A NUMERICAL APPROACH TO CAUDREY DODD GIBBON EQUATION VIA COLLOCATION METHOD USING QUINTIC B-SPLINE BASIS

Abstract

In this manuscript, a numerical approach is investigated to Caudrey-Dodd- Gibbon CDG equation. The nonlinear CDG equation is reduced to a system of partial dierential equation using uxxx = v. The new numerical solutions are obtained with a combination of collocation method with nite element method which is one of the most important methods among all numerical approaches. In order to proceed the method, solution for each unknown is written as a linear combination of time parameters and quintic B-spline basis. Then, with the advantage of the collocation method, a system of algebraic equation systems is formulated easily. Solving the system iteratively by a method results in numerical solutions of the CDG equation. The numerical solutions together with the error norms L2;L1 are tabulated. Additionally, graphical simulations of the solutions are depicted by gures.

Keywords

• Finite element method,collocation,quintic B-spline basis,Caudrey-Dodd- Gibbon equation.

References

1. Bulut, H., Sulaiman, T. A., Baskonus, H. M.,(2016), New solitary and optical wave structures to the Korteweg–de Vries equation with dual-power law nonlinearity, Optical and Quantum Electronics, 48.12:564, pp. 3-14.
2. Fakhar-Izadi, F., Dehghan M.,(2018), Fully spectral collocation method for nonlinear parabolic partial integro-differential equations, Applied Numerical Mathematics, 123, pp. 99-120.
3. Gnitchogna, R., Atangana, A.,(2018), New two step Laplace Adam-Bashforth method for integer a noninteger order partial differential equations, Numerical Methods for Partial Differential Equations, 34.5, pp. 1739-1758.
4. Baskonus, H. M., Sulaiman, T. A., Bulut H., (2017), On the novel wave behaviors to the coupled nonlinear Maccari’s system with complex structure, Optik-International Journal for Light and Electron Optics, 131, pp. 1036-1043.
5. Baseri, A., Abbasbandy, S., Babolian E.,(2018), A collocation method for fractional diffusion equation in a long time with Chebyshev functions, Applied Mathematics and Computation, 322, pp. 55-65.
6. Oru¸c, ¨O., Bulut, F., Eseni A.,(2016),Numerical solutions of regularized long wave equation by haar wavelet method, Mediterranean Journal of Mathematics 13.5, pp. 3235-3253.
7. Ba¸shan, A., Yagmurlu, M. N., Ucar, Y., Esen, A., (2018), A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schr¨odinger equation, The European Physical Journal Plus, 133.1, pp. 2-15.
8. Atangana, A., Owolabi, K. M., (2018), New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena 13.1:3, pp. 1-21.

9. Oru¸c, ¨O., Bulut, F., Esen A.,(2017), A numerical treatment based on Haar wavelets for coupled KdV equation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 7.2, pp. 195-204.
10. Aiyer, R.N., Fuchssteiner, B., Oevel, W.,(1986), Solitons and discrete eigenfunctions of the recursion operator of non-linear evolution equations: the Caudrey–Dodd–Gibbon–Sawada–Kotera equations, J. Phys. A: Math. Gen. 19, pp. 3755–3770.
11. Salas, A., (2008), Some exact solutions for the Caudrey-Dodd-Gibbon equation, arXiv preprint, arXiv:0805.2969.
12. Abdollahzadeh, M. et al., (2010), Exact travelling solutions for fifth order Caudrey-Dodd-Gibbon equation, International Journal of Applied Mathematics and Computation, 2.4, 81-90.
13. Naher, H., Abdullah, F. A., Akbar, M. A.,(2011), The (G’/G)-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation, Mathematical Problems in Engineering, vol:2011, pp. 1-11.
14. Abdollahzadeh, M., Hosseini, M., Ghanbarpour, M., Shirvani, H., (2010), Exact travelling solutions for fifth order Caudrey-Dodd-Gibbon equation, International Journal of Applied Mathematics and Computation, 2.4, pp. 81-90.
15. Jiang, B., Bi Q., (2010), A study on the bilinear Caudrey–Dodd–Gibbon equation, Nonlinear Analysis: Theory, Methods & Applications, 72.12, pp. 4530-4533.
16. Salas, A., Castillo, J. E., (2011), Computing multi-soliton solutions to Caudrey-Dodd-Gibbon equation by Hirotas method, International Journal of Physical Sciences, 6.34, pp. 7729-7737.
17. Chen, H., Xu,Z., Dai, Z., (2015), Breather soliton and cross two-soliton solutions for the fifth order Caudrey-Dodd-Gibbon (CDG) equation, International Journal of Numerical Methods for Heat & Fluid Flow, 25.3, pp. 651-655.
18. Wazwaz, A., (2008), Multiple-soliton solutions for the fifth order Caudrey–Dodd–Gibbon (CDG) equation, Applied Mathematics and Computation, 197.2, pp. 719-724.
19. Prenter, P. M., (2008), Splines and variational methods. Courier Corporation.
20. Kutluay, S., Yagmurlu, N. M., (2012), The modified Bi-quintic B-splines for solving the two- dimensional unsteady Burgers’ equation, European International Journal of Science and Technology, 1.2, pp. 23-39.
21. Yagmurlu, N. M., Tasbozan, O., Ucar, Y., (2016), Numerical solutions of the Combined KdV-mKdV Equation by a Quintic B-spline Collocation Method, Appl. Math. Inf. Sci. Lett 4.1, pp. 19-24.