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Explicit Soliton Solutions to the Fractional Order Nonlinear Models through the Atangana Beta Derivative

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Abstract

The nonlinear space-time fractional Cahn-Allen and space-time fractional Benjamin-Bona-Mahony equations have significant applications in the fusion and fission phenomena of solitons, electromagnetic interactions, quantum relativistic atom theory, signal processing, quantum relativistic properties, and phase isolation with an atom in several components bass system. The fractional wave transformation has been used to convert space-time fractional nonlinear equations to integer order equations through the extended Tanh-function method in the sense of Atangana beta derivatives. We have achieved numerous soliton solutions as polynomials of hyperbolic function including solitons solutions like kink type, single soliton, singular kink, spike, periodic, dark soliton, bell type, periodic, and so on by setting arbitrary values of the parameter by using the computational software namely Maple and Mathematica. The different shapes of the solutions are shown graphically in 3D, contour plots, and vector plots. In the beginning, a power series in tanh was used as an ansatz to get analytical solutions of the traveling wave type to some nonlinear evolution equations. Numerous non-rectangular domains are used to solve these nonlinear fractional partial differential equations. The exact solutions indicate that the proposed method is effective, simple, and capable of creating comprehensive soliton solutions for nonlinear models in engineering and mathematical physics.

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References

  1. Rosenthal, Arthur: The history of calculus. Am. Math. Mon. 58(2), 75–86 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  2. David, S.A., Linares, J.L., Pallone, E.M.J.A.: Fractional order calculus: historical apologia, basic concepts and some applications. Revista Brasileira de Ensino de Física 33(4), 4302–4302 (2011)

    Article  Google Scholar 

  3. Okyere, S., Ackora-Prah, J.: Modeling and analysis of monkeypox disease using fractional derivatives. Results Eng. 17, 100786 (2023)

    Article  Google Scholar 

  4. Joshi, T., Parkash, O., Murthy, A.A., Krishan, G.: Numerical investigation of Bi-model slurry transportation in a straight pipe. Results Eng. 17, 100858 (2023)

    Article  Google Scholar 

  5. Gobran, Y., McClure, G., Aboshosha, H.: Determination of aerodynamic derivative for one degree of freedom square cylinder using large eddy simulation. Results Eng. 16, 100620 (2022)

    Article  Google Scholar 

  6. Huzni, S., Ibrahim, I.B., Fonna, S., Pidaparti, R.: Physics-based surrogate model for reinforced concrete corrosion simulation. Results Eng. 16, 100659 (2022)

    Article  Google Scholar 

  7. Chowdhury, M.A., Miah, M.M., Ali, H.S., Chu, Y.M., Osman, M.S.: An investigation to the nonlinear (2+ 1)-dimensional soliton equation for discovering explicit and periodic wave solutions. Results Phys. 23, 104013 (2021)

    Article  Google Scholar 

  8. Zabusky, N.J., Kruskal, M.D.: Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15(6), 240 (1965)

    Article  ADS  MATH  Google Scholar 

  9. Abdul Kayum, M., Ali Akbar, M., Osman, M. S.: Stable soliton solutions to the shallow water waves and ion-acoustic waves in a plasma. Waves in Random and Complex Media. 32(4), 1672–1693 (2020)

  10. Kumar, D., Kaplan, M., Haque, M., Osman, M.S., Baleanu, D.: A variety of novel exact solutions for different models with the conformable derivative in shallow water. Front. Phys. 8, 177 (2020)

    Article  Google Scholar 

  11. Faridi, W.A., Asjad, M.I., Jhangeer, A., Yusuf, A., Sulaiman, T.A.: The weakly non-linear waves propagation for Kelvin-Helmholtz instability in the magnetohydrodynamics flow impelled by fractional theory. Opt. Quant. Electron. 55(2), 172 (2023)

    Article  Google Scholar 

  12. Faridi, W.A., Asghar, U., Asjad, M.I., Zidan, A.M., Eldin, S.M.: Explicit propagating electrostatic potential waves formation and dynamical assessment of generalized Kadomtsev-Petviashvili modified equal width-Burgers model with sensitivity and modulation instability gain spectrum visualization. Results Phys. 44, 106167 (2023)

    Article  Google Scholar 

  13. Majid, S.Z., Faridi, W.A., Asjad, M.I., Abd El-Rahman, M., Eldin, S.M.: Explicit soliton structure formation for the riemann wave equation and a sensitive demonstration. Fractal Fractional 7(2), 102 (2023)

    Article  Google Scholar 

  14. Rezazadeh, H., Kumar, D., Sulaiman, T.A., Bulut, H.: New complex hyperbolic and trigonometric solutions for the generalized conformable fractional Gardner equation. Mod. Phys. Lett. B 33(17), 1950196 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  15. Ur Rahman, R., Faridi, W.A., El-Rahman, M.A., Taishiyeva, A., Myrzakulov, R., Az-Zo’bi, E.A.: The sensitive visualization and generalized fractional solitons’ construction for regularized long-wave governing model. Fractal Fractional 7(2), 136 (2023)

    Article  Google Scholar 

  16. Khatun, M. A., Arefin, M. A., Hafiz Uddin, M., Inc, M: Abundant explicit solutions to fractional order nonlinear evolution equations. Math. Probl. Eng. 1–16, (2021)

  17. Arefin, M. A., Khatun, M. A., Uddin, M. H., & Inc, M: Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations. J. Ocean Eng. Sci. 7(3), 292–303 (2021)

  18. Khatun, M.A., Arefin, M.A., Uddin, M.H., Baleanu, D., Akbar, M.A., Inc, M.: Explicit wave phenomena to the couple type fractional order nonlinear evolution equations. Results Phys. 28, 104597 (2021)

    Article  Google Scholar 

  19. Asjad, M.I., Faridi, W.A., Alhazmi, S.E., Hussanan, A.: The modulation instability analysis and generalized fractional propagating patterns of the Peyrard-Bishop DNA dynamical equation. Opt. Quant. Electron. 55(3), 1–34 (2023)

    Article  Google Scholar 

  20. Al Alwan, B., Abu Bakar, M., Faridi, W.A., Turcu, A.C., Akgül, A., Sallah, M.: The Propagating Exact Solitary Waves Formation of Generalized Calogero–Bogoyavlenskii–Schiff Equation with Robust Computational Approaches. Fractal Fractional 7(2), 191 (2023)

    Article  Google Scholar 

  21. Senol, M., Tasbozan, O., Kurt, A.: Comparison of two reliable methods to solve fractional Rosenau-Hyman equation. Math. Meth. Appl. Sci. 44(10), 7904–7914 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Malfliet, W., Hereman, W.: The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys. Scr. 54(6), 563 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Wazwaz, A.M.: New solitary wave solutions to the modified forms of Degasperis-Procesi and Camassa-Holm equations. Appl. Math. Comput. 186(1), 130–141 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Yusufoglu, E., Bekir, A.: On the extended tanh method applications of nonlinear equations. Int. J. Nonlinear Sci. 4(1), 10–16 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Wazwaz, A.M.: Nonlinear variants of KdV and KP equations with compactons, solitons and periodic solutions. Commun. Nonlinear Sci. Numer. Simul. 10(4), 451–463 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Uddin, M.H., Khan, M.A., Akbar, M.A., Haque, M.A.: Multi-solitary wave solutions to the general time fractional Sharma–Tasso–Olver equation and the time fractional Cahn-Allen equation. Arab. J. Basic Appl. Si. 26(1), 193–201 (2019)

    Google Scholar 

  27. Akagi, G., Schimperna, G., Segatti, A.: Fractional Cahn-Hilliard, Allen-Cahn and porous medium equations. J. Differ. Equ. 261(6), 2935–2985 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Bekir, A., Aksoy, E., Cevikel, A.C.: Exact solutions of nonlinear time fractional partial differential equations by sub-equation method. Math. Meth. Appl. Sci. 38(13), 2779–2784 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Esen, A.L.A.A.T.T.İN., Yagmurlu, N.M., Tasbozan, O.: Approximate analytical solution to time-fractional damped Burger and Cahn-Allen equations. Appl. Math. Inf. Sci. 7(5), 1951 (2013)

    Article  MathSciNet  Google Scholar 

  30. Ullah, M.S., Ali, M.Z., Noor, N.F.M.: Novel dynamics of wave solutions for Cahn-Allen and diffusive predator–prey models using MSE scheme. Partial Diff. Equat. Appl. Math. 3, 100017 (2021)

    Article  Google Scholar 

  31. Katugampola, U. N.: A new approach to generalized fractional derivatives. Bulletin of Mathematical Analysis and Applications. 3(4), 1–15 (2011)

  32. Duangpan, A., & Boonklurb, R.: Numerical solution of time-fractional Benjamin-Bona-Mahony-Burgers equation via finite integration method by using Chebyshev expansion. Songklanakarin J. Sci.  Techno. 43(3) (2020)

  33. Kaabar, M.K., Kaplan, M., Siri, Z.: New exact soliton solutions of the (3+ 1)-dimensional conformable wazwaz-benjamin-bona-mahony equation via two novel techniques. J. Funct. Spaces 2021, 13 (2021)

    MathSciNet  MATH  Google Scholar 

  34. Akram, U., Seadawy, A.R., Rizvi, S.T.R., Younis, M., Althobaiti, S., Sayed, S.: Traveling wave solutions for the fractional Wazwaz-Benjamin-Bona-Mahony model in arising shallow water waves. Results Phys. 20, 103725 (2021)

    Article  Google Scholar 

  35. Atangana, A., & Doungmo Goufo, E. F. : Extension of matched asymptotic method to fractional boundary layers problems. Math. Probl. Eng. 2014, (2014)

  36. Atangana, A., Alqahtani, R.T.: Modelling the spread of river blindness disease via the Caputo fractional derivative and the beta-derivative. Entropy 18(2), 40 (2016)

    Article  ADS  Google Scholar 

  37. Hosseini, K., Mirzazadeh, M., Ilie, M., Gómez-Aguilar, J.F.: Biswas-Arshed equation with the beta time derivative: optical solitons and other solutions. Optik 217, 164801 (2020)

    Article  ADS  Google Scholar 

  38. Benjamin,  T.B., & Benjamin, T.B., Lectures on nonlinear wave motion.  1974

  39. Amick, C.J., Bona, J.L., Schonbek, M.E.: Decay of solutions of some nonlinear wave equations. J. Differ. Equ. 81(1), 1–49 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Goldstein, J.A., Wichnoski, B.J.: On the Benjamin-Bona-Mahony equation in higher dimensions. Nonlinear Anal: Theor Meth Applic 4(4), 665–675 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  41. Manafianheris, J.: Exact solutions of the BBM and MBBM equations by the generalized G′/G expansion method equations 19, 1789-1796 (2012)

Download references

Acknowledgement

The authors would like to express their sincere thanks to the anonymous referees for their valuable comments and suggestions to improve the article. The authors would also like to thank the Ministry of Science and Technology (MoST), Govt. of Bangladesh, for supporting the research.

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Authors and Affiliations

  1. Department of Mathematics, Jashore University of Science and Technology, Jashore, 7408, Bangladesh

    Mohammad Asif Arefin, M. Ayesha Khatun, Mohammad Shaiful Islam & M. Hafiz Uddin

  2. Department of Applied Mathematics, University of Rajshahi, Rajshahi, 6205, Bangladesh

    M. Ali Akbar

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  1. Mohammad Asif Arefin
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Contributions

Mohammad Asif Arefin: Data Curation, Software, Writing, Investigation.

M. Ayesha Khatun: Software, Data Curation, Writing, Formal Analysis.

Mohammad Shaiful Islam: Conceptualization, Writing-Reviewing Editing.

M. Ali Akbar: Conceptualization, Validation.

M. Hafiz Uddin: Conceptualization, Supervision, Writing-Reviewing Editing, Validation.

Corresponding author

Correspondence to M. Hafiz Uddin.

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Arefin, M.A., Khatun, M.A., Islam, M.S. et al. Explicit Soliton Solutions to the Fractional Order Nonlinear Models through the Atangana Beta Derivative. Int J Theor Phys 62, 134 (2023). https://doi.org/10.1007/s10773-023-05400-1

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