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The finite element method in anomalous diffusion problems

  • A. N. Bondarenko
  • D. S. Ivashchenko
Article

Abstract

Under consideration are some aspects of application of the finite element method to numerical solution of the initial boundary value problems for a multidimensional time-fractional diffusion equation. Some survey of the available results is given, the algorithms for constructing meshes are discussed, and a few numerical examples are presented.

Keywords

finite element method anomalous diffusion fractional derivative automatic mesh generation 

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References

  1. 1.
    R. Metzler and J. Klafter, “The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach,” Phys. Rep. 339, 1–77 (2000).CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    A. N. Bondarenko, “Inverse Scattering Problems for an Equation of Lipmann-Shvinger Type,” Sibirsk. Zh. Industr. Mat. 6(3), 18–33 (2003).MATHMathSciNetGoogle Scholar
  3. 3.
    A. N. Bondarenko, “Feynman’s Diagram Approach for the Lipmann-Shvinger Equation with a Singular Potential,” Sibirsk.Zh. Industr. Mat. 6(4), 3–10 (2003).MathSciNetGoogle Scholar
  4. 4.
    O. C. Zienkiewicz and K. Morgan, Finite Elements and Approximations (Wiley, New York, 1983; Mir, Moscow, 1986).Google Scholar
  5. 5.
    E. Mitchell and R. Wait, The Finite Element Method in Partial Differential Equations (Wiley, Chichester, 1977; Mir, Moscow, 1981).MATHGoogle Scholar
  6. 6.
    G. Strang and G. Fix, An Analysis of The Finite Element Method (Prentice Hall, 1973; Mir, Moscow, 1977).MATHGoogle Scholar
  7. 7.
    M. Ciesielski and J. Leszczynski, “Numerical simulations of anomalous diffusion,” in Proceedings of Conference on Computer Methods in Mechanics (CMM-2003), URL: http://arxiv.org/ftp/math-ph/papers/0309/0309007.pdf.
  8. 8.
    H. G. Sun, W. Chenb, and K. Y. Szea, “A Semi-Analytical Finite Element Method for a Class of Time-Fractional Diffusion Equations,” URL: http://arxiv.org/pdf/1109.0641.pdf.
  9. 9.
    P.-O. Persson and G. Strang, “A SimpleMesh Generator in Matlab,” SIAM Review 46, 329–345 (2004).CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    P.-O. Persson, Mesh Generation for Implicit Geometries, URL: http://persson.berkeley.edu/thesis/persson-thesis.pdf.
  11. 11.
    J. Alberty, C. Carstensen, and S. A. Funken, “Remarks around 50 Lines of Matlab: Short Finite Element Implementation,” Numer. Algorithms 20, 117–137 (1999).CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.RN-UfaNIPIneftUfaRussia

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