Boundary Value Problems for Fractional PDE and Their Numerical Approximation

  • Boško S. Jovanović
  • Lubin G. Vulkov
  • Aleksandra Delić
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)


Fractional order partial differential equations are considered. The main attention is devoted to fractional in time diffusion equation. An interface problem for this equation is studied and its well posedness in the corresponding Sobolev like spaces is proved. Analogous results are obtained for a transmission problem in disjoint intervals.


Fractional derivative fractional PDE initial-boundary value problem interface problem transmission problem finite differences 


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  1. 1.
    Alikhanov, A.A.: Boundary value problems for the diffusion equation of the variable order in differential and difference settings (May 10, 2011); arXiv:1105.2033v1 [math.NA]Google Scholar
  2. 2.
    Benson, D.A., Wheatcraft, S.W., Meerschaeert, M.M.: The fractional order governing equations of Levy motion. Water Resour. Res. 36, 1413–1423 (2000)CrossRefGoogle Scholar
  3. 3.
    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(12), 5096–5103 (2001)CrossRefGoogle Scholar
  4. 4.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differential Equations 23, 558–576 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Henry, B., Wearne, S.: Existence of turing instabilities in a two-species fractional reactiondiffusion system. SIAM J. Appl. Math. 62, 870–887 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ilic, M., Turner, I.W., Liu, F., Anh, V.: Analytical and numerical solutions to one-dimensional fractional-in-space diffusion in a composite medim. Appl. Math. Comp. 216, 2248–2262 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jovanović, B.S., Vulkov, L.G.: Operator’s approach to the problems with concentrated factors. In: Vulkov, L.G., Waśniewski, J., Yalamov, P. (eds.) NAA 2000. LNCS, vol. 1988, pp. 439–450. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Jovanović, B.S., Vulkov, L.G.: On the convergence of finite difference schemes for the heat equation with concentrated capacity. Numer. Math. 89(4), 715–734 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jovanović, B.S., Vulkov, L.G.: Numerical solution of a two-dimensional parabolic transmission problem. Int. J. Numer. Anal. Model. 7(1), 156–172 (2010)MathSciNetGoogle Scholar
  10. 10.
    Jovanović, B.S., Vulkov, L.G.: Numerical solution of a parabolic transmission problem. IMA J. Numer. Anal. 31, 233–253 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kirchner, J.W., Feng, X., Neal, C.: Fractal stream chemistry and its implications for contaminant transport in catchments. Nature 403, 524–526 (2000)CrossRefGoogle Scholar
  12. 12.
    Kiryakova, V.: A survey on fractional calculus, fractional order differential and integral equations and related special functions. Rousse (2012)Google Scholar
  13. 13.
    Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lions, J.L., Magenes, E.: Non homogeneous boundary value problems and applications. Springer, Berlin (1972)CrossRefGoogle Scholar
  15. 15.
    Love, E.R., Young, L.C.: On fractional integration by parts. Proc. London Math. Soc., Ser. 2 44, 1–35 (1938)CrossRefGoogle Scholar
  16. 16.
    Mainardi, F.: Fractional diffusive waves in viscoelastic solids. In: Wegner, J.L., Norwood, F.R. (eds.) Nonlinear Waves in Solids, pp. 93–97. Fairfield (1995)Google Scholar
  17. 17.
    Mainardi, F.: Fractional calculus. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)Google Scholar
  18. 18.
    Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Müller, H.P., Kimmich, R., Weis, J.: NMR flow velocity mapping in random percolation model objects: Evidence for a power-law dependence of the volume-averaged velocity on the probe-volume radius. Phys. Rev. E 54, 5278–5285 (1996)CrossRefGoogle Scholar
  20. 20.
    Nigmatullin, R.R.: The realization of the generalized transfer equation in a medium with fractal geometry. Physica Status Solidi. B 133(1), 425–430 (1986)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Podlubny, I.: Fractional Differential Equations. Academic Press (1999)Google Scholar
  22. 22.
    Samarskii, A.A.: Theory of difference schemes. Nauka, Moscow (1989) Russian; English edition: Pure and Appl. Math., vol. 240. Marcel Dekker, Inc., (2001)Google Scholar
  23. 23.
    Sokolov, I.M., Klafter, J., Blumen, A.: Fractional kinetics. Physics Today 55(11), 48–54 (2002)CrossRefGoogle Scholar
  24. 24.
    Taukenova, F.I., Shkhanukov-Lafishev, M.K.: Difference methods for solving boundary value problems for fractional differential equations. Comput. Math. Math. Phys. 46(10), 1785–1795 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Vladimirov, V.S.: Equations of mathematical physics. Nauka, Moscow (1988) (Russian)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Boško S. Jovanović
    • 1
  • Lubin G. Vulkov
    • 2
  • Aleksandra Delić
    • 1
  1. 1.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia
  2. 2.Center of Applied MathematicsUniversity of RousseRousseBulgaria

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