Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Maximum principle for certain generalized time and space fractional diffusion equations


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 | References | Similar Articles | Additional Information

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.


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  • [1] 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 (98i:86001), https://doi.org/10.1002/(SICI)1097-0363(19971230)25:12$ \langle $1373::AID-FLD620$ \rangle $3.3.CO;2-Y
  • [2] Oleg G. Bakunin, Turbulence and diffusion, scaling versus equations, Springer Series in Synergetics, Springer-Verlag, Berlin, 2008. MR 2450437 (2009j:76001)
  • [3] 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.
  • [4] Krzysztof Bogdan, Krzysztof Burdzy, and Zhen-Qing Chen, Censored stable processes, Probab. Theory Related Fields 127 (2003), no. 1, 89-152. MR 2006232 (2004g:60068), https://doi.org/10.1007/s00440-003-0275-1
  • [5] 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.
  • [6] 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 (2001f:91001)
  • [7] D. Del-Castillo-Negrete, B.A. Carreras and V.E. Lynch, Fractional diffusion in plasma turbulence. Physics of Plasmas 11, (2004), 3854.
  • [8] 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).
  • [9] 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).
  • [10] D. del-Castillo-Negrete, Fractional diffusion models of nonlocal transport, Phys. Plasmas 13 (2006), no. 8, 082308, 16. MR 2249732 (2007c:76081), https://doi.org/10.1063/1.2336114
  • [11] J. I. Diaz, T. Pierantozi and L. Vázquez, On the finite time extinction phenomena for some nonlinear fractional evolution equations. Preprint.
  • [12] 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 (93f:28001)
  • [13] B. R. Gadjiev, Disorder and critical phenomena, arxiv.org/pdf/0809.5036, 2008.
  • [14] P. Grosfils, and J. B. Boon, Nonextensive statistics in viscous fingering, Phys. A 362 (1) (2006) 168.
  • [15] G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Z. 27 (1928), no. 1, 565-606. MR 1544927, https://doi.org/10.1007/BF01171116
  • [16] Applications of fractional calculus in physics, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. Edited by R. Hilfer. MR 1890104 (2002j:00009)
  • [17] Rudolf Gorenflo and Sergio Vessella, Abel integral equations, analysis and applications, Lecture Notes in Mathematics, vol. 1461, Springer-Verlag, Berlin, 1991. MR 1095269 (92e:45003)
  • [18] 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 (2007a:34002)
  • [19] A. N. Kochubeĭ, The Cauchy problem for evolution equations of fractional order, Differentsialnye Uravneniya 25 (1989), no. 8, 1359-1368, 1468 (Russian); English transl., Differential Equations 25 (1989), no. 8, 967-974 (1990). MR 1014153 (90g:34063)
  • [20] N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027 (50 #2520)
  • [21] Yury Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl. 351 (2009), no. 1, 218-223. MR 2472935 (2009m:35274), https://doi.org/10.1016/j.jmaa.2008.10.018
  • [22] 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 (2010k:35515), https://doi.org/10.1016/j.camwa.2009.08.015
  • [23] 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, https://doi.org/10.2478/s13540-012-0010-7
  • [24] F. Mainardi, P. Paradisi, and R. Gorenflo, Probabililty distributions generated by fractional diffusion equation, Preprint, 2004.
  • [25] 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 (2001k:82082), https://doi.org/10.1016/S0370-1573(00)00070-3
  • [26] Tilak Raj Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J. 19 (1971), 7-15. MR 0293349 (45 #2426)
  • [27] Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993. Edited and with a foreword by S. M. Nikolskiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689 (96d:26012)
  • [28] D. W. Stroock, An introduction to the theory of large deviations, Universitext, Springer-Verlag, New York, 1984. MR 755154 (86h:60067a)
  • [29] 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 (2008k:82136), https://doi.org/10.1016/j.cnsns.2006.07.008
  • [30] N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), no. 2, 240-260. MR 803094 (87a:31011), https://doi.org/10.1016/0022-1236(85)90087-4
  • [31] Bruce J. West and Theo Nonnenmacher, An ant in a gurge, Phys. Lett. A 278 (2001), no. 5, 255-259. MR 1826567 (2002a:82066), https://doi.org/10.1016/S0375-9601(00)00781-7
  • [32] G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. Rep. 371 (2002), no. 6, 461-580. MR 1937584 (2003i:70030), https://doi.org/10.1016/S0370-1573(02)00331-9

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Additional Information

Ahmed Alsaedi
Affiliation: Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Email: aalsaedi@kau.edu.sa

Bashir Ahmad
Affiliation: Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Email: bashirahmad_qau@yahoo.com

Mokhtar Kirane
Affiliation: Laboratoire de Mathématiques, Image et Applications, EA 3165, Université de La Rochelle, Pôle Sciences et Technologies, Avenue Michel Crépeau, 17000 La Rochelle, France, and Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Email: mkirane@univ-lr.fr

DOI: https://doi.org/10.1090/S0033-569X-2015-01386-2
Keywords: Maximum principle, time fractional differential equation, time and space fractional differential equation, fractional porous medium equation
Received by editor(s): April 11, 2013
Published electronically: January 29, 2015
Additional Notes: The third author is the corresponding author
Article copyright: © Copyright 2015 Brown University

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