Product rule for vector fractional derivatives

Fractional Calculus and Applied Analysis

Volumn 15, Issue 3, 2012, Pages 463-478

  Bolster, Diogo ^a   Meerschaert, Mark M ^b   Sikorskii, Alla ^b  

^a University of Notre Dame   (United States)

^b MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Fourier transform; Fractional advection dispersion equation; Fractional derivative; Vector fractional derivative

Indexed keywords

EID: 84869179719     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-012-0033-0     Document Type: Article

References (19)

1. B. Baeumer and M. M. Meerschaert, Tempered stable L'evy motion and transient super-diffusion. J. Comput. Appl. Math. 233 (2010), 2438-2448.
2. D. A. Benson, R. Schumer, M. M. Meerschaert and S. W. Wheatcraft, Fractional dispersion, Ĺevy motion, and the MADE tracer tests. Transp. Porous Media 42 (2001), 211-240.
3. D. Bolster, D. A. Benson, T. LeBorgne and M. Dentz, Anomalous mixing and reaction induced by superdiffusive nonlocal transport. Phys. Rev. E 82, 021119.
4. A. K. Grünwald, Über begrenzte Derivationen und deren Anwedung. Z. Math. Phys. 12 (1867), 441-480.
5. M. G. Hahn, K. Kobayashi, and S. Umarov, Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion. Proc. Amer. Math. Soc. 139, No 2 (2011), 691-705.
6. E. Hille, Analytic Function Theory. Vol. 1. Ginn, Boston (1959).
7. V. N. Kolokoltsov, Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics. Theory Probab. Appl. 53 (2009), 594-609.
8. A. V. Letnikov, Theory of differentiation of fractional order. Mat. Sb. 3 (1868), 1-68.
9. J. Liouville, Memoire sur le calcul des differentielles a indices quelconques. J. de L'Ecole Polytechnique 13 (1832), 71-162.
10. M. M. Meerschaert, D. A. Benson, and B. Baeumer, Multidimensional advection and fractional dispersion. Phys. Rev. E 59 (1999), 5026-5028.
11. M. M. Meerschaert and H.-P. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice. John Wiley and Sons, New York (2001).
12. M. M. Meerschaert, J. Mortensen and S. W. Wheatcraft, Fractional vector calculus for fractional advection-dispersion. Physica A 367 (2006), 181-190.
13. M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus. De Gruyter Studies in Mathematics 43, De Gruyter, Berlin (2012); http://www.degruyter.com/view/product/129781
14. T. J. Osler, Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM. J. Appl. Math. 18 (1970), 658-674.
15. I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).
16. J. Rosínski, Tempering stable processes. Stoch. Proc. Appl. 117 (2007), 677-707.
17. W. Rudin, Principles of Mathematical Analysis. 3rd edition, McGraw-Hill (1976).
18. A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications. Chaos 7 (1997), 753-764.
19. K. Yosida, Functional Analysis. 6th edition, Springer (1980).