Computational and Periodic Wave Solutions for the Time-Fractional Calogero-Bogoyavlenskii-Schiff Equation

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DOI: https://doi.org/10.28924/2291-8639-23-2025-316
Published: Nov 28, 2025
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Ahmed A. Gaber, Computational and Periodic Wave Solutions for the Time-Fractional Calogero-Bogoyavlenskii-Schiff Equation, Int. J. Anal. Appl., 23 (2025), 316.

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Ahmed A. Gaber

Abstract

The time-fractional differential equation of Calogero-Bogoyavlenskii-Schiff (CBS) has an important role in plasma waves and shallow water and ocean waves. By using symbolic computation, we applied the generalized Kudryashov method (GKM) and the generalized He’s Exp-function method (GHEFM) For the purpose of creating novel time-fractional CBS equation solutions. Utilizing suitable fractional transformation the governing equation reduced to ordinary differential equation. New different wave solutions are obtained using the methodology of both GKM and GHEFM. Kink wave, single wave, and solitary wave solutions are represented by these solutions, which also include trigonometric and hyperbolic functions.

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References

  1. M. Shakeel, Attaullah, N.A. Shah, J.D. Chung, Modified Exp-Function Method to Find Exact Solutions of Microtubules Nonlinear Dynamics Models, Symmetry 15 (2023), 360. https://doi.org/10.3390/sym15020360.
  2. H.Ç. Yaslan, A. Girgin, Exp-Function Method for the Conformable Space-Time Fractional STO, ZKBBM and Coupled Boussinesq Equations, Arab. J. Basic Appl. Sci. 26 (2019), 163–170. https://doi.org/10.1080/25765299.2019.1580815.
  3. J. He, M. Abdou, New Periodic Solutions for Nonlinear Evolution Equations Using Exp-Function Method, Chaos Solitons Fractals 34 (2007), 1421–1429. https://doi.org/10.1016/j.chaos.2006.05.072.
  4. A.E. Ebaid, Generalization of He’s Exp-Function Method and New Exact Solutions for Burgers Equation, Z. Naturforsch. 64 (2009), 604–608. https://doi.org/10.1515/zna-2009-9-1010.
  5. A.A. Gaber, M.H. Shehata, New Approach of MHD Boundary Layer Flow Towards a Porous Stretching Sheet via Symmetry Analysis and the Generalized Exp-Function Method, Int. J. Anal. Appl. 18 (2020), 738–747. https://doi.org/10.28924/2291-8639-18-2020-738.
  6. M. Mamun Miah, H.M. Shahadat Ali, M. Ali Akbar, A. Majid Wazwaz, Some Applications of the (G'/G, 1/G)-Expansion Method to Find New Exact Solutions of NLEEs, Eur. Phys. J. Plus 132 (2017), 252. https://doi.org/10.1140/epjp/i2017-11571-0.
  7. P. Ye, G. Cai, An Extended (G'/G)–Expansion Method and Travelling Wave Solutions to Nonlinear Klein-Gordon Equation, Int. J. Nonlinear Sci. 11 (2011), 225–229.
  8. A. Biswas, A. Sonmezoglu, M. Ekici, M. Mirzazadeh, Q. Zhou, et al., Optical Soliton Perturbation with Fractional Temporal Evolution by Extended G'/G-Expansion Method, Optik 161 (2018), 301–320. https://doi.org/10.1016/j.ijleo.2018.02.051.
  9. J. Manafian, M.F. Aghdaei, M. Khalilian, R. Sarbaz Jeddi, Application of the Generalized G'/G-Expansion Method for Nonlinear PDEs to Obtaining Soliton Wave Solution, Optik 135 (2017), 395–406. https://doi.org/10.1016/j.ijleo.2017.01.078.
  10. Y.S. Özkan, E. Yaşar, On the Exact Solutions of Nonlinear Evolution Equations by the Improved $tan (varphi /2)$-Expansion Method, Pramana 94 (2020), 37. https://doi.org/10.1007/s12043-019-1883-3.
  11. D. Lu, A. Seadawy, M. Arshad, J. Wang, New Solitary Wave Solutions of (3 + 1)-Dimensional Nonlinear Extended Zakharov-Kuznetsov and Modified KdV-Zakharov-Kuznetsov Equations and Their Applications, Results Phys. 7 (2017), 899–909. https://doi.org/10.1016/j.rinp.2017.02.002.
  12. M. Iqbal, A.R. Seadawy, D. Lu, X. Xianwei, Construction of a Weakly Nonlinear Dispersion Solitary Wave Solution for the Zakharov–Kuznetsov–Modified Equal Width Dynamical Equation, Indian J. Phys. 94 (2019), 1465–1474. https://doi.org/10.1007/s12648-019-01579-4.
  13. M.H.M. Moussa, A.A. Gaber, Symmetry Analysis and Solitary Wave Solutions of Nonlinear Ion-Acoustic Waves Equation, Int. J. Anal. Appl. 18 (2020), 448–460. https://doi.org/10.28924/2291-8639-18-2020-448.
  14. A.A. Gaber, A.F. Aljohani, A. Ebaid, J.T. Machado, The Generalized Kudryashov Method for Nonlinear Space–Time Fractional Partial Differential Equations of Burgers Type, Nonlinear Dyn. 95 (2018), 361–368. https://doi.org/10.1007/s11071-018-4568-4.
  15. S.T. Demiray, Y. Pandir, H. Bulut, The Analysis of the Exact Solutions of the Space Fractional Coupled KD Equations, AIP Conf. Proc. 1648 (2015), 370013. https://doi.org/10.1063/1.4912602.
  16. S.T. Demiray, Y. Pandir, H. Bulut, New Solitary Wave Solutions of Maccari System, Ocean. Eng. 103 (2015), 153–159. https://doi.org/10.1016/j.oceaneng.2015.04.037.
  17. A.A. Gaber, A. Wazwaz, M.M. Mousa, Similarity Reductions and New Exact Solutions for (3+1)-Dimensional B–B Equation, Mod. Phys. Lett. B 38 (2023), 2350243. https://doi.org/10.1142/s0217984923502433.
  18. A.A. Gaber, A. Wazwaz, Symmetries and Dynamic Wave Solutions for (3 + 1)-Dimensional Potential Calogero–Bogoyavlenskii–Schiff Equation, J. Ocean. Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.05.018.
  19. Y. Peng, New Types of Localized Coherent Structures in the Bogoyavlenskii-Schiff Equation, Int. J. Theor. Phys. 45 (2006), 1764–1768. https://doi.org/10.1007/s10773-006-9139-7.
  20. M.S. Bruzón, M.L. Gandarias, C. Muriel, J. Ramírez, S. Saez, et al., The Calogero–Bogoyavlenskii–Schiff Equation in 2+1 Dimensions, Theor. Math. Phys. 137 (2003), 1367–1377. https://doi.org/10.1023/a:1026040319977.
  21. A. Wazwaz, A New Integrable Equation Constructed via Combining the Recursion Operator of the Calogero-Bogoyavlenskiischiff (CBS) Equation and Its Inverse Operator, Appl. Math. Inf. Sci. 11 (2017), 1241–1246. https://doi.org/10.18576/amis/110501.
  22. A. Wazwaz, The (2+1) and (3+1)-Dimensional CBS Equations: Multiple Soliton Solutions and Multiple Singular Soliton Solutions, Z. Naturforsch. 65 (2010), 173–181. https://doi.org/10.1515/zna-2010-0304.
  23. G. Moatimid, R.M. El-Shiekh, A.A. Al-Nowehy, Exact Solutions for Calogero–Bogoyavlenskii–Schiff Equation Using Symmetry Method, Appl. Math. Comput. 220 (2013), 455–462. https://doi.org/10.1016/j.amc.2013.06.034.
  24. C.M. Khalique, L.D. Moleleki, Lagrangian Formulation of the Calogero-Bogoyavlenskii-Schiff Equation, AIP Conf. Proc. 2133 (2019), 190009. https://doi.org/10.1063/1.5114178.
  25. A. Gaber, T. M. Younis, M. F. Alharbi, Symmetries and Novel Exact Solutions for (2+1)-D QZK Equation via Lie-Symmetry and Kudryashov-Auxiliary Method, Eur. J. Pure Appl. Math. 18 (2025), 6030. https://doi.org/10.29020/nybg.ejpam.v18i3.6030.