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Fractional diffusion equation and Green function approach: Exact solutions

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  • Lenzi, E.K.
  • Mendes, R.S.
  • Gonçalves, G.
  • Lenzi, M.K.
  • da Silva, L.R.

Abstract

We investigate the solutions of a fractional diffusion equation with radial symmetry by using the Green function approach and by taking the N-dimensional case into account. In our analysis, a spatial time-dependent diffusion coefficient is considered, i.e., D(r,t)=Dtδ-1r-θ/Γ(α). The presence of external forces F(r)=Krε with ε=-1-θ and F(r)=-kr+Krε is also taken into account. In particular, we discuss the results obtained by employing boundary conditions defined on a finite interval, and afterwards the analysis is extended to a semi-infinite interval. Finally, we also discuss a rich class of diffusive processes that can be obtained from the results presented in this work.

Suggested Citation

  • Lenzi, E.K. & Mendes, R.S. & Gonçalves, G. & Lenzi, M.K. & da Silva, L.R., 2006. "Fractional diffusion equation and Green function approach: Exact solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 360(2), pages 215-226.
    • Handle: RePEc:eee:phsmap:v:360:y:2006:i:2:p:215-226
    DOI: 10.1016/j.physa.2005.06.073
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    References listed on IDEAS

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    1. Metzler, Ralf & Glöckle, Walter G. & Nonnenmacher, Theo F., 1994. "Fractional model equation for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(1), pages 13-24.
    2. Metzler, Ralf & Klafter, Joseph, 2000. "Boundary value problems for fractional diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(1), pages 107-125.
    3. Narahari Achar, B.N. & W. Hanneken, John & Clarke, T., 2004. "Damping characteristics of a fractional oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 339(3), pages 311-319.
    4. Marcello Pagnini, 2003. "Misura e determinanti dell’agglomerazione spaziale nei comparti industriali in Italia," Rivista di Politica Economica, SIPI Spa, vol. 93(2), pages 149-149, March-Apr.
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    Cited by:

    1. Guo, Gang & Li, Kun & Wang, Yuhui, 2015. "Exact solutions of a modified fractional diffusion equation in the finite and semi-infinite domains," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 417(C), pages 193-201.
    2. Povstenko, Y.Z., 2010. "Evolution of the initial box-signal for time-fractional diffusion–wave equation in a case of different spatial dimensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4696-4707.
    3. Lashkarian, Elham & Reza Hejazi, S., 2017. "Group analysis of the time fractional generalized diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 479(C), pages 572-579.

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