The well-posedness and exact solution of fractional magnetohydrodynamic equations

Abstract

For numerous fluids between elastic and viscous materials, the fractional magnetohydrodynamic models have an advantage over the integer-order models. We study the fractional magnetohydrodynamic equations with external force. On the basis of conformable fractional derivative and the respective useful properties, we study the well-posedness of the incompressible time fractional magnetohydrodynamic equations. In the qualitative study, the complicate nonlinear terms in time fractional magnetohydrodynamic equations are technical handled. The existence of the solution to the fractional incompressible magnetohydrodynamic equations is obtained. Moreover, the uniqueness and continuous dependence of the solution on the initial data are proved. In the quantitative study, using some mathematical transformation with practical physical significance, the exact solutions and the respective figures of the fractional magnetohydrodynamic equations are presented. The time fractional derivative system possesses property of time memory. The fractional derivative influences the control effect in complex systems with fluids between different elastic and viscous materials. The results are still valid if the time order is integer (integer-order magnetohydrodynamic equations case), or the magnetic field is zero (Navier–Stokes equations case).

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References

  1. 1.

    Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 1st edn. Springer, Netherlands (2007)

    Google Scholar 

  2. 2.

    Baleanu, D., Tenreiro Machado, J.A., Luo, A.C.J.: Fractional Dynamics and Control. Springer, New York (2011)

    Google Scholar 

  3. 3.

    Ostalczyk, P.: Discrete Fractional Calculus: Applications in Control and Image Processing. World Scientific, Singapore (2016)

    Google Scholar 

  4. 4.

    Ma, W.X., Mousa, M.M., Ali, M.R.: Application of a new hybrid method for solving singular fractional Lane-Emden-type equations in astrophysics. Mod. Phys. Lett. B. 34, 2050049 (2020)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Li, C.P., Deng, W.H.: Remarks on fractional derivatives. Appl. Math. Comput. 187, 777–784 (2007)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Dilmi, M., Dilmi, M., Benseridi, H.: Variational formulation and asymptotic analysis of viscoelastic problem with Rieman–Liouville fractional derivatives. Math. Method Appl. Sci. 3, 1–20 (2019)

    MATH  Google Scholar 

  7. 7.

    Camilli, F., Goffi, A.: Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative. NODEA-Nonlinear Diff. 27, 22 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

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

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chung, W.S.: Fractional Newton mechanics with conformable fractional derivative. J. Comput. Appl. Math. 290, 150–158 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Khodadad, F.S., Nazari, F., Eslami, M., Rezazadeh, H.: Soliton solutions of the conformable fractional Zakharov–Kuznetsov equation with dual-power law nonlinearity. Opt. Quant. Electron. 49, 384 (2017)

    Article  Google Scholar 

  11. 11.

    Hashemi, M.S.: Invariant subspaces admitted by fractional differential equations with conformable derivatives. Chaos Soliton Fract. 107, 161–169 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Acan, O., Firat, O., Keskin, Y.: Conformable variational iteration method, conformable fractional reduced differential transform method and conformable homotopy analysis method for non-linear fractional partial differential equations. Wave Random Complex. 30, 250–268 (2020)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Wang, G.T., Qin, J.F., Zhang, L.H., Baleanu, D.: Explicit iteration to a nonlinear fractional Langevin equation with non-separated integro-differential strip-multi-point boundary conditions. Chaos Soliton. Fract. 131, 109476 (2020)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Jardin, S.C.: Review of implicit methods for the magnetohydrodynamic description of magnetically confined plasmas. J. Comput. Phys. 231, 822–838 (2012)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Low, B.C.: Magnetohydrodynamic processes in the solar corona: Flares, coronal mass ejections, and magnetic helicity. Phys. Plasmas. 1, 1684–1690 (1994)

    Article  Google Scholar 

  16. 16.

    Bodin, H.A.B., Keen, B.E.: Experimental studies of plasma confinement in toroidal systems. Rep. Prog. Phys. 40, 1415–1565 (1977)

    Article  Google Scholar 

  17. 17.

    Hammond, R.T., Davis, J., Bobb, L.: Reflection, absorption, and transmission of ultra-low-frequency electromagnetic waves through a Gaussian conductor. J. Appl. Phys. 81, 1619–1622 (1997)

    Article  Google Scholar 

  18. 18.

    Yousofvand, R., Derakhshan, S., Ghasemi, K., Siavashi, M.: MHD transverse mixed convection and entropy generation study of electromagnetic pump including a nanofluid using 3D LBM simulation. Int. J. Mech. Sci. 133, 73–90 (2017)

    Article  Google Scholar 

  19. 19.

    Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation. Commun. Nonlinear Sci. 59, 222–234 (2018)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Rogers, C., Schief, W.K.: Bäcklund and Darboux transformations: geometry and modern applications in soliton theory. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  21. 21.

    Ma, W.X.: Symbolic computation of lump solutions to a combined equation involving three types of nonlinear terms. E. Asian J. Appl. Math. 10, 732–745 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Ma, W.X., Huang, Y.H., Wang, F.D.: Inverse scattering transforms and soliton solutions of nonlocal reverse-space nonlinear Schrödinger hierarchies. Stud. Appl. Math. 145, 563–585 (2020)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Hayat, T., Naeem, I., Ayub, M., Siddiqui, A.M., Asghar, S., Khalique, C.M.: Exact solutions of second grade aligned MHD fluid with prescribed vorticity. Nonlinear Anal-Real. 10, 2117–2126 (2009)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Adem, A.R., Moawad, S.M.: Exact solutions to several nonlinear cases of generalized Grad-Shafranov equation for ideal magnetohydrodynamic flows in axisymmetric domain. Z. Naturforsch. A. 73, 371–383 (2018)

    Article  Google Scholar 

  25. 25.

    Chen, C., Jiang, Y.L.: Simplest equation method for some time-fractional partial differential equations with conformable derivative. Comput. Math. Appl. 75, 2978–2988 (2018)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Holm, D.D., Marsden, J.E., Ratiu, T., Weinstein, A.: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116 (1985)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Temam, R.: On the Euler equations of incompressible perfect fluids. J. Funct. Anal. 20, 32–43 (1975)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Huang, X., Wang, Y.: Global strong solution with vacuum to the two dimensional density-dependent Navier–Stokes system. SIAM J. Math. Anal. 46, 1771–1788 (2014)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Li, J., Xin, Z.: Global existence of regular solutions with large oscillations and vacuum. In: Giga, Y., Novotny, A. (eds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham (2016)

    Google Scholar 

  30. 30.

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

    MathSciNet  Article  Google Scholar 

  31. 31.

    Li, Z.T., Zhang, X.F.: New exact kink solutions and periodic form solutions for a generalized Zakharov–Kuznetsov equation with variable coefficients. Commun. Nonlinear Sci. 15, 3418–3422 (2010)

    MathSciNet  Article  Google Scholar 

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Correspondence to Huanhe Dong.

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This work was supported by National Natural Science Foundation of China (No. 11975143), Natural Science Foundation of Shandong Province (No. ZR2019QD018)

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Liu, M., Fang, Y. & Dong, H. The well-posedness and exact solution of fractional magnetohydrodynamic equations. Z. Angew. Math. Phys. 72, 54 (2021). https://doi.org/10.1007/s00033-021-01483-0

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Keywords

  • Magnetohydrodynamic equation
  • Conformable fractional derivative
  • Well-posedness
  • Generalized Riccati equation method

Mathematics Subject Classification

  • 49K40
  • 35C07
  • 76W05