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On the Semi-analytical and Hybrid Methods for the Drinfeld-Sokolov-Wilson System Modelling Dispersive Water Waves

Year 2025, Volume: 46 Issue: 1, 98 - 108, 25.03.2025

Emre Aydın , İnci Çilingir Süngü

https://doi.org/10.17776/csj.1510043

Abstract

In this study, modified variational iteration method (MVIM), modified variational iteration Laplace transform method (MVILTM) and modified variational iteration Sumudu transform method (MVISTM) are used to examine the Drinfeld-Sokolov-Wilson (DSW) system. Semi-analytical solutions have been obtained and compared with the analytical solutions. Moreover, it illustrates the effect of wave parameter on the approximate solutions. The exact solutions and semi-analytical solutions of the DSW system are compared with each other. Tables give maximum errors of semi-analytical solutions for various iteration values. The comparison of relative errors for various iteration values and the effect of change of wave constant is visualized by figures. Also, it commented on the effectiveness and usefulness of the methods when applied to the DSW system.

Keywords

Modified Variational İteration Method (MVIM); Modified Variational İteration Laplace Transform Method (MVILTM); Modified Variational İteration Sumudu Transform Method (MVISTM); The Drinfeld-Sokolov-Wilson (DSW) System.

References

  • [1] Drinfeld V.G., Sokolov V.V., Equations of Korteweg-de Vries type and simple lie algebras, Dokl. Akad. Nauk, 258 (1) (1981) 11-16.
  • [2] Wilson G., The affine lie algebra C_2^1 and an equation of Hirota and Satsuma, Phys. Lett. A, 89 (7) (1982) 332-334.
  • [3] Drinfeld V.G., Sokolov V.V., Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math., 24 (1984) 81-180.
  • [4] Gao W., Veeresha P., Prakasha D. G., Baskonus, H. M., Yel, G., A powerful approach for fractional Drinfeld-Sokolov-Wilson equation with Mittag-Leffler law, Alex. Eng. J., 58 (4) (2019) 1301-1311.
  • [5] Saifullah S., Ali A., Shah K., Promsakon C., Investigation of fractal-fractional nonlinear Drinfeld-Sokolov-Wilson system with non-singular operators, Results Phys., 33 (2022) 105145.
  • [6] Arora R., Kumar A., Solution of the coupled Drienfeld’s-Sokolov-Wilson (DSW) system by homotopy analysis method, Adv. Sci. Eng. Med., 5 (10) (2013) 1105-1111.
  • [7] Azizi N., Pourgholi R., Applications of Sine-Cosine wavelets method for solving Drinfel’d-Sokolov-Wilson system, Adv. Syst. Sci. Appl., 21 (3) (2021) 75-90.
  • [8] Salim B. J., Jasim O. A., Ali Z. Y., Numerical solution of Drinfeld-Sokolov-Wilson system by using modified Adomian decomposition method, Indones. J. Electr. Eng. Comput. Sci., 21 (1) (2021) 590-599.
  • [9] Eskondari E. M., Taghizadeh N., Applications of two methods in exact wave solutions in the space-time fractional Drinfeld-Sokolov-Wilson system, Int. J. Differ. Equ., 2022 (2022) 4470344.
  • [10] Ali N., Yassen M. F., Asiri S. A., Nawaz R., Zada L., Alam M. M., Sene N., New iterative method for solving a coupled system of fractional order Drinfeld-Sokolov-Wilson (FDSW) and fractional shallow water (FSW) equations, J. Nanomater., 2022 (2022) 8370107.
  • [11] Taghizadeh N., Neirameh A., Complex solutions for Drinfeld’s-Sokolov-Wilson system, Int. J. Eng. Sci. Technol., 1 (1) (2011) 1-6.
  • [12] Raslan A., Entesar A., Enhancing Banach’s contraction method using the particle swarm optimization to solve the system Drinfeld-Sokolov-Wilson, J. Phys. Conf. Ser., 2322 (1) (2022) 012031.
  • [13] Lindeberg L., Dao T., Mattson, K., A high order accurate finite difference method for the Drinfel’d-Sokolov-Wilson equation, J. Sci. Comput., 88 (1) (2021) 18.
  • [14] Zhang Y., Zhao Z., Lie symmetry analysis, Lie- Backlund symmetries, explicit solutions and conservation laws of Drinfeld-Sokolov-Wilson system, Bound. Value Probl., 154, (2017).
  • [15] Singh P. K., Vishal K., Som T., Solution of fractional Drinfeld-Sokolov-Wilson equation using homotopy perturbation transform method, Appl. Appl. Math., 10 (1) (2015) 460-472.
  • [16] Al-Rozbayani A. M. A., Ali A. H., Applied Sumudu transform with Adomian decomposition method to the coupled Drinfeld-Sokolov-Wilson system, Rafidain J. Comput. Sci. Math., 15 (2) (2021) 139-147.
  • [17] Alam M. N., Bonyah E., Fayz-Al-Asad M., Reliable analysis for the Drinfeld-Sokolov-Wilson equation in mathematical physics, Palest. J. Math., 11 (1) (2022) 397-407.
  • [18] Usman M. H. A., Zaman F. D., Eldin S. M., Symmetry analysis and exact Jacobi elliptic solutions for the nonlinear couple Drinfeld Sokolov Wilson dynamical system arising in shallow water waves, Results Phys., 51 (2023) 106613.
  • [19] Shahzad M. U., Ur-Rehman H., Awan A. U., Zafar Z., Hassan A. M., Iqbal I., Analysis of the exact solutions of nonlinear coupled Drinfeld-Sokolov-Wilson equation through Φ^6-model expansion method, Results Phys., 52 (2023) 106671.
  • [20] Iqbal M., Seadawy A. R., Lu D., Zhang Z., Computational approachand dynamical analysis of multiple solitary wave solutions for nonlinear coupled Drinfeld-Sokolov-Wilson equation, Results Phys., 55 (2023) 107099.
  • [21] Younis I. T., Al-Rawi E. S., Improved finite difference technique via Adominian polynomial to solve the coupled Drinfeld’s-Sokolov-Wilson system, Int. J. Math. Math. Sci., 2023 (2023) 6916596.
  • [22] Aydemir T., New exact solutions of the Drinfeld-Sokolov-Wilson system by the generalized unified method, J. New Theory, 44 (2023) 10-19.
  • [23] Hakkaev S., Spectral stability of periodic waves for the Drinfeld-Sokolov-Wilson equation, J. Math. Anal. Appl., 533 (1) (2024) 128016.
  • [24] Abassy T. A., El Tawil M. A., El Zoheiry H., Toward a Modified Variational Iteration Method, J. Comput. Appl. Math., 207 (1) (2007) 137-147.
  • [25] Abassy T. A., El Tawil M. A., El Zoheiry H., Solving Nonlinear Partial Differential Equations Using the Modified Variational Iteration Padé Technique, J. Comput. Appl. Math. 207 (1) (2007) 73-91.
  • [26]Abassy T. A., Modified Variational Iteration Method (Nonlinear Homogeneous Initial Value Problem), Comput. Math. Appl., 59 (2) (2010) 912-918.
  • [27] Abbasbandy, S., Shivanian, E., Application of the Variational Iteration Method for System of Nonlinear Volterra’s Integro-Differential Equations, Math. Comput. Appl., 14 (2) (2009) 147-158.
  • [28] Aggarwal S., Chaudhary R., A comparative study of Mohand and Laplace transforms, J. Emerg. Technol. Innov. Res., 6 (2) (2019) 230-240.
  • [29] Aggarwal S., Sharma S.D., Sumudu transform of error function, J. Appl. Sci. Comput., 6 (6) (2019) 1222-1231.
  • [30] Zhang W. M., Solitary solutions and singular periodic solutions of the Drinfeld-Sokolov-Wilson equation by variational approach, Appl. Math. Sci., 5 (38) (2011) 1887-1894.

Year 2025, Volume: 46 Issue: 1, 98 - 108, 25.03.2025

Emre Aydın , İnci Çilingir Süngü

https://doi.org/10.17776/csj.1510043

Abstract

References

  • [1] Drinfeld V.G., Sokolov V.V., Equations of Korteweg-de Vries type and simple lie algebras, Dokl. Akad. Nauk, 258 (1) (1981) 11-16.
  • [2] Wilson G., The affine lie algebra C_2^1 and an equation of Hirota and Satsuma, Phys. Lett. A, 89 (7) (1982) 332-334.
  • [3] Drinfeld V.G., Sokolov V.V., Lie algebras and equations of Korteweg-de Vries type, J. Sov. Math., 24 (1984) 81-180.
  • [4] Gao W., Veeresha P., Prakasha D. G., Baskonus, H. M., Yel, G., A powerful approach for fractional Drinfeld-Sokolov-Wilson equation with Mittag-Leffler law, Alex. Eng. J., 58 (4) (2019) 1301-1311.
  • [5] Saifullah S., Ali A., Shah K., Promsakon C., Investigation of fractal-fractional nonlinear Drinfeld-Sokolov-Wilson system with non-singular operators, Results Phys., 33 (2022) 105145.
  • [6] Arora R., Kumar A., Solution of the coupled Drienfeld’s-Sokolov-Wilson (DSW) system by homotopy analysis method, Adv. Sci. Eng. Med., 5 (10) (2013) 1105-1111.
  • [7] Azizi N., Pourgholi R., Applications of Sine-Cosine wavelets method for solving Drinfel’d-Sokolov-Wilson system, Adv. Syst. Sci. Appl., 21 (3) (2021) 75-90.
  • [8] Salim B. J., Jasim O. A., Ali Z. Y., Numerical solution of Drinfeld-Sokolov-Wilson system by using modified Adomian decomposition method, Indones. J. Electr. Eng. Comput. Sci., 21 (1) (2021) 590-599.
  • [9] Eskondari E. M., Taghizadeh N., Applications of two methods in exact wave solutions in the space-time fractional Drinfeld-Sokolov-Wilson system, Int. J. Differ. Equ., 2022 (2022) 4470344.
  • [10] Ali N., Yassen M. F., Asiri S. A., Nawaz R., Zada L., Alam M. M., Sene N., New iterative method for solving a coupled system of fractional order Drinfeld-Sokolov-Wilson (FDSW) and fractional shallow water (FSW) equations, J. Nanomater., 2022 (2022) 8370107.
  • [11] Taghizadeh N., Neirameh A., Complex solutions for Drinfeld’s-Sokolov-Wilson system, Int. J. Eng. Sci. Technol., 1 (1) (2011) 1-6.
  • [12] Raslan A., Entesar A., Enhancing Banach’s contraction method using the particle swarm optimization to solve the system Drinfeld-Sokolov-Wilson, J. Phys. Conf. Ser., 2322 (1) (2022) 012031.
  • [13] Lindeberg L., Dao T., Mattson, K., A high order accurate finite difference method for the Drinfel’d-Sokolov-Wilson equation, J. Sci. Comput., 88 (1) (2021) 18.
  • [14] Zhang Y., Zhao Z., Lie symmetry analysis, Lie- Backlund symmetries, explicit solutions and conservation laws of Drinfeld-Sokolov-Wilson system, Bound. Value Probl., 154, (2017).
  • [15] Singh P. K., Vishal K., Som T., Solution of fractional Drinfeld-Sokolov-Wilson equation using homotopy perturbation transform method, Appl. Appl. Math., 10 (1) (2015) 460-472.
  • [16] Al-Rozbayani A. M. A., Ali A. H., Applied Sumudu transform with Adomian decomposition method to the coupled Drinfeld-Sokolov-Wilson system, Rafidain J. Comput. Sci. Math., 15 (2) (2021) 139-147.
  • [17] Alam M. N., Bonyah E., Fayz-Al-Asad M., Reliable analysis for the Drinfeld-Sokolov-Wilson equation in mathematical physics, Palest. J. Math., 11 (1) (2022) 397-407.
  • [18] Usman M. H. A., Zaman F. D., Eldin S. M., Symmetry analysis and exact Jacobi elliptic solutions for the nonlinear couple Drinfeld Sokolov Wilson dynamical system arising in shallow water waves, Results Phys., 51 (2023) 106613.
  • [19] Shahzad M. U., Ur-Rehman H., Awan A. U., Zafar Z., Hassan A. M., Iqbal I., Analysis of the exact solutions of nonlinear coupled Drinfeld-Sokolov-Wilson equation through Φ^6-model expansion method, Results Phys., 52 (2023) 106671.
  • [20] Iqbal M., Seadawy A. R., Lu D., Zhang Z., Computational approachand dynamical analysis of multiple solitary wave solutions for nonlinear coupled Drinfeld-Sokolov-Wilson equation, Results Phys., 55 (2023) 107099.
  • [21] Younis I. T., Al-Rawi E. S., Improved finite difference technique via Adominian polynomial to solve the coupled Drinfeld’s-Sokolov-Wilson system, Int. J. Math. Math. Sci., 2023 (2023) 6916596.
  • [22] Aydemir T., New exact solutions of the Drinfeld-Sokolov-Wilson system by the generalized unified method, J. New Theory, 44 (2023) 10-19.
  • [23] Hakkaev S., Spectral stability of periodic waves for the Drinfeld-Sokolov-Wilson equation, J. Math. Anal. Appl., 533 (1) (2024) 128016.
  • [24] Abassy T. A., El Tawil M. A., El Zoheiry H., Toward a Modified Variational Iteration Method, J. Comput. Appl. Math., 207 (1) (2007) 137-147.
  • [25] Abassy T. A., El Tawil M. A., El Zoheiry H., Solving Nonlinear Partial Differential Equations Using the Modified Variational Iteration Padé Technique, J. Comput. Appl. Math. 207 (1) (2007) 73-91.
  • [26]Abassy T. A., Modified Variational Iteration Method (Nonlinear Homogeneous Initial Value Problem), Comput. Math. Appl., 59 (2) (2010) 912-918.
  • [27] Abbasbandy, S., Shivanian, E., Application of the Variational Iteration Method for System of Nonlinear Volterra’s Integro-Differential Equations, Math. Comput. Appl., 14 (2) (2009) 147-158.
  • [28] Aggarwal S., Chaudhary R., A comparative study of Mohand and Laplace transforms, J. Emerg. Technol. Innov. Res., 6 (2) (2019) 230-240.
  • [29] Aggarwal S., Sharma S.D., Sumudu transform of error function, J. Appl. Sci. Comput., 6 (6) (2019) 1222-1231.
  • [30] Zhang W. M., Solitary solutions and singular periodic solutions of the Drinfeld-Sokolov-Wilson equation by variational approach, Appl. Math. Sci., 5 (38) (2011) 1887-1894.
There are 30 citations in total.

Details

Primary Language English
Subjects Numerical Analysis, Partial Differential Equations
Journal Section Natural Sciences
Authors

Emre Aydın 0000-0001-7480-0965

İnci Çilingir Süngü 0000-0001-7788-181X

Publication Date March 25, 2025
Submission Date July 4, 2024
Acceptance Date December 30, 2024
Published in Issue Year 2025Volume: 46 Issue: 1

Cite

APA Aydın, E., & Çilingir Süngü, İ. (2025). On the Semi-analytical and Hybrid Methods for the Drinfeld-Sokolov-Wilson System Modelling Dispersive Water Waves. Cumhuriyet Science Journal, 46(1), 98-108. https://doi.org/10.17776/csj.1510043
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