Semi-analytic solution of time-fractional Korteweg-de Vries equation using fractional residual power series method

Abstract

In this paper, we have solved the non-linear Korteweg-de Vries equation by considering it in time-fraction Caputo sense and offered intrinsic properties of solitary waves. The fractional residual power series method is used to obtain the approximate solution of the aforesaid equation and compared the obtained results with Adomian Decomposition Method. Obtained results are efficient, reliable, and simple to execute on most of the non-linear fractional partial differential equations, which arise in various dynamical systems.

Keywords

• Fractional differential equation
• KdV equation
• Residual Power series method
• Caputo derivative

References

1. [1] J. V. Boussinesq, Essai sur la theorie des eaux courantes, Memoires presentes par divers savants. lead. des Sci. Inst. Nat. France, XXIII, (1877) pp, 1680.
2. [2] D. J. Korteweg and G. D. Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 39(240): (1895) 422-443.
3. [3] T. Xiang, A summary of the Korteweg-de Vries equation, (2015).
4. [4] M. Alquran, M. Ali, and H. Jadallah, New topological and non-topological unidirectional-wave solutions for the modified- mixed KdV equation and bidirectional-waves solutions for the benjamin ono equation using recent techniques. Journal of Ocean Engineering and Science, (2021).
5. [5] Y. Bakir, Numerical solution of the non-linear Korteweg-de Vries equation by using the Chebyshev wavelet collocation method. Honam Mathematical Journal, 43(3): (2021), 373-383.
6. [6] A. Althobaiti, S. Althobaiti, K. E. Rashidy, and A. R. Seadawy, Exact solutions for the non-linear extended KdV equation in a stratified shear flow using the modified exponential rational method. Results in Physics, 29:104723, (2021).
7. [7] P. Liu, B. Huang, B. Ren, and J.R. Yang, Consistent Riccati expansion solvability, symmetries and analytic solutions of a forced variable-coeficient extended Korteveg-de Vries the equation in fluid dynamics of internal solitary waves. Chinese Physics B, (2021).
8. [8] M. Alejo, C. Muñoz, and L. Vega, The Gardner equation and the L 2 − stability of the N− soliton solution of the Korteweg- de Vries equation. Transactions of the American Mathematical Society, 365(1): (2013), 195-212.

9. [9] O. Abu Arqub, A.E. Ajou, A.S. Bataineh, and I. Hashim, A representation of the exact solution of generalized Lane-Emden equations using a new analytical method. In Abstract and Applied Analysis, volume 2013. Hindawi, (2013).
10. [10] B.A. Mahmood and M.A. Yousif, A residual power series technique for solving Boussinesq-Burgers equations. Cogent Mathematics, 4(1):1279398, (2017).
11. [11] K.M. Owolabi, A. Atangana, and A. Akgul, Modelling and analysis of fractal-fractional partial differential equations: application to reaction-di?usion model. Alexandria Engineering Journal, 59(4): (2020), 2477-2490.
12. [12] M.S. Hashmi, M. Wajiha, S.W. Yao, A. Gha?ar, and Mustafa Inc, Cubic spline-based differential quadrature method: A numerical approach for fractional burger equation. Results in Physics, (2021), page 104415.
13. [13] M. Modanli, S.T. Abdulazeez, and A.M. Husien, A residual power series method for solving pseudo hyperbolic partial differential equations with nonlocal conditions. Numerical Methods for Partial Di?erential Equations, 37(3): (2021), 2235-2243
14. [14] A. Kumar, S. Kumar, and M. Singh, Residual power series method for fractional Sharma-Tasso-Olver equation. Commun. Numer. Anal, 10: (2016), 1-10.
15. [15] R.M. Jena and S. Chakraverty, Residual power series method for solving time-fractional model of vibration equation of large membranes. Journal of Applied and Computational Mechanics, 5(4): (2019), 603-615.
16. [16] I. Komashynska, M.A. Smadi, A. Ateiwi, and S.A. Obaidy, Approximate analytical solution by residual power series method for a system of Fredholm integral equations. Appl. Math. Inf. Sci., 10(3): (2016), 1-11.
17. [17] H. Ahmad, T.A. Khan, H. Durur, G.M. Ismail, and A. Yokus, Analytic approximate solutions of diffusion equations arising in oil pollution. Journal of Ocean Engineering and Science, 6(1): (2021), 62-69.
18. [18] M. Alquran, Analytical solutions of fractional foam drainage equation by residual power series method. Mathematical sciences, 8(4): (2014), 153-160.
19. [19] L. Wang and X. Chen, Approximate analytical solutions of time-fractional Whitham-Broer?-equations by a residual power series method. Entropy, 17(9): (2015), 6519-6533.
20. [20] Y.S. Özkan, E. Ya³ar, and N. Çelik, On the exact and numerical solutions to a new (2+ 1)-dimensional Korteweg-de Vries equation with conformable derivative. Non-linear Engineering, 10(1): (2021), 46-65.
21. [21] M.M. Khader and K.M. Saad, Numerical studies of the fractional Korteweg-de Vries, Korteweg-de Vries-Burgers and Burgers equations. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 91(1): (2021), 67-77.
22. [22] F. Sjölander, Numerical solutions to the Boussinesq equation and the Korteweg-de Vries equation, (2021).
23. [23] Q.M. Al-Mdallal, M.A. Hajji, T. Abdeljawad, On the iterative methods for solving fractional initial value problems: a new perspective, Journal of Fractional Calculus and Nonlinear Systems, 2(1) (2021):76-81.
24. [24] Q.M. Al-Mdallal, H. Yusuf, and A. Ali, A novel algorithm for time-fractional foam drainage equation, Alexandria Engineering Journal, 59.3 (2020): 1607-1612.
25. [25] F. Haq, K. Shah, Q.M. Al-Mdallal, F. Jarad, Application of a hybrid method for systems of fractional order partial di?erential equations arising in the model of the one-dimensional Keller-Segel equation, The European Physical Journal Plus, 134.9 (2019): 1-11.
26. [26] T. Abdeljawad, R. Amin, K. Shah, Q.M. Al-Mdallal, E?cient sustainable algorithm for numerical solutions of systems of fractional order di?erential equations by Haar wavelet collocation method, Alexandria Engineering Journal, 59.4 (2020): 2391-2400.
27. [27] M. Alaroud, Application of Laplace residual power series method for approximate solutions of fractional IVP. Alexandria Engineering Journal, (2021).matics and Computation, 162(3): (2005), 1465–1473.
28. [28] S. Kumar and B. Kour, Residual power series solution of fractional bi-Hamiltonian Boussinesq system. In Proceedings of International Conference on Trends in Computational and Cognitive Engineering, (2021), pages 163?172. Springer.
29. [29] K.S. Miller and B. Ross, An introduction to fractional calculus and fractional di?erential equations, (1993), Wiley.
30. [30] I. Podlubny, Fractional di?erential equations: an introduction to fractional derivatives, fractional di?erential equations, to methods of their solution and some of their applications. Elsevier, (1998).
31. [31] A.E. Ajou, O.A. Arqub, Z.A. Zhou, and S. Momani, New results on fractional power series: theories and applications. Entropy, 15(12): (2013), 5305-5323.
32. [32] M.I. Syam, Adomian decomposition method for approximating the solution of the Korteweg-de Vries equation. Applied Mathematics and Computation, 162(3): (2005), 1465-1473.