A Study on the Two Forms of (2+1)-Fractional Order Burgers Equation

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DOI: https://doi.org/10.28924/2291-8639-22-2024-134
Published: Aug 12, 2024
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Fathea M. O. Birkea, Ali Satty, Abaker A. Hassaballa, Ahmed M. A. Adam, Elzain A. E. Gumma, Emad A-B. Abdel-Salam, Eltayeb A. Yousif, Mohamed I. Nouh, A Study on the Two Forms of (2+1)-Fractional Order Burgers Equation, Int. J. Anal. Appl., 22 (2024), 134.

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Fathea M. O. Birkea, Ali Satty, Abaker A. Hassaballa, Ahmed M. A. Adam, Elzain A. E. Gumma, Emad A-B. Abdel-Salam, Eltayeb A. Yousif, Mohamed I. Nouh

Abstract

This study examines the dynamics of two forms of (2+1)-dimensional fractional order Burgers equation, a paramount structure within the realm of nonlinear fractional calculus. The main objective is to acquire solutions for this equation through the application of the singular manifold (SM) method. The study successfully derives diverse solutions corresponding to varied fractional orders. Additionally, multiple-kink solutions are systematically derived and illustrated with graphical representations to highlight their intrinsic physical properties. Overall, the results demonstrate the effectiveness and reliability of the SM method in yielding precise solutions for the two forms of (2+1)-dimensional fractional order Burgers equation.

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References

  1. R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York, 1982.
  2. P.J. Olver, Application of Lie Group to Differential Equation, Springer, New York, 1986.
  3. V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer-Verlag, Berlin, 1991.
  4. X.Y. Gao, In the Shallow Water: Auto-Bäcklund, Hetero-Bäcklund and Scaling Transformations via a (2+1)-Dimensional Generalized Broer-Kaup System, Qual. Theory Dyn. Syst. 23 (2024), 184. https://doi.org/10.1007/s12346-024-01025-9.
  5. J. Weiss, M. Tabor, G. Carnevale, The Painlevé Property for Partial Differential Equations, J. Math. Phys. 24 (1983), 522–526. https://doi.org/10.1063/1.525721.
  6. J.M. Burgers, The Nonlinear Diffusion Equation, Reidel, Dordrecht, 1974.
  7. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.
  8. T. Abdeljawad, On Conformable Fractional Calculus, J. Comp. Appl. Math. 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016.
  9. R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, J. Comp. Appl. Math. 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002.
  10. A.A. Alexeyev, Some Notes on the Singular Manifold Method: Several Manifolds and Constraints, J. Phys. A: Math. Gen. 33 (2000), 1873–1894. https://doi.org/10.1088/0305-4470/33/9/311.
  11. Y. Peng, E. Yomba, New Applications of the Singular Manifold Method to the (2+1)-Dimensional Burgers Equations, Appl. Math. Comp. 183 (2006), 61–67. https://doi.org/10.1016/j.amc.2006.06.025.
  12. I.G. Murtaza, N. Raza, S. Arshed, Painlevé Integrability and a Collection of New Wave Structures Related to an Important Model in Shallow Water Waves, Comm. Theor. Phys. 75 (2023), 075008. https://doi.org/10.1088/1572-9494/acd999.
  13. R. Saleh, M. Kassem, S.M. Mabrouk, Exact Solutions of Nonlinear Fractional Order Partial Differential Equations via Singular Manifold Method, Chinese J. Phys. 61 (2019), 290–300. https://doi.org/10.1016/j.cjph.2019.09.005.
  14. H.Q. Zhang, B. Tian, J. Li, T. Xu, Y.X. Zhang, Symbolic-Computation Study of Integrable Properties for the (2 + 1)-Dimensional Gardner Equation with the Two-Singular Manifold Method, IMA J. Appl. Math. 74 (2008), 46–61. https://doi.org/10.1093/imamat/hxn024.
  15. J.D. Cole, On a Quasi-Linear Parabolic Equation Occurring in Aerodynamics, Quart. Appl. Math. 9 (1951), 225–236. https://doi.org/10.1090/qam/42889.
  16. H. Eberhard, The Partial Differential Equation ut+uux=μxx, Comm. Pure Appl. Math. 3 (1950), 201-230. https://doi.org/10.1002/cpa.3160030302.
  17. A.M. Wazwaz, A Study on the (2+1)-Dimensional and the (2+1)-Dimensional Higher-Order Burgers Equations, Appl. Math. Lett. 25 (2012), 1495–1499. https://doi.org/10.1016/j.aml.2011.12.034.