
- J. L. Bona and P. J. Pryant, A mathematical model for long wave generated by wave makers in nonlinear dispersive systems, Proc. Cambridge Phil. Soc. V.73, 1973, pp.391-405.
- J. C. Eilbeck, G. R. McGuire, Numerical study of the regularized long wave equation, II:Interaction of solitary wave, J. Comput. Phys. V.19, N.1, 1975, pp. 43-57.
- P. C. Jain, R. Shankar, T. V. Singh, Numerical solution of regularized long wave equation, Commun. Numer. Meth. Eng. V.9, N.7, 1993, pp. 579-586.
- D. Bhardwaj, R. Shankar, A computational method for regularized long wave equation, Comput. Math. Appl., V.40, N.12, 2000, pp. 1397-1404.
- Q. Chang, G. Wang, B. Guo, Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion, J. Comput. Phys. V.93, N.2, 1995, pp. 360-375.
- L. R. T. Gardner, G. A. Gardner, Solitary waves of the regularized long wave equation, J. Comput. Phys. V.91, 1990, pp.441-459.
- L. R. T. Gardner, G. A. Gardner, A. Dogan, A least-squares Şnite element scheme for the RLW equation, Commun. Numer. Meth. Eng. V.12, N.11, 1996, pp.795-804.
- L. R. T. Gardner, G. A. Gardner, I. Dag, A B-spline Şnite element method for the regularized long wave equation, Commun. Numer. Meth. Eng. V.11, N.1,1995, pp.59-68.
- M. E. Alexander, J. L. Morris, Galerkin method applied to some model equations for nonlinear dispersive waves, J. Comput. Phys. V.30, N.3, 1979, pp. 428-451.
- J. M. Sanz Serna, I. Christie, Petrov Galerkin methods for nonlinear dispersive wave, J. Comput. Phys. V.39 ,1981, pp.94-102.
- A.Dogan, Numerical solution of RLW equation using linear Şnite elements within Galerkin’s method, Appl. Math. Model., V.26, N.7, 2002, pp.771-783.
- A. Esen, S. Kutluay, Application of lumped Galerkin method to the regularized long wave equation, Appl. Math. Comput. V.174,N.2,2006, pp.833-845
- A. A. Soliman, K. R. Raslan, Collocation method using quadratic B-spline for the RLW equation, Int. J. Comput. Math. V.78, N.3, 2001, pp.399-412.
- A. A. Soliman, M. H. Hussien, Collocation solution for RLW equation with septic spline, Appl. Math. Comput. V.161, N.2, 2005, pp.623-636.
- K. R. Raslan, A computational method for the regularized long wave (RLW) equation, Appl. Math. Comput. V.167, N.2, 2005, pp.1101-1118.
- B. Saka, I. Dag and A. Dogan, Galerkin method for the numerical solution of the RLW equation using quadratic B-splines, Int. J. Comput. Math. V.81, N.6, 2004, pp.727-739.
- I. Dag, B. Saka, D. Irk, Application of cubic B-splines for numerical solution of the RLW equation, Appl. Math. Comput. V.159, N.2, 2004, pp.373-389.
- I. Dag, M. N. Ozer, Approximation of RLW equation by least-square cubic B-spline Şnite element method, Appl. Math. Model. V.25, N.3, 2001, pp.221-231.
- S. I .Zaki, Solitary waves of the splitted RLW equation, Comput. Phys. Commun. V.138, N.1, 2001, pp. 80-91.
- B. Y. Gou, W. M. Cao, The Fourier pseudo-spectral method with a restrain operator for the RLW equation, J. Comput. Phys. V.74, N.1, 1988, pp.110-126.
- L. Zhang, A Şnite diﬀerence scheme for generalized long wave equation, Appl. Math. Comput. V.168,N.2 2005,pp. 962-972.
- D. Kaya, S. M. El-Sayed, An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons and Fractals, V.17, N.5, 2003, pp.869-877.
- J. I. Ramos, Solitary wave interactions of the GRLW equation, Chaos, Solitons and Fractals, V.33, N.2, 2007, pp.479-491.
- T. Roshan, A Petrov-Galerkin method for solving the generalized regularized long wave (GRLW) equation, Comput. Math. Appl. V.63, N.5, 2012, pp.943-956.
- L. R. T. Gardner, G. A. Gardner, F. A. Ayoup, N. K. Amein, Simulations of solitary waves of the MRLW equation by B-spline Şnite element, Arab. J. Sci. Eng. V.22, 1997, pp. 183-193.
- A. K. Khalifa, K. R. Raslan, H. M. Alzubaidi, A collocation method with cubic B- splines for solving the MRLW equation, J. Comput. Appl. Math. V.212, N.2, 2008, pp.406-418.
- A. K. Khalifa, K. R. Raslan, H. M. Alzubaidi, A Şnite diﬀerence scheme for the MRLW and solitary wave interactions, Appl. Math. Comput. V.189, N.1, 2007, pp.346-354.
- K. R. Raslan, Numerical study of the modiŞed regularized long wave equation, Chaos, Solitons and Fractals, V.42, N.3, 2009, pp.1845-1853.
- K. R. Raslan and S. M. Hassan, Solitary waves for the MRLW equation, Appl. Math. Lett. V.22, N.7, , pp.984-989. S. B. Gazi Karakoc and T. Geyikli, Petrov-Galerkin Şnite element method for solving the MRLW equation, Mathematical Sciences, 7:25, 2013.
- F. Haq, S. Islam and I. A. Tirmizi, A numerical technique for solution of the MRLW equation usibng quartic B-splines, Appl. Math. Model. V.34, N.12, 2010, pp.4151-4160.
- P. J. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Cambridge Philos.Soc. V.85, 1979, pp.143-159.
- P. M. Prenter, Splines and Variational Methods, (New York:John Wiley), (1975).