Comptes Rendus
Model reduction, data-based and advanced discretization in computational mechanics
On the physical interpretation of fractional diffusion
Comptes Rendus. Mécanique, Volume 346 (2018) no. 7, pp. 581-589.

Even if the diffusion equation has been widely used in physics and engineering, and its physical content is well understood, some variants of it escape fully physical understanding. In particular, anormal diffusion appears in the so-called fractional diffusion equation, whose main particularity is its non-local behavior, whose physical interpretation represents the main part of the present work.

Published online:
DOI: 10.1016/j.crme.2018.04.004
Keywords: Fractional calculus, Anomalous diffusion, Non-local models

Enrique Nadal 1; Emmanuelle Abisset-Chavanne 2; Elias Cueto 3; Francisco Chinesta 4

1 Research Centre on Mechanical Engineering (CIIM), Universitat Politècnica de València, Camino de Vera s/n, 46071 Valencia, Spain
2 High Performance Computing Institute & ESI GROUP Chair, École centrale de Nantes, 1, rue de la Noë, BP 92101, 44321 Nantes cedex 3, France
3 I3A, University of Zaragoza, Maria de Luna s/n, 50018 Zaragoza, Spain
4 PIMM Laboratory & ESI GROUP Chair, ENSAM ParisTech, 151, boulevard de l'Hôpital, 75013 Paris, France
     author = {Enrique Nadal and Emmanuelle Abisset-Chavanne and Elias Cueto and Francisco Chinesta},
     title = {On the physical interpretation of fractional diffusion},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {581--589},
     publisher = {Elsevier},
     volume = {346},
     number = {7},
     year = {2018},
     doi = {10.1016/j.crme.2018.04.004},
     language = {en},
AU  - Enrique Nadal
AU  - Emmanuelle Abisset-Chavanne
AU  - Elias Cueto
AU  - Francisco Chinesta
TI  - On the physical interpretation of fractional diffusion
JO  - Comptes Rendus. Mécanique
PY  - 2018
SP  - 581
EP  - 589
VL  - 346
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crme.2018.04.004
LA  - en
ID  - CRMECA_2018__346_7_581_0
ER  - 
%0 Journal Article
%A Enrique Nadal
%A Emmanuelle Abisset-Chavanne
%A Elias Cueto
%A Francisco Chinesta
%T On the physical interpretation of fractional diffusion
%J Comptes Rendus. Mécanique
%D 2018
%P 581-589
%V 346
%N 7
%I Elsevier
%R 10.1016/j.crme.2018.04.004
%G en
%F CRMECA_2018__346_7_581_0
Enrique Nadal; Emmanuelle Abisset-Chavanne; Elias Cueto; Francisco Chinesta. On the physical interpretation of fractional diffusion. Comptes Rendus. Mécanique, Volume 346 (2018) no. 7, pp. 581-589. doi : 10.1016/j.crme.2018.04.004.

[1] A. Jaishankar; G. McKinley Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations, Proc. R. Soc. A, Volume 469 (2012)

[2] J. Klafter; A. Blumen; M.F. Shlesinger Stochastic pathway to anomalous diffusion, Phys. Rev. A, Volume 35 (1987) no. 7, pp. 3081-3085

[3] R. Metzler; J. Klafter The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., Volume 339 (2000), pp. 1-77

[4] F. Chinesta; E. Abisset-Chavanne A Journey Around the Different Scales Involved in the Description of Matter and Complex Systems, SpringerBriefs in Applied Science and Technology, Springer, 2018

[5] J.V. Aguado; E. Abisset; E. Cueto; F. Chinesta; R. Keunings Fractional modelling of functionalized CNT suspensions, Rheol. Acta, Volume 54 (2015) no. 2, pp. 109-119

[6] A. Ma; F. Chinesta; M. Mackley The rheology and modelling of chemically treated carbon nanotube suspensions, J. Rheol., Volume 53 (2009) no. 3, pp. 547-573

[7] E. Nadal; J.V. Aguado; E. Abisset-Chavanne; F. Chinesta; R. Keunings; E. Cueto A physically-based fractional diffusion model for semi-dilute suspensions of rods in a Newtonian fluid, Appl. Math. Model., Volume 51 (2017), pp. 58-67

[8] B. Li; J. Wang Anomalous heat conduction and anomalous diffusion in one-dimensional systems, Phys. Rev. Lett., Volume 91 (2003) no. 4

[9] V.E. Tarsal Elasticity of fractal materials using the continuum model with non-integer dimensional space, C. R. Mecanique, Volume 343 (2015) no. 1, pp. 57-73

[10] A. Carpinteri; B. Chiaia; P. Cornetti A fractional calculus approach to the mechanics of fractal media, Rend. Semin. Mat., Volume 1 (2000), pp. 57-68

[11] I. Podlubny Fractional Differential Equations, Academic Press, San Diego, CA, USA, 1999

[12] A. Kilbas; H.M. Srivastava; J.J. Trujillo Theory and Applications of Fractional Differential Equations, Elsevier, 2006

[13] Q. Yang; F. Liu; I. Turner Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model., Volume 34 (2010), pp. 200-218

[14] F.J. Molz; G.J. Fix; S. Lu A physical interpretation for the fractional derivative in Levy diffusion, Appl. Math. Lett., Volume 15 (2002), pp. 907-911

[15] O. Narayan; S. Ramaswamy Anomalous heat conduction in one-dimensional momentum-conserving systems, Phys. Rev. Lett., Volume 89 (2002)

[16] R. Livi; S. Lepri Heat in one dimension, Nature, Volume 421 (2003) no. 23, p. 327

Cited by Sources:

Comments - Policy