A maximum principle for fractional diffusion differential equations
Authors:
C. Y. Chan and H. T. Liu
Journal:
Quart. Appl. Math. 74 (2016), 421-427
MSC (2010):
Primary 35R11
DOI:
https://doi.org/10.1090/qam/1433
Published electronically:
June 16, 2016
MathSciNet review:
3518222
Full-text PDF Free Access
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Additional Information
Abstract: A weak maximum principle is established for a fractional diffusion equation involving the Riemann-Liouville fractional derivative. As applications, it is used to prove the uniqueness and the continuous dependence of a solution on the initial data.
References
- Ahmed Alsaedi, Bashir Ahmad, and Mokhtar Kirane, Maximum principle for certain generalized time and space fractional diffusion equations, Quart. Appl. Math. 73 (2015), no. 1, 163–175. MR 3322729, DOI 10.1090/S0033-569X-2015-01386-2
- A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, Fractional diffusion in inhomogeneous media, J. Phys. A 38 (2005), no. 42, L679–L684. MR 2186196, DOI 10.1088/0305-4470/38/42/L03
- Rudolf Gorenflo and Francesco Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal. 1 (1998), no. 2, 167–191. MR 1656314
- Colleen M. Kirk and W. Edward Olmstead, Thermal blow-up in a subdiffusive medium due to a nonlinear boundary flux, Fract. Calc. Appl. Anal. 17 (2014), no. 1, 191–205. MR 3146656, DOI 10.2478/s13540-014-0162-8
- Yury Luchko, Maximum principle and its application for the time-fractional diffusion equations, Fract. Calc. Appl. Anal. 14 (2011), no. 1, 110–124. MR 2782248, DOI 10.2478/s13540-011-0008-6
- W. E. Olmstead and Catherine A. Roberts, Thermal blow-up in a subdiffusive medium, SIAM J. Appl. Math. 69 (2008), no. 2, 514–523. MR 2465853, DOI 10.1137/080714075
- W. E. Olmstead and Catherine A. Roberts, Dimensional influence on blow-up in a superdiffusive medium, SIAM J. Appl. Math. 70 (2009/10), no. 5, 1678–1690. MR 2587775, DOI 10.1137/090753280
- Igor Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR 1658022
- Mohammed Al-Refai and Yuri Luchko, Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications, Fract. Calc. Appl. Anal. 17 (2014), no. 2, 483–498. MR 3181067, DOI 10.2478/s13540-014-0181-5
- Mohammed Al-Refai and Yuri Luchko, Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives, Appl. Math. Comput. 257 (2015), 40–51. MR 3320647, DOI 10.1016/j.amc.2014.12.127
References
- Ahmed Alsaedi, Bashir Ahmad, and Mokhtar Kirane, Maximum principle for certain generalized time and space fractional diffusion equations, Quart. Appl. Math. 73 (2015), no. 1, 163–175. MR 3322729, DOI 10.1090/S0033-569X-2015-01386-2
- A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, Fractional diffusion in inhomogeneous media, J. Phys. A 38 (2005), no. 42, L679–L684. MR 2186196 (2006h:82090), DOI 10.1088/0305-4470/38/42/L03
- Rudolf Gorenflo and Francesco Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal. 1 (1998), no. 2, 167–191. MR 1656314 (99m:60117)
- Colleen M. Kirk and W. Edward Olmstead, Thermal blow-up in a subdiffusive medium due to a nonlinear boundary flux, Fract. Calc. Appl. Anal. 17 (2014), no. 1, 191–205. MR 3146656, DOI 10.2478/s13540-014-0162-8
- Yury Luchko, Maximum principle and its application for the time-fractional diffusion equations, Fract. Calc. Appl. Anal. 14 (2011), no. 1, 110–124. MR 2782248 (2012a:35349), DOI 10.2478/s13540-011-0008-6
- W. E. Olmstead and Catherine A. Roberts, Thermal blow-up in a subdiffusive medium, SIAM J. Appl. Math. 69 (2008), no. 2, 514–523. MR 2465853 (2009i:80005), DOI 10.1137/080714075
- W. E. Olmstead and Catherine A. Roberts, Dimensional influence on blow-up in a superdiffusive medium, SIAM J. Appl. Math. 70 (2009/10), no. 5, 1678–1690. MR 2587775 (2011e:35162), DOI 10.1137/090753280
- Igor Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. MR 1658022 (99m:26009)
- Mohammed Al-Refai and Yuri Luchko, Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications, Fract. Calc. Appl. Anal. 17 (2014), no. 2, 483–498. MR 3181067, DOI 10.2478/s13540-014-0181-5
- Mohammed Al-Refai and Yuri Luchko, Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives, Appl. Math. Comput. 257 (2015), 40–51. MR 3320647, DOI 10.1016/j.amc.2014.12.127
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Additional Information
C. Y. Chan
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
MR Author ID:
203257
Email:
chan@louisiana.edu
H. T. Liu
Affiliation:
Department of Applied Mathematics, Tatung University, 40 Chung Shan North Road, Sec. 3, Taipei, Taiwan 104
Email:
tliu@ttu.edu.tw
Keywords:
Maximum principle,
fractional diffusion equations,
fractional derivatives
Received by editor(s):
December 11, 2014
Published electronically:
June 16, 2016
Article copyright:
© Copyright 2016
Brown University
