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Maximum principle for the generalized time-fractional diffusion equation. (English) Zbl 1172.35341
In this interesting paper a maximum type principle for the Caputo time-fractional diffusion equation is presented. Also, some applications of such a principle are given.

35B50 Maximum principles in context of PDEs
26A33 Fractional derivatives and integrals
35S05 Pseudodifferential operators as generalizations of partial differential operators
Full Text: DOI
[1] Bazhlekova, E.G., Duhamel-type representation of the solutions of nonlocal boundary value problems for the fractional diffusion-wave equation, (), 32-40 · Zbl 0926.35027
[2] Chechkin, A.V.; Gorenflo, R.; Sokolov, I.M., Fractional diffusion in inhomogeneous media, J. phys. A, 38, 679-684, (2005) · Zbl 1082.76097
[3] Dubbeldam, J.L.A.; Milchev, A.; Rostiashvili, V.G.; Vilgis, T.A., Polymer translocation through a nanopore: A showcase of anomalous diffusion, Phys. rev. E, 76, 010801(R), (2007)
[4] Eidelman, S.D.; Kochubei, A.N., Cauchy problem for fractional diffusion equations, J. differential equations, 199, 211-255, (2004) · Zbl 1129.35427
[5] Freed, A.; Diethelm, K.; Luchko, Yu., Fractional-order viscoelasticity (FOV): constitutive development using the fractional calculus, (2002), NASA’s Glenn Research Center Ohio
[6] Gorenflo, R.; Luchko, Yu.; Umarov, S., On the Cauchy and multi-point problems for partial pseudo-differential equations of fractional order, Fract. calc. appl. anal., 3, 249-277, (2000) · Zbl 1033.35160
[7] Gorenflo, R.; Mainardi, F., Random walk models for space-fractional diffusion processes, Fract. calc. appl. anal., 1, 167-191, (1998) · Zbl 0946.60039
[8] ()
[9] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equation, (2006), Elsevier Amsterdam · Zbl 1092.45003
[10] Kochubei, A.N., A Cauchy problem for evolution equations of fractional order, Differ. equ., 25, 967-974, (1989) · Zbl 0696.34047
[11] Luchko, Yu.; Gorenflo, R., An operational method for solving fractional differential equations with the Caputo derivatives, Acta math. Vietnam., 24, 207-233, (1999) · Zbl 0931.44003
[12] Luchko, Yu., Operational method in fractional calculus, Fract. calc. appl. anal., 2, 463-489, (1999) · Zbl 1030.26009
[13] Mainardi, F., Fractional relaxation – oscillation and fractional diffusion-wave phenomena, Chaos solitons fractals, 7, 1461-1477, (1996) · Zbl 1080.26505
[14] Mainardi, F.; Tomirotti, M., Seismic pulse propagation with constant Q and stable probability distributions, Ann. geofisica, 40, 1311-1328, (1997)
[15] Mainardi, F.; Luchko, Yu.; Pagnini, G., The fundamental solution of the space – time fractional diffusion equation, Fract. calc. appl. anal., 4, 153-192, (2001) · Zbl 1054.35156
[16] Metzler, R.; Klafter, J., The random Walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. rep., 339, 1-77, (2000) · Zbl 0984.82032
[17] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[18] Pskhu, A.V., Partial differential equations of fractional order, (2005), Nauka Moscow, (in Russian) · Zbl 1095.33010
[19] Voroshilov, A.A.; Kilbas, A.A., The Cauchy problem for the diffusion-wave equation with the Caputo partial derivative, Differ. equ., 42, 638-649, (2006) · Zbl 1123.35302
[20] Zhang, S., Existence of solution for a boundary value problem of fractional order, Acta math. sci. ser. B, 26, 220-228, (2006) · Zbl 1106.34010
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