THERMAL SCIENCE

International Scientific Journal

Ervin K. Lenzi

,

Denner S. Vieira

,

Marcelo K. Lenzi

,

Giane Gonçalves

,

Delano P. Leitoles

SOLUTIONS FOR A FRACTIONAL DIFFUSION EQUATION WITH RADIAL SYMMETRY AND INTEGRO-DIFFERENTIAL BOUNDARY CONDITIONS

ABSTRACT
The solutions for a dimensional system with radial symmetry and governed by a fractional diffusion equation have been investigated. More specifically, a spherical system was considered, being defined in the semi - infinity interval [R, ¥) and subjected to surface effects described in terms of integro - differential boundary conditions which has many practical applications. The analytical solutions were obtained by using the Green function approach, showing a broad range of different behaviors which can be related to anomalous diffusion. The analyses also considered the influence of the parameters of the analytical solution in order to describe a more realistic scenario.
KEYWORDS
fractional derivative, diffusion equation, anomalous diffusion
PAPER SUBMITTED: 2015-01-14
PAPER REVISED: 2015-01-15
PAPER ACCEPTED: 2015-03-05
PUBLISHED ONLINE: 2015-04-04
DOI REFERENCE: https://doi.org/10.2298/TSCI150114045L
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Supplement 1, PAGES [S1 - S6]
REFERENCES
  1. Hilfer, R.,Applications of Fractional Calculus in Physics, World Scientific, Singapore,2000
  2. Shlesinger, M.F., et al.,Lévy Flights and Related Topics in Physics,Spring-Verlag, Berlin,1994
  3. Podlubny, I.,Fractional Differential Equations,Academic Press, New York,1999
  4. Mathai, A.M.,et al.,The H-Function: Theory and Applications,Springer,New York,2009
  5. Debnath, L. Recent Applications of Fractional Calculus to Science and Engineering, International Journal of Mathematics and Mathematical Sciences, 2003(2003), pp. 3413-3442
  6. Silva, F.R.G.B.,et al.,An Extension of the Linear Luikov System Equations of Heat and Mass Transfer,International Journal of Heat and Mass Transfer,63(2013), pp. 233-238
  7. Lenzi, E.K.,et al.,Reaction on a Solid Surface Supplied by an Anomalous Mass Transfer Source,Physica A,410(2014), pp. 399-406
  8. Akil, K.A.K.,et al.,Fractional Reaction-Diffusion Equations for Modelling Complex Biological Patterns,Malaysian Journal of Fundamental & Applied Sciences,8(2012), pp. 122-126
  9. Losa, G.A.,et al.,Fractals in Biology and Medicine: Volume IV,Birkhauser,Basel, 2005
  10. Metzler, R., Klafter, J., The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach,Physics Report,339 (2000),pp. 1-77
  11. Pekalski, A., Sznajd-Weron, K.,Anomalous Diffusion: From Basics to Applications,Springer, Heidelberg,1999
  12. Gardiner, C.,Stochastic Methods: A Handbook for the Natural and Social Sciences,Springer, New York,2009
  13. Risken, H.,The Fokker-Planck Equation: Methods of Solutions and Applications,Springer,New York,1989
  14. Lenzi, E.K.,et al.,Some Results for a Fractional Diffusion Equation with Radial Symmetry in a Confined Region,Physica A, 388 (2009), pp. 806-810
  15. Santoro, P.A.,et al.,Anomalous Diffusion Governed by a Fractional Diffusion Equation and the Electrical Response of an Electrolytic Cell,Journal of Chemical Physics,135 (2011),article number: 114704
  16. Lenzi, E.K.,et al.,Non-Markovian Diffusion an the Adsorption-Desorption Process,Physical Review E, 81(2010), article number:011116
  17. Lenzi, E.K.,et al.,Fractional Diffusion Equation, Boundary Conditions and Surface Effects,Journal of Statistical Mechanics-Theory and Experiment,2014(2014), article number:P08019
  18. Tateishi, A.A.,et al.,First Passage Time for a Diffusive Process Under a Geometric Constraint,Journal of Statistical Mechanics-Theory and Experiment,2013 (2013), article number.P09017
  19. Leijnse, N.,et al.,Diffusion Inside Living Human Cells,The European Physical Journal Special Topics,204 (2012),75-84, 2012
  20. Yang, X.J.,Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Iteration Method,Thermal Science, 17 (2013), pp. 625-628, 2013
  21. Hristov, J., Heat-Balance Integral to Fractional (Half-Time) Heat Diffusion Sub-Model, Thermal Science, 14 (2010), pp. 291-316, 2010
  22. Caspi, A.,et al.,Enhanced Diffusion in Active Intracellular Transport, Physical Review Letters, 85 (2000), pp. 5655
  23. Bruno, L.,et al.,Transition to Superdiffusive Behavior in Intracellular Actin-Based Transport Mediated by Molecular Motors, Physical Review E, 80 (2009), article number:011912
  24. Brangwynne, C. P.,et al.,Intracellular Transport by Active Diffusion, Trends in Cell Biology, 19 (2009), pp. 423-427
  25. Weber S. C.,et al.,Nonthermal ATP-Dependent Fluctuations Contribute to the in vivo Motion of Chromosomal loci, Proceedings of the National Academy of Sciences of the United States of America, 109 (2012) , pp. 7338-7343
  26. Sokolov, I. M., Models of Anomalous Diffusion in Crowded Environments,Soft Matter,8(2012) pp. 9043-9052
  27. Weigel, A. V.,et al.,Ergodic and Nonergodic Processes Coexist inthe Plasma Membrane as Observed by Single-Molecule Tracking, Proceedings of the National Academy of Sciences of the United States of America, 108 (2011), pp. 6439-6443
  28. Rice,S. A., Diffusion - Limited Reactions,Elsevier,New York, 1985
  29. Yao, B.,et al.,Fractal Analysis of Porous Polycrystalline Titania by Adsorption, Chemistry of Materials, 12 (2000), pp. 3740 - 3744
  30. Samuel, J.,et al.,Diffusion Limited Reactions at Solid-Liquid Interfaces: Effects of Surface Geometry, Journal of Physical Chemistry, 95 (1991), pp. 1890 - 1895
  31. Samuel, J.,et al.,Diffusion Controlled Reactions on Porous Silicas: Mechanisms, Surface Diffusion Coefficients and Effects of Geometry, Journal of Physical Chemistry,96 (1992), pp. 6398-6405
PDF VERSION [DOWNLOAD]
Solutions for a fractional diffusion equation with radial symmetry and integro-differential boundary conditions

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence