Anomalous diffusion and fractional diffusion equations

  • Giane Gonçalves UEM
  • Marcelo Kaminski Lenzi UEM
  • Luciana de Souza Moraes UEM
  • Ervin Kaminski Lenzi Física - UEM
  • Marcelo Freitas de Andrade UEM

Abstract

In this work we investigate the anomalous diffusion equations, usually applied to describe the anomalous diffusion, which employ fractional derivatives for the time or the spatial variables. In particular, we obtain exact solutions by taking a generic initial condition into account and developing a perturbation theory to investigate complex situations. We also verify that the fractional derivatives, when applied to the time variable, lead us to a anomalous diffusion with second moment finite, i.e., <χ2> ∝ tα (0 < α < 1, and α > 1, corresponding to sub and superdifusive behavior, respectively). By way of contrast, the fractional derivative applied to the spatial variable results in a anomalous diffusion where the second moment is not finite. These equations generalize the usual diffusion equation in order to incorporate several situations. We also employ the continuous time random walking formalism to investigate the implications obtained by using fractional derivatives in the diffusion equation

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Author Biography

Ervin Kaminski Lenzi, Física - UEM
Possui graduação em Física pela Universidade Estadual de Maringá (1996) , mestrado em Física pelo Centro Brasileiro de Pesquisas Físicas (1998) , doutorado em Física pelo Centro Brasileiro de Pesquisas Físicas (2002) , ensino-fundamental-primeiro-grau pelo Colégio Marista de Maringá (1989) e ensino-medio-segundo-grau pelo Colégio Marista de Maringá (1992) . Atualmente é Professor Adjunto da Universidade Estadual de Maringá, Revisor de periódico da Physical Review E - Statistical Physics, Plasmas, Fluids and Related Interd, Revisor de periódico da Physica. A, Revisor de periódico da Physical Review Letters, Revisor de periódico da Physics Letters A, Membro de corpo editorial da Acta Scientiarum (UEM) e Revisor de periódico da Journal of Physics. A, Mathematical and Theoretical. Tem experiência na área de Física , com ênfase em Física Geral. Atuando principalmente nos seguintes temas: não extensivo, transporte, difusão anômala. Currículo Lattes
Published
2008-03-27
How to Cite
Gonçalves, G., Lenzi, M. K., Moraes, L. de S., Lenzi, E. K., & Andrade, M. F. de. (2008). Anomalous diffusion and fractional diffusion equations. Acta Scientiarum. Technology, 27(2), 123-131. https://doi.org/10.4025/actascitechnol.v27i2.1476
Section
Chemical Engineering

 

0.8
2019CiteScore
 
 
36th percentile
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0.8
2019CiteScore
 
 
36th percentile
Powered by  Scopus