Advertisement

Existence of nontrivial solutions for a system of fractional advection–dispersion equations

  • Dexiang Ma
  • Lishan Liu
  • Yonghong Wu
Original Paper
  • 56 Downloads

Abstract

In this paper, we investigate the existence of nontrivial solutions for a class of fractional advection–dispersion systems. The approach is based on the variational method by introducing a suitable fractional derivative Sobolev space. We take two examples to demonstrate the main results.

Keywords

Fractional advection–dispersion equation Weak solution Critical point theory Anomalous diffusion Variational method 

Mathematics Subject Classification

58E05 34B15 26A33 

Notes

Acknowledgements

Lishan Liu was supported financially by the National Natural Science Foundation of China (11371221). Dexiang Ma was supported financially by the Fundamental Research Funds for the Central Universities (2014MS62).

References

  1. 1.
    Ervin, V., Roop, J.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carreras, B.A., Lynch, V.E., Zaslavsky, G.M.: Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence models. Phys. Plasmas 8, 5096–5103 (2001)CrossRefGoogle Scholar
  3. 3.
    Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: applications to turbulence. Phys. Rev. Lett. 58, 1100–1103 (1987)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Zaslavsky, G.M., Stevens, D., Weitzner, H.: Self-similar transport in incomplete chaos. Phys. Rev. E 48, 1683–1694 (1993)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: The fractional order governing equations of Levy motion. Water Resour. Res. 36, 1413–1423 (2000)CrossRefGoogle Scholar
  6. 6.
    Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M.: Application of a fractional advection–dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)CrossRefGoogle Scholar
  7. 7.
    Fix, G., Roop, J.: Least squares finite-element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48, 1017–1033 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Benson, D.A., Schumer, R., Meerschaert, M.M., Wheatcraft, S.W.: Fractional dispersion, Levy flights, and the MADE tracer tests. Transp. Porous Media 42, 211–240 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Lu, S., Molz, F.J., Fix, G.J.: Possible problems of scale dependency in applications of the three-dimensional fractional advection–dispersion equation to natural porous media. Water Resour. Res. 38(9), 1165–1171 (2002)CrossRefGoogle Scholar
  10. 10.
    Meerschaert, M.M., Benson, D.A., Baeumer, B.: Multidimensional advection and fractional dispersion. Phys. Rev. E 59(5), 5026–5028 (1999)CrossRefGoogle Scholar
  11. 11.
    Jiao, F., Zhou, Y.: Existence of solutions for a class of fractional boundary value problems via critical point theory. Comput. Math. Appl. 62, 1181–1199 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Teng, K., Jia, H., Zhang, H.: Existence and multiplicity results for fractional differential inclusions with Dirichlet boundary conditions. Appl. Math. Comput. 220, 792–801 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Zhang, X., Liu, L., Wu, Y.: Variational structure and multiple solutions for a fractional advection dispersion equation. Comput. Math. Appl. 68, 1794–1805 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bonanno, G.: A critical points theorem and nonlinear differential problems. J. Global Optim. 28, 249–258 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bonanno, G., Candito, P.: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ. 244, 3031–3059 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chen, J., Tang, X.: Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory. Abstr. Appl. Anal. (2012).  https://doi.org/10.1155/2012/648635
  17. 17.
    Zhu, B., Liu, L., Wu, Y.: Existence and uniqueness of global mild solutions for a class of nonlinear fractional reaction diffusion equations with delay. Comput. Math. Appl.  https://doi.org/10.1016/j.camwa.2016.01.028
  18. 18.
    Zhang, X., Liu, L., Wu, Y., Wiwatanapataphee, B.: Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion. Appl. Math. Lett. 66, 1–8 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ge, B.: Multiple solutions for a class of fractional boundary value problems. Abstr. Appl. Anal., art. ID 468-980 (2012)Google Scholar
  20. 20.
    Kong, L.: Existence of solutions to boundary value problems arising from the fractional advection dispersion equation. Electron. J. Differ. Equ. 106, 1–15 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Heidarkhani, S.: Infinitely many solutions for nonlinear perturbed fractional boundary value problems. Ann. Univ. Craiova Math. Comput. Sci. Ser. 41(1), 88–103 (2014)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Bai, C.: Existence of solutions for a nonlinear fractional boundary value problem via a local minimum theorem. Electron. J. Differ. Equ. 176, 1–9 (2012)MathSciNetGoogle Scholar
  23. 23.
    Li, Y., Sun, H., Zhang, Q.: Existence of solutions to fractional boundary-value problems with a parameter. Electron. J. Differ. Equ. 141, 1–12 (2013)MathSciNetGoogle Scholar
  24. 24.
    Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS, vol. 65. American Mathematical Society, New York (1986)Google Scholar
  25. 25.
    Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)CrossRefzbMATHGoogle Scholar
  26. 26.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  27. 27.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)Google Scholar
  28. 28.
    Bonanno, G., D’Aguì, G.: A critical point theorem and existence results for a nonlinear boundary value problem. Nonlinear Anal. 72, 1977–1982 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNorth China Electric Power UniversityBeijingChina
  2. 2.School of Mathematical SciencesQufu Normal UniversityQufuChina
  3. 3.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia

Personalised recommendations