Septic B-Spline Collocation Method for the Numerical Solution of the Modified Equal Width Wave Equation

ABSTRACT

Numerical solutions of the modified equal width wave equation are obtained by using collocation method with septic B-spline finite elements with three different linearization techniques. The motion of a single solitary wave, interaction of two solitary waves and birth of solitons are studied using the proposed method. Accuracy of the method is discussed by computing the numerical conserved laws error norms L2 and L∞. The numerical results show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis shows that this numerical scheme, based on a Crank Nicolson approximation in time, is unconditionally stable.

Numerical solutions of the modified equal width wave equation are obtained by using collocation method with septic B-spline finite elements with three different linearization techniques. The motion of a single solitary wave, interaction of two solitary waves and birth of solitons are studied using the proposed method. Accuracy of the method is discussed by computing the numerical conserved laws error norms L2 and L∞. The numerical results show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis shows that this numerical scheme, based on a Crank Nicolson approximation in time, is unconditionally stable.

Cite this paper

nullT. Geyikli and S. Karakoc, "Septic B-Spline Collocation Method for the Numerical Solution of the Modified Equal Width Wave Equation,"*Applied Mathematics*, Vol. 2 No. 6, 2011, pp. 739-749. doi: 10.4236/am.2011.26098.

nullT. Geyikli and S. Karakoc, "Septic B-Spline Collocation Method for the Numerical Solution of the Modified Equal Width Wave Equation,"

References

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[2] L. R. T. Gardner and G. A. Gardner, “Solitary Waves of the Regularized Long Wave Equation,” Journal of Computational Physics, Vol. 91, No. 2, 1990, pp. 441-459. doi:10.1016/0021-9991(90)90047-5

[3] L. R. T. Gardner and G. A. Gardner, “Solitary Waves of the Equal Width Wave Equation,” Journal of Computational Physics, Vol. 101, No. 1, 1992, pp. 218-223. doi:10.1016/0021-9991(92)90054-3

[4] L. R. T. Gardner, G. A. Gardner, F. A. Ayoup and N. K. Amein, “Simulations of the EW Undular Bore,” Communications in Numerical Methods in Engineering, Vol. 13, No. 7, 1997, pp. 583-592. doi:10.1002/(SICI)1099-0887(199707)13:7<583::AID-CNM90>3.0.CO;2-E

[5] S. I. Zaki, “Solitary Wave Interactions for the Modified Eual Width Equation,” Computer Physics Communications, Vol. 126, No. 3, 2000, pp. 219-231. doi:10.1016/S0010-4655(99)00471-3

[6] S. I. Zaki, “A Least-Squares Finite Element Scheme for the Ew Equation,” Computer Methods in Applied Mechanics and Engineering, Vol. 189, No. 2, 2000, pp. 587- 594. doi:10.1016/S0045-7825(99)00312-6

[7] A. M. Wazwaz, “The tanh and sine-cosine Methods for a Reliable Treatment of the Modified Equal Width Equation and Its Variants,” Communications in Nonlinear Science and Numerical Simulation, Vol. 11, No. 2, 2006, pp. 148-160. doi:10.1016/j.cnsns.2004.07.001

[8] A. Esen, “A Numerical Solution of the Equal Width Wave Equation by a Lumped Galerkin Method,” Applied Mathematics and Computational, Vol. 168, No. 1, 2005, pp. 270-282. doi:10.1016/j.amc.2004.08.013

[9] A. Esen, “A Lumped Galerkin Method for the Numerical Solution of the Modified Equal Width Wave Equation Using Quadratic B Splines,” International Journal of Computer Mathematics, Vol. 83, No. 5-6, 2006, pp. 449- 459. doi:10.1080/00207160600909918

[10] B. Saka, “Algorithms for Numerical Solution of the Modified Equal Width Wave Equation Using Collocation Method,” Mathematical and Computer Modelling, Vol. 45, No. 9-10, 2007, pp. 1096-1117. doi:10.1016/j.mcm.2006.09.012

[11] B. Saka and ?. Dag, “Quartic B-Spline Collocation Method to the Numerical Solutions of the Burgers Equation,” Chaos, Solitons and Fractals, Vol. 32, No. 3, May 2007, pp. 1125-1137. doi:10.1016/j.chaos.2005.11.037

[12] J. F. Lu, “He’s Variational Method for the Modified Equal Width Wave Equation,” Chaos Solitons and Fractals, Vol. 39, No. 5, 2009, pp. 2102-2109. doi:10.1016/j.chaos.2007.06.104

[13] S. Hamdi, W. E. Enright, W. E. Schiesser, J. J. Gottlieb and A. Alaal, “Exact Solutions of the Generalized Equal Width Wave Equation,” Proceedings of the International Conference on Computational Science and Its Application, LNCS, Springer-Verlog, Berlin, Vol. 2668, 2003, pp. 725-734.

[14] D. J. Evans, K. R. Raslan and A. Alaal, “Solitary Waves for the Generalized Equal Width (Gew) Equation,” International Journal of Computer Mathematics, Vol. 82, No. 4, 2005, pp. 445-455.

[15] T. Geyikli, “Modelling Solitary Waves of a Fifth-Order Non-Linear Wave Equation,” International Journal of Computer Mathematics, Vol. 84, No. 7, July 2007, pp. 1079-1087. doi:10.1080/00207160701294368

[16] S. G. Rubin and R. A. Graves, “A Cubic Spline Approximation for Problems in Fluid Mechanics,” Nasa TR R-436, Washington, DC, 1975.

[17] J. Caldwell and P. Smith, “Solution of Burgers’ Equation with a Large Reynolds Number,” Applied Mathematical Modelling, Vol. 6, No. 5, October 1982, pp. 381-386. doi:10.1016/S0307-904X(82)80102-9

[18] A. Esen and S. Kutluay, “Solitary Wave Solutions of the Modified Equal Width Wave Equation,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 2008, pp. 1538-1546. doi:10.1016/j.cnsns.2006.09.018

[19] K. R. Raslan, “Collocation Method Using Cubic B-Spline for the Generalized Equal Width Equation,” International Journal of Simulation and Process Modelling, Vol. 2, No. 1-2, 2006, pp. 37-44.

[1] P. J. Morrison, J. D. Meiss and J. R. Carey, “Scattering of Regularized-Long-Wave Solitary Waves,” Physica D: Non- linear Phenomena, Vol. 11, No. 3, 1984, pp. 324-336. doi:10.1016/0167-2789(84)90014-9

[2] L. R. T. Gardner and G. A. Gardner, “Solitary Waves of the Regularized Long Wave Equation,” Journal of Computational Physics, Vol. 91, No. 2, 1990, pp. 441-459. doi:10.1016/0021-9991(90)90047-5

[3] L. R. T. Gardner and G. A. Gardner, “Solitary Waves of the Equal Width Wave Equation,” Journal of Computational Physics, Vol. 101, No. 1, 1992, pp. 218-223. doi:10.1016/0021-9991(92)90054-3

[4] L. R. T. Gardner, G. A. Gardner, F. A. Ayoup and N. K. Amein, “Simulations of the EW Undular Bore,” Communications in Numerical Methods in Engineering, Vol. 13, No. 7, 1997, pp. 583-592. doi:10.1002/(SICI)1099-0887(199707)13:7<583::AID-CNM90>3.0.CO;2-E

[5] S. I. Zaki, “Solitary Wave Interactions for the Modified Eual Width Equation,” Computer Physics Communications, Vol. 126, No. 3, 2000, pp. 219-231. doi:10.1016/S0010-4655(99)00471-3

[6] S. I. Zaki, “A Least-Squares Finite Element Scheme for the Ew Equation,” Computer Methods in Applied Mechanics and Engineering, Vol. 189, No. 2, 2000, pp. 587- 594. doi:10.1016/S0045-7825(99)00312-6

[7] A. M. Wazwaz, “The tanh and sine-cosine Methods for a Reliable Treatment of the Modified Equal Width Equation and Its Variants,” Communications in Nonlinear Science and Numerical Simulation, Vol. 11, No. 2, 2006, pp. 148-160. doi:10.1016/j.cnsns.2004.07.001

[8] A. Esen, “A Numerical Solution of the Equal Width Wave Equation by a Lumped Galerkin Method,” Applied Mathematics and Computational, Vol. 168, No. 1, 2005, pp. 270-282. doi:10.1016/j.amc.2004.08.013

[9] A. Esen, “A Lumped Galerkin Method for the Numerical Solution of the Modified Equal Width Wave Equation Using Quadratic B Splines,” International Journal of Computer Mathematics, Vol. 83, No. 5-6, 2006, pp. 449- 459. doi:10.1080/00207160600909918

[10] B. Saka, “Algorithms for Numerical Solution of the Modified Equal Width Wave Equation Using Collocation Method,” Mathematical and Computer Modelling, Vol. 45, No. 9-10, 2007, pp. 1096-1117. doi:10.1016/j.mcm.2006.09.012

[11] B. Saka and ?. Dag, “Quartic B-Spline Collocation Method to the Numerical Solutions of the Burgers Equation,” Chaos, Solitons and Fractals, Vol. 32, No. 3, May 2007, pp. 1125-1137. doi:10.1016/j.chaos.2005.11.037

[12] J. F. Lu, “He’s Variational Method for the Modified Equal Width Wave Equation,” Chaos Solitons and Fractals, Vol. 39, No. 5, 2009, pp. 2102-2109. doi:10.1016/j.chaos.2007.06.104

[13] S. Hamdi, W. E. Enright, W. E. Schiesser, J. J. Gottlieb and A. Alaal, “Exact Solutions of the Generalized Equal Width Wave Equation,” Proceedings of the International Conference on Computational Science and Its Application, LNCS, Springer-Verlog, Berlin, Vol. 2668, 2003, pp. 725-734.

[14] D. J. Evans, K. R. Raslan and A. Alaal, “Solitary Waves for the Generalized Equal Width (Gew) Equation,” International Journal of Computer Mathematics, Vol. 82, No. 4, 2005, pp. 445-455.

[15] T. Geyikli, “Modelling Solitary Waves of a Fifth-Order Non-Linear Wave Equation,” International Journal of Computer Mathematics, Vol. 84, No. 7, July 2007, pp. 1079-1087. doi:10.1080/00207160701294368

[16] S. G. Rubin and R. A. Graves, “A Cubic Spline Approximation for Problems in Fluid Mechanics,” Nasa TR R-436, Washington, DC, 1975.

[17] J. Caldwell and P. Smith, “Solution of Burgers’ Equation with a Large Reynolds Number,” Applied Mathematical Modelling, Vol. 6, No. 5, October 1982, pp. 381-386. doi:10.1016/S0307-904X(82)80102-9

[18] A. Esen and S. Kutluay, “Solitary Wave Solutions of the Modified Equal Width Wave Equation,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 2008, pp. 1538-1546. doi:10.1016/j.cnsns.2006.09.018

[19] K. R. Raslan, “Collocation Method Using Cubic B-Spline for the Generalized Equal Width Equation,” International Journal of Simulation and Process Modelling, Vol. 2, No. 1-2, 2006, pp. 37-44.