Application of a new method for nonlinear partial differential equations of fractional order arising in fluid mechanics

Document Type : Research Paper

Authors

1 Bryansk State University of Engineering and Technology, Russia.

2 M. Auezov South Kazakhstan University, Shymkent, Kazakhstan.

3 Russian State Social University, Moscow, Russia.

4 Kazan Federal University, Kazan, Russia.

5 Kuban State University, Krasnodar, Russia.

Abstract

In this work, we established some exact solutions for the (2+1)-dimensional Zakharov-Kuznetsov, KdV, and K(2,2) equations which are considered based on the improved Exp-function method, by utilizing Maple software. We use the fractional derivatives with fractional complex transform. We obtained new periodic solitary wave solutions. The obtained solutions include three classes of soliton wave solutions in terms of hyperbolic function, trigonometric function, and rational function solutions. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. Many other such types of nonlinear equations arising in fluid dynamics and nonlinear phenomena.

Keywords

Main Subjects

References

  • [1] M. A. Abdou, Generalized solitonary and periodic solutions for nonlinear partial differential equations by the Exp–function method, Nonlinear Dyn, 512 (2008), 1-9.
  • [2] N. H. Ali, S. A. Mohammed, and J. Manafian, Study on the simplified MCH equation and the combined KdV-mKdV equations with solitary wave solutions, Partial Diff. Eq. Appl. Math., 9 (2024), 100599.
  • [3] A. Bekir, O. Guner, A. H. Bhrawy, and A. Biswas, Solving nonlinear fractional differential equations using exp-function and (G’/G)-expansion methods, Rom. J. Phys., 60(3-4) (2015), 360-378.
  • [4] A. Boz and A. Bekir, Application of Exp–function method for (3+1)-dimensional nonlinear evolution equations, Comput. Math. Appl, 56 (2008), 1451-1456.
  • [5] R. Churchill, Operational Mathematics (3rd edition), McGraw-Hill, New York, 1972.
  • [6] L. Debanth, Recents applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci, 54 (2003), 3413-3442.
  • [7] M. Dehghan, J. Manafian, and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Num. Meth. Partial Differential Eq J., 26 (2010), 448-479.
  • [8] M. Dehghan, J. Manafian, and A. Saadatmandi, The solution of the linear fractional partial differential equations using the homotopy analysis method, Z. Naturforsch., 65(a) (2010), 935-949.
  • [9] M. Dehghan, J. Manafian Heris, and A. Saadatmandi, Analytical treatment of some partial differential equations arising in mathematical physics by using the Exp-function method, Int. J. Modern Phys. B, 25 (2011), 2965-2981.
  • [10] M. Dehghan, J. Manafian Heris, and A. Saadatmandi, Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics, Int. J. Num. Methods for Heat Fluid Flow, 21 (2011), 736-753.
  • [11] D. G. Duffy, Transform Methods for Solving Partial Differential, Chapman and Hall/CRC, 2004.
  • [12] V. D. Gejji and H. Jafari, Solving a multi-order fractional differential equation, Appl. Math. Comput, 189 (2007), 541-548.
  • [13] K. A. Gepreel and M. S. Mohamed, An optimal homotopy analysis method nonlinear fractional differential equation, Journal of Advanced Research in Dynamical and Control Systems, 6(1) (2014), 1-10.
  • [14] I. Grigorenko and E. Grigorenko, Chaotic dynamics of the fractional lorenz system, Phys. Rev. Lett, 91 (2003), 034101. 2007) 541-548.
  • [15] O. Guner, A. Bekir, and H. Bilgil, A note on exp-function method combined with complex transform method applied to fractional differential equations, Adv. Nonlinear Anal., 4(3) (2015), 201-208.
  • [16] Y. Gu, S. Malmir, J. Manafian, O. A. Ilhan, A. A. Alizadeh, and A. J. Othman, Variety interaction between k-lump and k-kink solutions for the (3+1)-D Burger system by bilinear analysis, Results Phys., 43 (2022), 106032.
  • [17] J. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering, Dalian, China, (1998), 288-291.
  • [18] J. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods. Appl. Mech. Eng, 167 (1998), 57-68.
  • [19] J. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol, 15 (1999), 86-90.
  • [20] J. H. He and X. H. Wu, Exp–function method for nonlinear wave equations, Chaos, Solitons Fractals, 30 (2006), 700-708.
  • [21] J. H. He, S. K. Elagan, and Z. B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376 (2012), 257-259.
  • [22] M. A. Herzallah and K. A. Gepreel, Approximate solution to time-space fractional cubic nonlinear Schrodinger equation, Applied Mathematical Modeling, 36(11) (2012), 5678-5685.
  • [23] H. Jafari and S. Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commu. Nonlinear Sci. Num. Simu, 14 (2009), 1962-1969.
  • [24] G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl, 51 (2006), 1367-1376.
  • [25] S. Kemple and H. Beyer, Global and causal solutions of fractional differential equations, Transform methods and special functions: Varna 96, Proceedings of 2nd international workshop (SCTP), Singapore, 96 (1997), 210-216.
  • [26] M. Lakestani and J. Manafian, Analytical treatment of nonlinear conformable timefractional Boussinesq equations by three integration methods, Opt Quant Electron, 50(4) (2018), 1-31.
  • [27] M. Lakestani, J. Manafian, A. R. Najafizadeh, and M. Partohaghighi, Some new soliton solutions for the nonlinear the fifth-order integrable equations, Comput. Meth. Diff. Equ., 10(2) (2022), 445-460.
  • [28] Z. B. Li and J. H. He, Application of the fractional complex transform to fractional differential equations, Nonlinear Sci. Lett. A, 2 (2011), 121-126.
  • [29] B. Lu, The first integral method for some time fractional differential equations, J. Math. Anal. Appl., 395 (2012), 684-693.
  • [30] J. Manafian Heris and M. Bagheri, Exact solutions for the modified KdV and the generalized KdV equations via Exp-function method, J. Math. Extension, 4 (2010), 77-98.
  • [31] J. Manafian Heris and M. Fazli Aghdaei, Application of the Exp-function method for solving the combined KdV-mKdV and Gardner-KP equations, Math. Science, 6 (2012), 68.
  • [32] J. Manafian Heris and I. Zamanpour, Analytical treatment of the coupled Higgs equation and the Maccari system via Exp-Function Method, Acta Universitatis Apulensis, 33 (2013), 203-216.
  • [33] J. Manafian Heris and M. Lakestani, Solitary wave and periodic wave solutions for variants of the KdVBurger and the K(n,n)-Burger equations by the generalized tanh-coth method, Commu. Num. Anal., 2013 (2013), 1-18.
  • [34] J. Manafian and M. Lakestani, Application of tan(φ/2)-expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearity, Optik, 127(4) (2016), 2040-2054.
  • [35] J. Manafian and M. Lakestani, Abundant soliton solutions for the Kundu-Eckhaus equation via tan(φ(ξ))expansion method, Optik, 127(14) (2016), 5543-5551.
  • [36] J. Manafian and M. Lakestani, Optical soliton solutions for the Gerdjikov-Ivanov model via tan(φ/2)expansion method, Optik, 127(20) (2016), 9603-9620.
  • [37] J. Manafian and M. Lakestani, A new analytical approach to solve some of the fractional-order partial differential equations, Indian J Phys., 91(3) (2016), 243-258.
  • [38] J. Manafian and M. Lakestani, N-lump and interaction solutions of localized waves to the (2+ 1)- dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation, J. Geom. Phys., 150 (2020), 103598.
  • [39] J. Manafian, L. A. Dawood, and M. Lakestani, New solutions to a generalized fifth-order KdV like equation with prime number p = 3 via a generalized bilinear differential operator, Partial Diff. Eq. Appl. Math., 9 (2024), 100600.
  • [40] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, New York, Wiley, 1993.
  • [41] M. S. Mohamed, Analytical treatment of Abel integral equations by optimal homotopy analysis transform method, Journal of Information and Computing Science, 10(1) (2015), 19-28.
  • [42] R. Y. Molliq and B. Batiha, Approximate analytic solutions of fractional Zakharov-Kuznetsov equations by fractional complex transform, Int. J. Eng. Tech., 1 (2012), 1-13.
  • [43] S. Momani and M. A. Noor, Numerical methods for fourth-order fractional integrodifferential equations, Appl. Math. Comput, 182 (2006), 754-760.
  • [44] S. Momani and N. T. Shawagfeh, Decomposition method for solving fractional Riccati differential equations, Appl.Math. Comput., 182 (2006), 1083-1092.
  • [45] S. Momani and Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A, 365 (2007), 345-350.
  • [46] S. R. Moosavi, N, Taghizadeh, and J. Manafian, Analytical approximations of one-dimensional hyperbolic equation with non-local integral conditions by reduced differential transform method, Comput. Meth. Diff. Equ., 8(3) (2020), 537-552.
  • [47] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, New York: Academic Press, 1999.
  • [48] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
  • [49] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract.Calculus. Appl. Anal., 5 (2002), 367-386.
  • [50] S. G. Samko, A. Kilbas, and O. Marichev, Fractional integrals and derivatives: theory and applications, Yverdon, Gordon and Breach, 1993.
  • [51] N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, Numerical studies for a multi-order fractional differential equation, Phys. Lett. A, 371 (2007), 26-33.
  • [52] V. E. Zakharov and E. A. Kuznetsov, On three–dimensional solitons, Sov. Phys, 39 (1974), 285-288.
  • [53] M. Zhang, X. Xie, J. Manafian, O. A. Ilhan, and G. Singh, Characteristics of the new multiple rogue wave solutions to the fractional generalized CBS-BK equation, J. Adv. Res., 38 (2022), 131-142.
  • [54] X. Zhao, H. Zhou, Y. Tang, and H. Jia, Travelling wave solutions for modified Zakharov–Kuznetsov equation, Appl. Math. Comput, 181 (2006), 634-648.
  • [55] B. Zheng and C. Wen, Exact solutions for fractional partial differential equations by a new fractional subequation method, Advances in Difference Equations, 2013 (2013), 199.
Computational Methods for Differential Equations
Volume 13, Issue 1
January 2025
Pages 1-12

Files

History

  • Receive Date: 14 February 2024
  • Revise Date: 24 March 2024
  • Accept Date: 24 March 2024

Share

How to cite

Statistics