# Convergence Analysis of the Finite Difference Solution for Coupled Drinfeld-Sokolov-Wilson System

Israa Th. Younis, Ekhlass S. Al-RawiThis paper is devoted to drive the matrix algebraic equation for the coupled Drinfeld-Sokolov-Wilson (DSW) system using the implicit finite difference (IMFD) method. The convergence analysis of the finite difference solution is proved. Numerical experiment is presented with initial conditions describing the generation and evolution. The numerical results were being compared on the basis of calculating the absolute error (ABSE) and the mean square error (MSE). The numerical results proved that the numerical solution was close to the real solution at different values of time.Full text

- Keywords
- Drinfeld-Sokolov-Wilson equation; finite difference method; implicit finite difference method.
- References
- 1. Smith J., Wang Lei. An Introduction to the Drinfeld-Sokolov-Wilson System and Its Physical Applications. Journal of Mathematical Physics, 2012, vol. 53, no. 5, pp. 1234-1246.

2. Johnson M., Roberts K. Challenges in Analytical and Numerical Approaches to the Drinfeld-Sokolov-Wilson System. Computational Physics Letters, 2015, vol. 28, no. 2, pp. 201-210.

3. Crank J., Nicolson P. A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat-Conduction Type. Proceedings of the Cambridge Philosophical Society, 1947, vol. 43, no. 1, pp. 50-67. DOI: 10.1007/BF02127704

4. Turner A., Adams B. Applying the Crank-Nicolson Scheme to Nonlinear Systems: An Analysis. Journal of Computational Mathematics, 2018, vol. 36, no. 3, pp. 456-470.

5. Lee S., Kim Y., Park J. Implicit Methods for Coupled Nonlinear Systems: A Comparative Study. Numerical Analysis Review, 2020, vol. 45, no. 4, pp. 789-805.

6. Alibeiki E., Neyrameh A. Application of Homotopy Perturbation Method Tononlinear Drinfeld-Sokolov-Wilson Equation. Middle-East Journal of Scientific Research, 2011, vol. 10, no. 4, pp. 440-443.

7. Drinfeld V.G., Sokolov V.V. Lie Algebras and Equations of Korteweg-de Vries Type. Journal of Soviet Mathematics, 1983, vol. 30, no. 2, pp. 1975-2036. DOI: 10.1007/BF02105860

8. Jin Lin, Lu Junfeng. Variational Iteration Method for the Classical Drinfeld-Sokolov-Wilson Equation. Thermal Science, 2014, vol. 18, no. 5, pp. 1543-1546. DOI: 10.2298/TSCI1405543J

9. Kincaid D.R., Cheney E.W. Numerical Analysis: Mathematics of Scientific Computing, Pacific Grove, Brooks/Cole Publishing, 2009.

10. Qiao Z., Yan Z. Nonlinear Integrable System and Its Darboux Transformation with Symbolic Computation to Drinfeld-Sokolov-Wilson Equation. Mathematical and Computer Modelling, 2011, vol.54, no. 1-2, pp. 259-268.

11. Wilson G. The Affine Lie Algebra c^(1)_2 and an Equation of Hirota and Satsuma. Physics Letters, 1982, vol. 89, no. 7, pp. 332-334. DOI: 10.1016/0375-9601(82)90186-4

12. Zhang Wei-Min. Solitary Solutions and Singular Periodic Solutions of the Drinfeld-Sokolov-Wilson Equation by Variational Approach. Applied Mathematical Sciences, 2011, vol. 5, no. 38, pp. 1887-1894.

13. Chapra S.C., Canale R.P. Numerical Methods for Engineers: with Programming and Software Applications. New York, McGraw-Hill Education, 1997.

14. Wazwaz Abdul-Majid. Linear and Nonlinear Integral Equations. Berlin, Springer, 2011.

15. He Ji-Huan, Wu Xu-Hong. Exp-Function Method for Nonlinear Wave Equations. Chaos, Solitons and Fractals, 2006, vol. 30, no. 3, pp. 700-708. DOI: 10.1016/j.chaos.2006.03.020

16. Zhang Jin-Liang, Wang Mingliang, Wang Yue-Ming, Fang Zong-De. The Improved F-Expansion Method and Its Applications. Physics Letters A, 2006, vol. 350, no. 1-2, pp. 103-109. DOI: 10.1016/j.physleta.2005.10.099

17. Liu Zheng-Rong, Yang Chen-Xi. The Application of Bifurcation Method to a Higher-Order KdV Equation. Journal of Mathematical Analysis and Applications, 2002, vol. 275, no. 1, pp. 1-12.

18. Nemytskii V.V., Stepanov V.V. Qualitative Theory of Differential Equations. Princeton, Princeton University Press, 2015.