Diffusion problems in fractal media defined on Cantor sets

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 90, Issue 3, 2010, Pages 203-210

  Carpinteri, Alberto ^a   Sapora, Alberto ^a  

^a POLITECNICO DI TORINO   (Italy)

Author keywords

Anomalous diffusion; Fractal media; Local fractional calculus

Indexed keywords

EID: 77049111800     PISSN: 00442267     EISSN: 15214001     Source Type: Journal    
DOI: 10.1002/zamm.200900376     Document Type: Article

References (35)

1. P.A. Alemany, Fractional diffusion equation for fractal-time-continuous-time random walks, Chaos Solitons Fractals 6, 7-19 (1995).
2. A. Carpinteri, Fractal nature of material microstructure and size effects on apparent mechanical properties, Mech. Mater. 18, 89-101 (1994).
3. A. Carpinteri, Scaling laws and renormalization groups for strength and toughness of disordered materials, Int. J. Solids Struct. 31,291-302 (1994).
4. A. Carpinteri, Structural Mechanics: A Unified Approach (Chapman & Hall, London, 1997).
5. A. Carpinteri, B. Chiaia, and P. Cornetti, Static-kinematic duality and the principle of virtual work in the mechanics of fractal media, Comput. Methods Appl. Mech. Eng. 191, 3-19 (2001).
6. A. Carpinteri, B. Chiaia, and P. Cornetti, The elastic problem for fractal media: basic theory and finite element formulation, Comput. Struct. 82, 499-508 (2004).
7. A. Carpinteri, B. Chiaia, and P. Cornetti, A disordered microstructure material model based on fractal geometry and fractional calculus, Z. Angew. Math. Mech. 84, 128-35 (2004).
8. A. Carpinteri and P. Cornetti, A fractional calculus approach to the description of stress and strain localization in fractal media, Chaos Solitons Fractals 13, 85-94 (2002).
9. A. Carpinteri, P. Cornetti, and A. Sapora, Static-kinematic fractional operators for fractal and non-local solids, Z. Angew. Math. Mech. 89, 207-217 (2009).
10. A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics (Springer-Verlag, Wien, 1997).
11. W. Chen, Time-space fabric underlying anomalous diffusion, Chaos Solitons Fractals 28, 923-929 (2006).
12. W. Cheng and S.F. Ren, Size effect of thermal conductivity of Si nanocrystals, Solid State Commun. 147, 274-277 (2008).
13. M. Giona and H. Roman, Fractional diffusion equation on fractals-one-dimensional case and asymptotic behavior, J. Phys. A 25, 2093-2105 (1992).
14. M. Giona, W. Schwalm, M.K. Schwalm, and A. Adorever, Exact solution of linear transport equations in fractal media-I. Renormalization analysis and general theory, Chem. Eng. Sci. 51, 4717-4729 (1996).
15. R. Gorenflo, F. Mainardi, M. Moretti, G. Pagnini, and P. Paradisi, Discrete random walk for space-time fractional diffusion, Chem. Phys. 284, 521-541 (2002).
16. R. Gorenflo and F. Mainardi, Some recent advances in theory and simulation of fractional diffusion process, J. Comput. Appl. Math. 229, 400-415 (2009).
17. S. Havlin and D. Ben-Avraham, Diffusion in disordered media, Adv. Phys. 51, 187-292 (2002).
18. K.M. Kolwankar and A. D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos 6, 505-23 (1996).
19. K.M. Kolwankar and A.D. Gangal, Local fractional Fokker-Plank equations, Phys. Rev. Lett. 80, 214-7 (1998).
20. G. Jumarie, Further results on Fokker-Planck equation of fractional order, Chaos Solitons Fractals 12, 1873-1886 (2001).
21. G. Jumarie, Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions Further results, Comput. Math. Appl. 51, 1367-1376 (2006).
22. G. Jumarie, Probability calculus of fractional order and fractional Taylors series application to Fokker-Planck equation and information of non-random functions, Chaos Solitons Fractals 40, 1428-1448 (2009).
23. F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solitons Fractals 7, 1461-1477 (1996).
24. F. Mainardi, Y. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus Appl. Anal. 4, 153-192 (2001).
25. M.M. Meerschaert, H.P. Scheffler, and C. Tadjeran, Finite difference methods for two dimensional fractional dispersion equation, J. Comput. Phys. 211, 249-261 (2006).
26. R. Metzler and T. F. Nonnenmacher, Space- and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation, Chem. Phys. 284, 67-90 (2002).
27. R.R. Nigmatullin, Fractional integral and its physical interpretation, Theor. Math. Phys. 90, 354-368 (1992)
28. K.B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, New York, 1974)
29. B. O'Shaughnessy and I. Procaccia, Analytical solutions for diffusion in fractal media, Phys. Rev. Lett. 54, 455-458 (1985).
30. S. Saha Ray, Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method, Commun. Nonlinear Sci. Numer. Simul. 14, 1295-1306 (2009).
31. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives (Gordon and Breach Science Publishers, Amsterdam, 1993).
32. S. Wang, Z.F. Ma, and H.Q. Yao, Fractal diffusion model used for diffusion in porous material within limited volume of stiff container, Chem. Eng. Sci. 64, 1318-1325 (2008).
33. B. Yu, Analysis of flow in fractal porous media, Appl. Mech. Rev. 61, 1-19 (2008).
34. 4He near Tλ, Physica B 329-333, 200-201 (2003).s
35. Q. Zeng, H. Li, and D. Liu, Anomalous fractional diffusion equation for transport phenomena, Commun. Nonlinear Sci. Numer. Simul. 4, 99-104 (1999).