Maximum principle for certain generalized time and space fractional diffusion equations

Alsaedi, Ahmed; Ahmad, Bashir; Kirane, Mokhtar

doi:10.1090/S0033-569X-2015-01386-2

Authors: Ahmed Alsaedi, Bashir Ahmad and Mokhtar Kirane
Journal: Quart. Appl. Math. 73 (2015), 163-175
MSC (2010): Primary 35R11, 35B50
DOI: https://doi.org/10.1090/S0033-569X-2015-01386-2
Published electronically: January 29, 2015
MathSciNet review: 3322729
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Abstract: We present an inequality for fractional derivatives related to the Leibniz rule that not only answers positively a conjecture raised by J. I. Diaz, T. Pierantozi, and L. Vázquez but also helps us to obtain a modern proof of the maximum principle for fractional differential equations. The inequality turns out to be versatile in nature as it can be used to obtain a priori estimates for many fractional differential problems.

References

Paul S. Addison, Bo Qu, Alistair Nisbet, and Gareth Pender, A non-Fickian, particle-tracking diffusion model based on fractional Brownian motion, Internat. J. Numer. Methods Fluids 25 (1997), no. 12, 1373–1384. MR 1601533, DOI https://doi.org/10.1002/%28SICI%291097-0363%2819971230%2925%3A12%3C1373%3A%3AAID-FLD620%3E3.3.CO%3B2-Y
Oleg G. Bakunin, Turbulence and diffusion, Springer Series in Synergetics, Springer-Verlag, Berlin, 2008. Scaling versus equations. MR 2450437

D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resources Research Vol. 36, No. 6 (2000), 1403–1412.

Krzysztof Bogdan, Krzysztof Burdzy, and Zhen-Qing Chen, Censored stable processes, Probab. Theory Related Fields 127 (2003), no. 1, 89–152. MR 2006232, DOI https://doi.org/10.1007/s00440-003-0275-1

M. Bologna, C. Tsallis, P. Grigolini, Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions, Phys. Review E 62 (2) (2000), 2213.

Jean-Philippe Bouchaud and Marc Potters, Theory of financial risks, Cambridge University Press, Cambridge, 2000. From statistical physics to risk management; With a foreword by Nick Dunbar. MR 1787145

D. Del-Castillo-Negrete, B.A. Carreras and V.E. Lynch, Fractional diffusion in plasma turbulence. Physics of Plasmas 11, (2004), 3854.

D. Del-Castillo-Negrete, B.A. Carreras and V.E. Lynch, Nondiffusive transport in plasma turbulence: a fractional diffusion approach. Physical Review Letters 94, 065003 (2005).

D. Del-Castillo-Negrete, Fractional diffusion models of transport in magnetically confined plasmas. Proceedings of the 32nd European Physical Society, Plasma Physics Conference. Tarragona Spain (2005).

D. del-Castillo-Negrete, Fractional diffusion models of nonlocal transport, Phys. Plasmas 13 (2006), no. 8, 082308, 16. MR 2249732, DOI https://doi.org/10.1063/1.2336114

J. I. Diaz, T. Pierantozi and L. Vázquez, On the finite time extinction phenomena for some nonlinear fractional evolution equations. Preprint.

Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660

B. R. Gadjiev, Disorder and critical phenomena, arxiv.org/pdf/0809.5036, 2008.

P. Grosfils, and J. B. Boon, Nonextensive statistics in viscous fingering, Phys. A 362 (1) (2006) 168.

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Z. 27 (1928), no. 1, 565–606. MR 1544927, DOI https://doi.org/10.1007/BF01171116

R. Hilfer (ed.), Applications of fractional calculus in physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1890104

Rudolf Gorenflo and Sergio Vessella, Abel integral equations, Lecture Notes in Mathematics, vol. 1461, Springer-Verlag, Berlin, 1991. Analysis and applications. MR 1095269

Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006. MR 2218073

A. N. Kochubeĭ, The Cauchy problem for evolution equations of fractional order, Differentsial′nye Uravneniya 25 (1989), no. 8, 1359–1368, 1468 (Russian); English transl., Differential Equations 25 (1989), no. 8, 967–974 (1990). MR 1014153

N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027

Yury Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl. 351 (2009), no. 1, 218–223. MR 2472935, DOI https://doi.org/10.1016/j.jmaa.2008.10.018

Yury Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl. 59 (2010), no. 5, 1766–1772. MR 2595950, DOI https://doi.org/10.1016/j.camwa.2009.08.015

Yuri Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fract. Calc. Appl. Anal. 15 (2012), no. 1, 141–160. MR 2872116, DOI https://doi.org/10.2478/s13540-012-0010-7

F. Mainardi, P. Paradisi, and R. Gorenflo, Probabililty distributions generated by fractional diffusion equation, Preprint, 2004.

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 https://doi.org/10.1016/S0370-1573%2800%2900070-3

Tilak Raj Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7–15. MR 293349

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

D. W. Stroock, An introduction to the theory of large deviations, Universitext, Springer-Verlag, New York, 1984. MR 755154

Vladimir V. Uchaikin and Renat T. Sibatov, Fractional theory for transport in disordered semiconductors, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), no. 4, 715–727. MR 2381497, DOI https://doi.org/10.1016/j.cnsns.2006.07.008

N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), no. 2, 240–260. MR 803094, DOI https://doi.org/10.1016/0022-1236%2885%2990087-4

Bruce J. West and Theo Nonnenmacher, An ant in a gurge, Phys. Lett. A 278 (2001), no. 5, 255–259. MR 1826567, DOI https://doi.org/10.1016/S0375-9601%2800%2900781-7

G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (2002), no. 6, 461–580. MR 1937584, DOI https://doi.org/10.1016/S0370-1573%2802%2900331-9