Some new exact solutions for a generalized variable coefficients KdV equation

Volume 29, Issue 1, pp 1--11 http://dx.doi.org/10.22436/jmcs.029.01.01

Publication Date: August 11, 2022 Submission Date: October 18, 2021 Revision Date: May 13, 2022 Accteptance Date: June 01, 2022

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Authors

R. Rajagopalan - Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia. A. H. Abdel Kader - Department of Mathematics and Engineering Physics, Engineering Faculty, Mansoura University, Mansoura, Egypt. M. S. Abdel Latif - Department of Mathematics and Engineering Physics, Engineering Faculty, Mansoura University, Mansoura, Egypt. - Department of Mathematics, Faculty of Science, New Mansoura University, New Mansoura City, Egypt. D. Baleanu - Mathematics Department, Arts and Sciences Faculty, Cankaya University, Ankara 06530, Turkey. - Instiute of Space Sciences, Magurele-Bucharest, P. O. Box MG-23, R 76900, Romania. A. E. Sonbaty - Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia. - Department of Mathematics and Engineering Physics, Engineering Faculty, Mansoura University, Mansoura, Egypt.

Abstract

In this paper, the variable coefficients KdV equation with general power nonlinearities is proposed. Firstly, it is transformed into a generalized KdV equation with constant coefficients using a point transformation. Then, the traveling wave transformation is utilized to transform the obtained constant coefficients generalized KdV equation into a generalized ordinary differential equation. We give a classification for the obtained generalized ordinary differential equation using a suitable integrating factor. Some new solutions are obtained for the generalized KdV equation with constant coefficients. All the obtained solutions in this paper for the variable coefficients KdV equation with general power nonlinearities are new.

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Keywords

• Exact solutions
• generalized KdV equation
• traveling wave
• solitons

MSC

•  35C05
•  35C07
•  35G20
•  35Q53

References

• [1] A. H. Abdel Kader, M. S. Abdel Latif, New soliton solutions of the CH–DP equation using Lie symmetry method, Modern Phys. Lett. B, 32 (2018), 9 pages
  • View Article
  • MathSciNet


• [2] A. H. Abdel Kader, M. S. Abdel Latif, New bright and dark soliton solutions for a generalized nonlinear Schrodinger equation, Optik, 176 (2019), 699--703
  • View Article
  • Google Scholar


• [3] M. S. Abdel Latif, A. H. Abdel Kader, D. Baleanu, The invariant subspace method for solving nonlinear fractional partial differential equations with generalized fractional derivatives, Adv. Difference Equ., 2020 (2020), 13 pages
  • View Article
  • MathSciNet


• [4] A. H. Abdel Kader, M. S. Abdel Latif, D. Baleanu, Some exact solutions of a variable coefficients fractional biological population model, Math. Methods Appl. Sci., 44 (2021), 4701--4714
  • View Article
  • MathSciNet


• [5] A. H. Abdel Kader, M. S. Abdel Latif, F. E. Bialy, A. Elsaid, Symmetry analysis and some new exact solutions of some nonlinear KdV-like equations, Asian-Eur. J. Math., 11 (2018), 14 pages
  • View Article
  • MathSciNet


• [6] A. H. Abdel Kader, M. S. Abdel Latif, H. M. Nour, Some New Exact Solutions of the Modified KdV Equation Using Lie Point Symmetry Method, Int. J. Appl. Comput. Math., 3 (2017), 1163--1171
  • View Article
  • MathSciNet


• [7] M. S. Abdel Latif, E. El-Shazly, D. Baleanu, A. Elsaid, H. M. Nour, Some new soliton-like and doubly periodic-like solutions of Fisher equation with time-dependent coefficients, Modern Phys. Lett. B, 12 (2018), 12 pages
  • View Article
  • MathSciNet


• [8] M. S. Abdel Latif, E. El-Shazly, A. Elsaid, H. M. Nour, Group classification and some new periodic-like and soliton-like solutions of the generalised Fisher equation with time-variable coefficients, Int. J. Dyn. Syst. Differ. Equ., 8 (2018), 313--325
  • View Article
  • MathSciNet


• [9] S. Ahmad, A. Ullah, A. Akgül, M. De la Sen, A novel homotopy perturbation method with applications to nonlinear fractional order KdV and Burger equation with exponential-decay kernel, J. Funct. Space, 2021 (2021), 11 pages
  • View Article
  • MathSciNet


• [10] A. Akgül, A. Kılıçman, M. Inc, Improved $(G'/G)$-expansion method for the space and time fractional foam drainage and KdV equations, Abstr. Appl. Anal., 2013 (2013), 7 pages
  • View Article
  • MathSciNet


• [11] D. J. Arrigo, Symmetry Analysis of Differential Equations, John Wiley & Sons, Hoboken (2015)
  • View Article
  • MathSciNet


• [12] M. I. Asjad, H. Ur Rehman, Z. Ishfaq, J. Awrejcewicz, A. Akgul, M. B. Riaz, On Soliton Solutions of Perturbed Boussinesq and KdV-Caudery-Dodd-Gibbon Equations, Coatings, 11 (2021), 14 pages
  • View Article
  • Google Scholar


• [13] M. Ekici, D. Duran, A. Sonmezoglu, Constructing of exact solutions to the $(2+1)$-dimensional breaking soliton equations by the multiple $(G'/G)$-expansion method, J. Adv. Math. Stud., 7 (2014), 27--44
  • View Article
  • MathSciNet


• [14] F. El-Bialy, M. S. Abdel Latif, A. Elsaid, Symmetry analysis and some new exact solutions of the $(2+1)$-dimensional Burgers equations, Acta Phys. Polon. B, 48 (2017), 2031--2043
  • View Article
  • MathSciNet


• [15] M. S. Hashemi, Z. Balmeh, A. Akgül, E. K. Akgül, D. Baleanu, Invariant investigation on the system of Hirota-Satsuma coupled KdV equation, AIP Conference Proc., 1978 (2018), 13 pages
  • View Article
  • Google Scholar


• [16] B. J. Hong, D. C. Lu, New exact solutions for the generalized variable-coefficient Gardner equation with forcing term, Appl. Math. Comput., 219 (2012), 2732--2738
  • View Article
  • MathSciNet


• [17] M. Inc, A. Akgül, A. A. Kiliçman, A novel method for solving KdV equation based on reproducing kernel Hilbert space method, Abstr. Appl. Anal., 2013 (2013), 11 pages
  • View Article
  • MathSciNet


• [18] H. Jafari, J. G. Prasad, P. Goswami, R. S. Dubey, Solution of the local fractional generalized KDV equation using homotopy analysis method, Fractals, 29 (2021), 10 pages
  • View Article
  • Google Scholar


• [19] H. Jafari, H. G. Sun, M. Azadi, Lie symmetry reductions and conservation laws for fractional order coupled KdV system, Adv. Difference Equ., 2020 (2020), 10 pages
  • View Article
  • MathSciNet


• [20] L. Li, Y. Xie, S. Zhu, New exact solutions for a generalized KdV equation, Nonlinear Dyn., 92 (2018), 215--219
  • View Article
  • Google Scholar


• [21] S. Zhang, A generalized new auxiliary equation method and its application to the $(2+1)$-dimensional breaking soliton equations, Appl. Math. Comput., 190 (2007), 510--516
  • View Article
  • MathSciNet


• [22] L.-H. Zhang, L.-H. Dong, L.-M. Yan, Construction of non-travelling wave solutions for the generalized variable-coefficient Gardner equation, Appl. Math. Comput., 203 (2008), 784--791
  • View Article
  • MathSciNet


• [23] S. Zhang, W. Li, F. Zheng, J. H. Yu, M. Ji, Z. Y. Lau, C. Z. Ma, A generalized F-expansion method and its application to $(2+1)$-dimensional breaking soliton equations, Int. J. Nonlinear Sci., 5 (2008), 25--32
  • View Article
  • MathSciNet