Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 737–741 | Cite as

Numerical Method for Fractional Advection-Diffusion Equation with Heredity



We propose a method of construction of difference schemes for fractional partial differential equations with delay in time. For the fractional equation with two-sided diffusion, fractional transfer in time, and a functional aftereffect, we construct an implicit difference scheme. We use the shifted Grünwald–Letnikov formulas for the approximation of fractional derivatives with respect to spatial variables and the L1-algorithm for the approximation of fractional derivatives in time. Also we use piecewise constant interpolation and extrapolation by extending the discrete prehistory of the model in time. The algorithm is a fractional analog of a purely implicit method; on each time step, it is reduced to the solution of linear algebraic systems. We prove the stability of the method and find its order of convergence.

Keywords and phrases

equation with fractional derivatives functional delay mesh scheme interpolation extrapolation order of convergence 

AMS Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin (2010).CrossRefMATHGoogle Scholar
  2. 2.
    F. Liu, P. Zhuang, and K. Burrage, “Numerical methods and analysis for a class of fractional advection-dispersional models,” Comput. Math. Appl., 64, 2990–3007 (2012).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    V. G. Pimenov, Difference Methods of Solution of Partial Differential Equations with Heredity [in Russian], Ural State Univ., Yekaterinburg (2014).Google Scholar
  4. 4.
    V. G. Pimenov and A. B. Lozhnikov, “Difference schemes for numerical solution of the heat transfer equation with aftereffect,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 17, No. 1, 178–189 (2011).MATHGoogle Scholar
  5. 5.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives and Their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).MATHGoogle Scholar
  6. 6.
    M. Kh. Shkhanukov-Lafishev and F. I. Taukenova, “Difference methods of solutions of boundaryvalue problems for fractional differential equations,” Zh. Vychisl. Mat. Mat. Fiz., 46, No. 10, 1871–1881 (2006).MathSciNetGoogle Scholar
  7. 7.
    C. Tadjeran, M. M. Meerschaert, and H. P. Scheffler, “A second-order accurate numerical approximation for the fractional diffusion equation,” J. Comput. Phys., 213, 205–214 (2006).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York (1996).CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and Mechanics, Ural Branch of RASUral Federal UniversityEkaterinburgRussia

Personalised recommendations