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Different wave structures in water wave mechanics with two conformable models

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Abstract

This paper investigates the exact wave solutions of the time-fractional modified Liouville equation (mLE) and time-fractional modified regularized long wave equation (mRLWE) which arise in water wave mechanics, via a new version of trial equation method (NVTEM). The present nonlinear models are reduced to nonlinear ordinary differential equations (NLODEs) by the traveling wave transform and the proposed solution by the NVTEM is used to evaluate the solutions of mLE and mRLWE. This analytical method is not applied before to these equations and novel wave solutions are acquired in the form of rational, exponential, hyperbolic, and Jacobi elliptic types. The solitary solutions of the equations under consideration make them essential models in shallow water dynamics, in liquids and gas bubbles, in magneto-hydrodynamics, and in plasma. This fact has become a motivation for this research.

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References

  1. Boakye, G., Hosseini, K., Hinçal, E., Sirisubtawee, S., Osman, M.S.: Some models of solitary wave propagation in optical fibers involving Kerr and parabolic laws. Opt. Quant. Electron. 56(3), 345 (2024)

    Article  MATH  Google Scholar 

  2. Arnous, A.H., Hashemi, M.S., Nisar, K.S., Shakeel, M., Ahmad, J., Ahmad, I., Jan, R., Ali, A., Kapoor, M., Shah, N.A.: Investigating solitary wave solutions with enhanced algebraic method for new extended Sakovich equations in fluid dynamics. Res. Phys. 57, 107369 (2024)

    Google Scholar 

  3. Mabrouk, S.M., Wazwaz, A.M., Rashed, A.S.: Monitoring dynamical behavior and optical solutions of space-time fractional order double-chain deoxyribonucleic acid model considering the Atangana’s conformable derivative. J. Appl. Comput. Mech. (2024). https://doi.org/10.22055/JACM.2024.45462.4371

    Article  MATH  Google Scholar 

  4. Şenol, M., Gençyiğit, M., Koksal, M.E., Qureshi, S.: New analytical and numerical solutions to the (2+ 1)-dimensional conformable cpKP-BKP equation arising in fluid dynamics, plasma physics, and nonlinear optics. Opt. Quant. Electron. 56(3), 352 (2024)

    Article  MATH  Google Scholar 

  5. Thabet, H., Kendre, S.: Conformable mathematical modeling of the COVID-19 transmission dynamics: a more general study. Math. Methods Appl. Sci. 46(17), 18126–18149 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  6. Martynyuk, A., Stamov, G., Stamova, I., Gospodinova, E.: Formulation of impulsive ecological systems using the conformable calculus approach: qualitative analysis. Mathematics 11(10), 2221 (2023)

    Article  MATH  Google Scholar 

  7. Şenol, M., Gençyiğit, M., Koksal, M.E., Qureshi, S.: New analytical and numerical solutions to the (2+ 1)-dimensional conformable cpKP-BKP equation arising in fluid dynamics, plasma physics, and nonlinear optics. Opt. Quant. Electron. 56(3), 352 (2024)

    Article  MATH  Google Scholar 

  8. Akinyemi, L., Şenol, M., Huseen, S.N.: Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma. Adv. Differ. Equ. 2021(1), 45 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, C., Xu, Z., Zhao, K., Xie, W.: Forecasting education expenditure with a generalized conformable fractional-order nonlinear grey system model. Heliyon (2023). https://doi.org/10.1016/j.heliyon.2023.e16499

    Article  MATH  Google Scholar 

  10. Shahen, N.H.M., Rahman, M.M., Alshomrani, A.S., Inc, M.: On fractional order computational solutions of low-pass electrical transmission line model with the sense of conformable derivative. Alex. Eng. J. 81, 87–100 (2023)

    Article  MATH  Google Scholar 

  11. Rehman, H.U., Iqbal, I., Zulfiqar, H., Gholami, D., Rezazadeh, H.: Stochastic soliton solutions of conformable nonlinear stochastic systems processed with multiplicative noise. Phys. Lett. A 486, 129100 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kırcı, Ö., Koç, D.A., Bulut, H.: Dynamics of the traveling wave solutions of conformable time-fractional ISLW and DJKM equations via a new expansion method. Opt. Quant. Electron. 56, 933 (2024)

    Article  Google Scholar 

  13. Gorenflo, R., Mainardi, F.: Fractional Calculus: Integral and Differential Equations of Fractional Order, pp. 223–276. Springer Vienna (1997)

  14. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  15. Khalil, R., Horani, M.A., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mamun, A.A., Ananna, S.N., An, T., Shahen, N.H.M., Asaduzzaman, M.: Dynamical behaviour of travelling wave solutions to the conformable time-fractional modified Liouville and mRLW equations in water wave mechanics. Heliyon. (2021). https://doi.org/10.1016/j.heliyon.2021.e07704

    Article  MATH  Google Scholar 

  17. Semenov, É.I.: New exact solutions to the nonautonomous Liouville equation. Sib. Math. J. 49, 166–174 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Khan, Y., Taghipour, R., Falahian, M., Nikkar, A.: A new approach to modified regularized long wave equation. Neural Comput. Appl. 23, 1335–1341 (2013)

    Article  MATH  Google Scholar 

  19. Salam, M.A.: Traveling-wave solution of modified Liouville equation by means of modified simple equation method. Int. Scholar. Res. Not. 2012(1), 565247 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Lu, D., Seadawy, A.R., Ali, A.: Applications of exact traveling wave solutions of modified Liouville and the symmetric regularized long wave equations via two new techniques. Res. Phys. 9, 1403–1410 (2018)

    MATH  Google Scholar 

  21. Zahran, E.H., Khater, M.M.: Exact traveling wave solutions for the system of shallow water wave equations and modified Liouville equation using extended Jacobian elliptic function expansion method. Am. J. Comput. Math. 4(05), 455 (2014)

    Article  MATH  Google Scholar 

  22. Abdelrahman, M.A., Khater, M.M.: Exact traveling wave solutions for Fitzhugh–Nagumo (FN) equation and modified liouville equation. Int. J. Comput. Appl. 113(3), 1–7 (2015)

    MATH  Google Scholar 

  23. Abdelrahman, M.A., Zahran, E.H., Khater, M.M.: Exact traveling wave solutions for modified Liouville equation arising in mathematical physics and biology. Int. J. Comput. Appl. 975(8887), 112 (2015)

    MATH  Google Scholar 

  24. Hossain, S., Roshid, M.M., Uddin, M., Ripa, A.A., Roshid, H.O.: Abundant time-wavering solutions of a modified regularized long wave model using the EMSE technique. Part. Differ. Equ. Appl. Math. 8, 100551 (2023)

    MATH  Google Scholar 

  25. Alquran, M., Najadat, O., Ali, M., Qureshi, S.: New kink-periodic and convex-concave-periodic solutions to the modified regularized long wave equation by means of modified rational trigonometric-hyperbolic functions. Nonlinear Eng. 12(1), 20220307 (2023)

    Article  MATH  Google Scholar 

  26. Ramli, M., Irsalina, D., Iwanisa, I.P., Halfiani, V.: Soliton solution of Benjamin–Bona–Mahony equation and modified regularized long wave equation. In: AIP Conference Proceedings, 1913:1, AIP Publishing (2017)

  27. Karakoç, S.B.G., Mei, L., Ali, K.K.: Two efficient methods for solving the generalized regularized long wave equation. Appl. Anal. 101(13), 4721–4742 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  28. Durur, H., Yokuş, A., Doğan, K.A.Y.A.: Hyperbolic type traveling wave solutions of regularized long wave equation. Bilecik Seyh Edebali Univ. J. Sci. 7(2), 815–824 (2020)

    MATH  Google Scholar 

  29. Mamun, A.A., Ananna, S.N., An, T., Shahen, N.H.M., Asaduzzaman, M.: Dynamical behaviour of travelling wave solutions to the conformable time-fractional modified Liouville and mRLW equations in water wave mechanics. Heliyon (2021). https://doi.org/10.1016/j.heliyon.2021.e07704

    Article  MATH  Google Scholar 

  30. Uddin, M.H., Khan, M.A., Akbar, M.A., Haque, M.A.: Analytical wave solutions of the space time fractional modified regularized long wave equation involving the conformable fractional derivative. Karbala Int. J. Mod. Sci. 5(1), 7 (2019)

    Article  MATH  Google Scholar 

  31. Ma, W.X., Fuchssteiner, B.: Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation. Int. J. Non-Linear Mech. 31(3), 329–338 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Liu, C.S.: Trial equation method to nonlinear evolution equations with rank inhomogeneous: mathematical discussions and its applications. Commun. Theor. Phys. 45(2), 219 (2006)

    Article  MATH  Google Scholar 

  33. Liu, C.S.: A new trial equation method and its applications. Commun. Theor. Phys. 45(3), 395 (2006)

    Article  MATH  Google Scholar 

  34. Liu, C.S.: Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations. Comput. Phys. Commun. 181(2), 317–324 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Du, X.H.: An irrational trial equation method and its applications. Pramana 75, 415–422 (2010)

    Article  MATH  Google Scholar 

  36. Liu, Y.: Exact solutions to nonlinear Schrödinger equation with variable coefficients. Appl. Math. Comput. 217(12), 5866–5869 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Pandir, Y., Gurefe, Y., Misirli, E.: A multiple extended trial equation method for the fractional Sharma-Tasso-Olver equation. In: AIP Conference Proceedings, 1558(1), 1927-1930, American Institute of Physics (2013)

  38. Kirci, O., Pandir, Y., Latifa, A., Bulut, H.: A new version of trial equation method for a complex nonlinear system arising in optical fibers. Opt. Quant. Electron. 56, 1019 (2024)

    Article  MATH  Google Scholar 

  39. Stokes, G.G.: Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form. Math. Phys. Pap. 1, 225 (1880)

    MATH  Google Scholar 

Download references

Acknowledgements

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Department of Mathematics, Kırklareli University, Kayalı Campus, 39000, Kırklareli, Turkey

    Özlem Kırcı

  2. Department of Mathematics, Bozok University, 66100, Yozgat, Turkey

    Yusuf Pandır

  3. Department of Mathematics, Fırat University, 23119, Elazig, Turkey

    Hasan Bulut

  4. Department of Mathematics and Informatics, Azerbaijan University, Baku, Azerbaijan

    Hasan Bulut

Authors
  1. Özlem Kırcı
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  2. Yusuf Pandır
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  3. Hasan Bulut
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Contributions

Conceptualization, software, methodology, supervision, writing—review and editing: Yusuf Pandır, Hasan Bulut; Formal analysis, investigation, software, writing—original draft and visualization: Özlem Kırcı

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Correspondence to Özlem Kırcı.

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Kırcı, Ö., Pandır, Y. & Bulut, H. Different wave structures in water wave mechanics with two conformable models. J. Appl. Math. Comput. 71, 49–68 (2025). https://doi.org/10.1007/s12190-024-02222-0

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  • DOI: https://doi.org/10.1007/s12190-024-02222-0

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