Efficient Application of the Two-Grid Technique for Solving Time-Fractional Non-linear Parabolic Problem

  • Miglena N. KolevaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)


In this paper we present numerical methods for solving a non-linear time-fractional parabolic model. To cope with non-local in time nature of the problem, we exploit the idea of the two-grid method and develop fast numerical algorithms. Moreover, we show that suitable modifications of the standard two-grid technique lead to significant reduction of the computational time. Numerical results are also discussed.



This research is supported by the Bulgarian National Fund of Science under the Project I02/20 - 2014.


  1. 1.
    Axelsson, O.: On mesh independence and Newton methods. Appl. Math. 38(4–5), 249–265 (1993)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bear, J.: Dynamics of Fluids in Porous Media. American Elsevier, New York (1972)zbMATHGoogle Scholar
  3. 3.
    Cui, M.: Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer. Algor. 62(3), 383–409 (2013)zbMATHCrossRefGoogle Scholar
  4. 4.
    El-Sayed, A.M.A., Rida, S.Z., Arafa, A.A.M.: Exact solutions of fractional-order biological population model. Commun. Theor. Phys. 52, 992–996 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)zbMATHCrossRefGoogle Scholar
  6. 6.
    Gurtin, M.E., Maccamy, R.C.: On the diffusion of biological population. Math. Biosci. 33, 35–49 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Ishimura, N., Koleva, M.N., Vulkov, L.G.: Numerical solution via transformation methods of nonlinear models in option pricing. Am. Inst. Phys. CP 1301, 387–394 (2010)Google Scholar
  8. 8.
    Jiang, X.Y., Xu, M.Y.: Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media. Int. J. Nonlinear Mech. 41, 156–165 (2006)CrossRefGoogle Scholar
  9. 9.
    Jin, J., Shu, S., Xu, J.: A two-grid discretization method for decoupling systems of partial differential equations. Math. Comput. 75, 1617–1626 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Koçak, H., Yildirim, A.: An efficient new iterative method for finding exact solutions of nonlinear time-fractional partial differential equations. Nonlinear Anal. Model. Control 16(4), 403–414 (2011)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Koleva, M.N., Vulkov, L.G.: A two-grid approximation of an interface problem for the nonlinear Poisson-Boltzmann equation. In: Margenov, S., Vulkov, L.G., Waśniewski, J. (eds.) NAA 2008. LNCS, vol. 5434, pp. 369–376. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  12. 12.
    Koleva, M.N., Vulkov, L.G.: Two-grid quasilinearization approach to ODEs with applications to model problems in physics and mechanics. Comput. Phys. Commun. 181, 663–670 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Liu, Y., Li, Z., Zhang, Y.: Homotopy perturbation method to fractional biological population equation. Fract. Diff. Calc. 1, 117–124 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lu, Y.G.: Hölder estimates of solutions of biological population equations. Appl. Math. Lett. 13, 123–126 (2000)zbMATHCrossRefGoogle Scholar
  15. 15.
    Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Okubo, A.: Diffusion and Ecological Problem: Mathematical Models. Biomathematics 10. Springer, Berlin (1980) Google Scholar
  17. 17.
    Patlashenko, I., Givoli, D., Barbone, P.: Time-stepping schemes for systems of Volterra integrodifferential equations. Comput. Methods Appl. Mech. Eng. 190, 5691–5718 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Roul, P.: Application of homotopy perturbation method to biological population model. Appl. Appl. Math. 5(10), 1369–1378 (2010)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Wang, P., Zheng, C., Gorelick, S.: A general approach to advective-dispersive transport with multirate mass transfer. Adv. Water. Res. 28, 33–42 (2005)CrossRefGoogle Scholar
  20. 20.
    Xu, J.: A novel two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15(1), 231–237 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.FNSEUniversity of RousseRousseBulgaria

Personalised recommendations