Transformation solution for Korteweg-de Vries equation with small delay


Keywords:
Solitary waves, KdV equation, Small delay, Lie group, Transformation solution.Abstract
In this paper we develop a new approach to get the transformation solution for the mathematical model of waves on shallow fluid; Korteweg-de Vries with a small delay without change the space variables. This method can be base to solve most of nonlinear higher order partial differential equation with time delay.
References
A. Bakheet, and E. A. Alnussairy. Numerical Simulation of Magnetohydrodynamic Influences on Casson Model for Blood Flow through an Overlapping Stenosed Artery. IJS 1016-1024(2021). https://doi.org/10.24996/ijs.2021.62.3.30.
D. J. Korteweg, and G. De Vries, On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New
Type of Long Stationary Waves, Philos. Mag., 39(240): 422-443 (1895). https://doi.org/10.1080/14786449508620739.
D. Abrahamsen, and B. Fornberg, Solving the Korteweg-de Vries equation with Hermite-based Finite differences. Appl. Math. Comput., 401: 126101(2021).https://doi.org/10.1016/j.amc.2021.126101.
T. Geyikli, Collocation Method for Solving the Generalized KdV Equation. J. Appl. Math. Phys. 8(6): 1123-1134 (2020). https://doi.org/10.4236/jamp.2020.86085.
H., Rezazadeh, A., Korkmaz, A. E., W. Achab, Adel, and A. Bekir, New travelling wave solution-based new Riccati Equation for solving KdV and modified KdV Equations. Appl. math. nonlinear sci. 6(1): 447-458(2021). https://doi.org/10.2478/amns.2020.2.00034.
S. R. Choudhury, G. Gaetana and A. R. Ranses. Stability and dynamics of regular and embedded solitons of a perturbed
Fifth-order KdV equation. Phys. D: Nonlinear Phenom. 460: 134056 (2024). https://doi.org/10.2139/ssrn.4601740.
Z. H. Zhao, and Y. T. Xu, Solitary waves for Korteweg-de Vries equation with small delay, J. Math. Anal. Appl. 368 : 43-53(2010). https://doi.org/10.1016/j.jmaa.2010.02.014.
L. Baudouin, E. Crépeau, and J. Valein, Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback. IEEE Trans Autom Control. 64(4): 1403-1414(2019). https://doi.org/10.1109/TAC.2018.2849564.
H. Parada, E. Crépeau, and C. Prieur, Delayed stabilization of the Korteweg–de Vries equation on a star-shaped network. Math. Control Signals Syst., 43(3): 1-47(2022). https://doi.org/10.1007/s00498-022-00319-0.
J. Valein, On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Math. Control Relat. Fields, (2021). https://doi.org/10.3934/mcrf.2021039.
Parada, Hugo, T. Chahnaz, and V. Julie, Stability results for the KdV equation with time-varying delay. Control Syst. Lett. 177: 105547(2023). https://doi.org/10.2139/ssrn.4247393.
P. J. Olver, Application of Lie Groups to Differential Equations. New York NY, USA: Springer. 1986.
G. W. Bluman, and S. Kumei, Symmetries and Differential Equations. New York: Sprinder, 1989.
N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations (1-3). London: Wiley, 1999.
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