Multidimensional fractional advection-dispersion equations and related stochastic processes

Electronic Journal of Probability

Volumn 19, Issue , 2014, Pages

  D'Ovidio, Mirko ^a   Garra, Roberto ^a  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

Author keywords

Directional derivatives; Fractional advection equation; Fractional vector calculus

Indexed keywords

EID: 84904284199     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v19-2854     Document Type: Article

References (32)

1. Applebaum, D.: Lévy processes and stochastic calculus, Second edition. Cambridge Studies in Advanced Mathematics, 116. Cambridge University Press, Cambridge, 2009. 30+460 pp. MR-2512800
2. Baleanu, D., Golmankhaneh, A.K., Golmankhaneh, A.K. and Baleanu, M.C.: Fractional Electromagnetic Equations Using Fractional Forms, International Journal of Theoretical Physics, 48(11), (2009), 3114-3123. MR-2546691
3. Ben Adda, F.: The differentiability in fractional calculus, Nonlinear Analysis, 47, (2001), 5423-5428. MR-1974748
4. Bertoin, J.: Lévy processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996. 10+265 pp. MR-1406564
5. Bolster, D., Benson, D.A., Meerschaert, M.M. and Baeumer, B.: Mixing-driven equilibrium reactions in multidimensional fractional advection-dispersion systems, Physica A, 392(10), (2013), 2513-2525. MR-3038326
6. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, Geophysical Journal of the Royal Astronomical Society, 13, (1967), 529-539. MR-2379269
7. Dalla Riva, M. and Yakubovich, S.: On a Riemann-Liouville fractional analog of the Laplace operator with positive energy, Integral Transform and Special Functions, 23(4), (2012), 277-295. MR-2903489
8. D'Ovidio, M.: Wright functions governed by fractional directional derivatives and fractional advection diffusion equations, Methods and Applications of Analysis, (2014). To appear.
9. D'Ovidio, M.: Continuous random walks and fractional powers of operators, Journal of Mathematical Analysis and Applications, 411, (2014), 362-371. MR-3118491
10. Ervin, V.J. and Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in Rd, Numerical Methods for Partial Differential Equations, 23(2), (2007), 256-281. MR-2289452
11. GadElkarim, J.J., Magin, R.M., Meerschaert, M.M., Capuani, S., Palombo, M., Kumar, A., Leow, A.D.: Fractional order generalization of anomalous diffusion as a multidimensional extension of the transmission line equation, IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 3(3), (2013), 432-441.
12. Hille, E. and Phillips, R.S.: Functional analysis and semi-groups, Third printing of the revised edition of 1957. American Mathematical Society Colloquium Publications, Vol. 31. American Mathematical Society, Providence, R. I., 1974. 12+808 pp. MR-0423094
13. Jacob, N.: Pseudo Differential Operators And Markov Processes, Vol. 3: Markov Processes and Applications, Imperial College Press, London, 2005. 28+474 pp. MR-2158336
14. Kampé de Fériet, J.: Random Solutions of Partial Differential Equations, Proc. Third Berkeley Symp. on Math. Statist. and Prob., 3, (1956), 199-208, Univ. of Calif. Press. MR-0084927
15. Komatsu, H.: Fractional powers of operators, Pacific Journal of Mathematics, 19, (1966), 285-346. MR-0201985
16. Mainardi, F., Luchko, Y. and Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis, 4(2), (2001), 153-192. MR-1829592
17. Metzler, R. and Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, Journal of Physics A, 37, (2004), 161-208. MR-2090004
18. Meerschaert, M.M., Benson, D.A. and Baeumer, B.: Multidimensional advection and fractional dispersion, Physical Review E, 59, (1999), 5026-5028.
19. Meerschaert, M.M., Mortensen, J. and Scheffler, H.P.: Vector Grunwald formula for fractional derivatives, Fractional Calculus and Applied Analysis, 7, (2004), 61-81. MR-2077400
20. Meerschaert, M.M., Mortensen, J. and Wheatcraft, S.W.: Fractional vector calculus for fractional advection-dispersion, Physica A, 367, (2006), 181-190.
21. Miskinis, P.: On Integral Representation of the Translation Operator, Mathematical Modelling and Analysis, 17(1), (2012), 100-112. MR-2904379
22. Orsingher, E. and Beghin, L.: Fractional diffusion equations and processes with randomly varying time, Annals of Probability, 37(1), (2009), 206-249. MR-2489164
23. Ostoja-Starzewski, M.: Electromagnetism on anisotropic fractal media, ZAMP, 64(2), (2013), 381-390. MR-3041577
24. Paradisi, P., Cesari, R., Mainardi, F. and Tampieri, F.: The fractional Fick's law for non-local transport processes, Physica A, 293 (1/2), (2001), 130-142
25. Samko, S.G., Kilbas, A.A. and Marichev, O.I.: Fractional integrals and derivatives. Theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993. 36+976 pp. MR-1347689
26. Samorodnitsky, G. and Taqqu, M.S.: Stable non-Gaussian random processes, Stochastic models with infinite variance, Stochastic Modeling, Chapman and Hall, New York, 1994. 22+632 pp. MR-1280932
27. Schumer, R., Meerschaert, M.M. and Baeumer, B.: Fractional advection-dispersion equations for modeling transport at the Earth surface, Journal of Geophysical Research: Earth Surface, 114(F4), (2009)
28. Tarasov, V.E.: Fractional dynamics. Applications of fractional calculus to dynamics of particles, fields and media. Nonlinear Physical Science. Springer, Heidelberg; Higher Education Press, Beijing, 2010. 16+504 pp. MR-2796453
29. Tarasov, V.E.: Fractional vector calculus and fractional Maxwell's equations, Annals of Physics, 323(11), (2008), 2756-2778. MR-2463217
30. Tarasov, V.E.: Fractional generalization of gradient systems, Letters in Mathematical Physics, 73(1), (2005), 49-58. MR-2168006
31. Traple, J.: Markov semigroups generated by Poisson driven differential equation, Bulletin of the Polish academy of Sciences Mathematics, 44, (1996), 161-182. MR-1416422
32. Wheatcraft, S.W. and Meerschaert, M.M.: Fractional conservation of mass, Advances in Water Resources, 31, (2008), 1377-1381.