Space-time fractional derivative operators

Baeumer, Boris; Meerschaert, Mark; Mortensen, Jeff

doi:10.1090/S0002-9939-05-07949-9

Space-time fractional derivative operators
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by Boris Baeumer, Mark M. Meerschaert and Jeff Mortensen

Proc. Amer. Math. Soc. 133 (2005), 2273-2282

DOI: https://doi.org/10.1090/S0002-9939-05-07949-9

Published electronically: March 14, 2005

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Abstract:

Evolution equations for anomalous diffusion employ fractional derivatives in space and time. Linkage between the space-time variables leads to a new type of fractional derivative operator. This paper develops the mathematical foundations of those operators.

References

Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, vol. 96, Birkhäuser Verlag, Basel, 2001. MR 1886588, DOI 10.1007/978-3-0348-5075-9
Boris Baeumer and Mark M. Meerschaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal. 4 (2001), no. 4, 481–500. MR 1874479
Peter Becker-Kern, Mark M. Meerschaert, and Hans-Peter Scheffler, Limit theorems for coupled continuous time random walks, Ann. Probab. 32 (2004), no. 1B, 730–756. MR 2039941, DOI 10.1214/aop/1079021462
D. Benson, S. Wheatcraft and M. Meerschaert, The fractional-order governing equation of Lévy motion. Water Resources Research 36 (2000), 1413–1424.
A. Blumen, G. Zumofen and J. Klafter, Transport aspects in anomalous diffusion: Lévy walks. Phys Rev. A 40 (1989), 3964–3973.
Jean-Philippe Bouchaud and Antoine Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Phys. Rep. 195 (1990), no. 4-5, 127–293. MR 1081295, DOI 10.1016/0370-1573(90)90099-N
A. S. Chaves, A fractional diffusion equation to describe Lévy flights, Phys. Lett. A 239 (1998), no. 1-2, 13–16. MR 1616103, DOI 10.1016/S0375-9601(97)00947-X
William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, Revised edition, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. Translated from the Russian, annotated, and revised by K. L. Chung; With appendices by J. L. Doob and P. L. Hsu. MR 0233400
Rudolf Gorenflo, Francesco Mainardi, Enrico Scalas, and Marco Raberto, Fractional calculus and continuous-time finance. III. The diffusion limit, Mathematical finance (Konstanz, 2000) Trends Math., Birkhäuser, Basel, 2001, pp. 171–180. MR 1882830
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
Zbigniew J. Jurek and J. David Mason, Operator-limit distributions in probability theory, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1993. A Wiley-Interscience Publication. MR 1243181
J. Klafter, A. Blumen, and M. F. Shlesinger, Stochastic pathway to anomalous diffusion, Phys. Rev. A (3) 35 (1987), no. 7, 3081–3085. MR 884309, DOI 10.1103/PhysRevA.35.3081
M.M. Meerschaert, D. Benson and B. Bäumer, Multidimensional advection and fractional dispersion. Phys. Rev. E 59 (1999), 5026–5028.
Mark M. Meerschaert and Hans-Peter Scheffler, Limit distributions for sums of independent random vectors, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 2001. Heavy tails in theory and practice. MR 1840531
M.M. Meerschaert, D. Benson and B. Bäumer, Operator Lévy motion and multiscaling anomalous diffusion. Phys. Rev. E 63 (2001), 1112–1117.
Peter Becker-Kern, Mark M. Meerschaert, and Hans-Peter Scheffler, Limit theorem for continuous-time random walks with two time scales, J. Appl. Probab. 41 (2004), no. 2, 455–466. MR 2052584, DOI 10.1017/s002190020001442x
Mark M. Meerschaert, David A. Benson, Hans-Peter Scheffler, and Boris Baeumer, Stochastic solution of space-time fractional diffusion equations, Phys. Rev. E (3) 65 (2002), no. 4, 041103, 4. MR 1917983, DOI 10.1103/PhysRevE.65.041103
M.M. Meerschaert, D.A. Benson, H.P. Scheffler and P. Becker-Kern, Governing equations and solutions of anomalous random walk limits. Phys. Rev. E 66 (2002), 102-105.
Ralf Metzler and Joseph Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 77. MR 1809268, DOI 10.1016/S0370-1573(00)00070-3
Kenneth S. Miller and Bertram Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993. MR 1219954
Elliott W. Montroll and George H. Weiss, Random walks on lattices. II, J. Mathematical Phys. 6 (1965), 167–181. MR 172344, DOI 10.1063/1.1704269
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
M. Raberto, E. Scalas and F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study. Physica A 314 (2002), 749–755.
L. Sabatelli, S. Keating, J. Dudley, and P. Richmond, Waiting time distributions in financial markets, Eur. Phys. J. B Condens. Matter Phys. 27 (2002), no. 2, 273–275. MR 1930869, DOI 10.1140/epjb/e20020151
Alexander I. Saichev and George M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos 7 (1997), no. 4, 753–764. MR 1604710, DOI 10.1063/1.166272
Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
Enrico Scalas, Rudolf Gorenflo, and Francesco Mainardi, Fractional calculus and continuous-time finance, Phys. A 284 (2000), no. 1-4, 376–384. MR 1773804, DOI 10.1016/S0378-4371(00)00255-7
H. Scher and M. Lax, Stochastic transport in a disordered solid. I. Theory, Phys. Rev. B (3) 7 (1973), no. 10, 4491–4502. MR 391854
Vladimir V. Uchaikin and Vladimir M. Zolotarev, Chance and stability, Modern Probability and Statistics, VSP, Utrecht, 1999. Stable distributions and their applications; With a foreword by V. Yu. Korolev and Zolotarev. MR 1745764, DOI 10.1515/9783110935974
G. M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Phys. D 76 (1994), no. 1-3, 110–122. Chaotic advection, tracer dynamics and turbulent dispersion (Gavi, 1993). MR 1295881, DOI 10.1016/0167-2789(94)90254-2