Solvability in Hölder Space of an Initial Boundary Value Problem for the Time-Fractional Diffusion Equation

Автор(и)

  • M. V. Krasnoschok Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 1 Dobrovol'skogo Str., Slov'yansk 84100, Ukraine

DOI:

https://doi.org/10.15407/mag12.01.048

Анотація

Розглянуто початково-крайову задачу для рівняння дифузії  з дробовою  похідною за часом зі змішаними крайовими умовами. Доведено теорему існування та єдиності розв’язку цієї задачі в просторах Гьольдера.

Mathematics Subject Classification: 35B65, 26A33.

Ключові слова:

Дифузія дробового порядку, простір Гьольдера, похідна Капуто

Посилання

M. Caputo, Models of Flux in Porous Media with Memory. — Water Resources Research 36 (2000), No. 3, 693–705.

Ph. Clément, S.-O. Londen, and G. Simonett, Quasilinear Evolutionary Equations and Continuous Interpolation Spaces. — J. Diff. Eqs. 196 (2004), No. 2, 418–447.

S.D. Eidelman and A.N. Kochubei, Cauchy Problem for Fractional Diffusion Equations. — J. Diff. Eqs. 199 (2004), 211–255. https://doi.org/10.1016/j.jde.2003.12.002

H. Engler, Strong Solutions of Quasilinear Integro-Differential Equations with Singular Kernles in Several Space Dimensions. — Electronic J. Diff. Eqs. (1995), No. 2, 1–16.

I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Sums, Series and Products. Fizmatgiz, Moscow, 1963. (Russian)

J. Kemppainen, Existence and Uniqueness of the Solution for a Time-Fractional Diffusion Equation with Robin Boundary Condition. Abstract and Applied Analysis 2011, Article ID 321903, 11 pages.

A.A. Kilbas, Fractional Calculus of the Generalized Wright Functions. — Fractional Calculus and Applied Analysis 8 (2005), No. 2, 113–126.

A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North Holland, Mathematical studies, 204, Elsevier Science B.V., Amsterdam, 2006.

A.N. Kochubei, Fractional-Order Diffusion. — Diff. Eqs. 26 (1990), 485–492.

A.N. Kochubei, Fractional Parabolic Systems. — Potential Analysis 37 (2012), 1–30. https://doi.org/10.1007/s11118-011-9243-z

M. Krasnoschok and N. Vasylyieva, On a Solvability of Nonlinear Fractional Reaction-Diffusion System in the Hölder Spaces. — Nonlinear Studies 20 (2013), No. 4, 591–621.

O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural’tseva, Linear and Quasilinear Parabolic Equations. Transl. Math. Monogr. 23 AMS, Providence, RI, 1968.

H.P. Lopushans’ka and A.O. Lopushans’kyi, Space-Time Fractional Cauchy Problem in Spaces of Generalized Functions. — Ukr. Math. J. 64 (2013), No. 8, 1215– 1230.

R. Metzler and J. Klafter, The Random Walk’s Guide to Anomalous Diffusion: a Fractional Dynamics Approach. — Phys. Rep. 339 (2000), 1–77. https://doi.org/10.1016/S0370-1573(00)00070-3

G.M. Mophou and G.M. N’Guérékata, On a Class of Fractional Differential Equations in a Sobolev Space. — Applicable Analysis 91 (2012), No. 1, 15–34.

A.M. Nahushev, Fractional Calculus and Its Applications. Nauka, Moscow, 2003. (Russian)

I. Podlubny, Fracrional Differential Equations. Academic Press, San-Diego, 1999.

R. Ponce, Hölder Continuous Solutions for Fractional Differential Equations and Maximal Regularity. — J. Diff. Eqs. 255 (2013), 3284–3304. https://doi.org/10.1016/j.jde.2013.07.035

A.V. Pskhu, A Fundamental Solution for a Fractional Diffusion Wave Equation. — Izvestia RAN 73 (2009), 141–181. (Russian) https://doi.org/10.1070/im2009v073n02abeh002450

A.V. Pskhu, Partial Differential Equations of the Fractional Order. Nauka, Moscow, 2005. (Russian)

K. Sakamoto and M. Yamamoto, Initial Value Boundary Value Problems for Fractional Diffusion-Wave Equations and Applications to Some Inverse Problems. — J. Math. Anal. Appl. 382 (2011), 426–447. https://doi.org/10.1016/j.jmaa.2011.04.058

W.R. Schneider and W. Wyss, Fractional Diffusion and Wave Equations. — J. Math. Phys. 30 (1989), 134–144. https://doi.org/10.1063/1.528578

V. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles. Fields and Media. Higher Education Press, Beijing and Springer-Verlag, Berlin, Heidelberg, 2010.

R. Zacher, Quasilinear Parabolic Problems with Nonlinear Boundary Conditions.Ph. D. Thesis, Martin-Luther-Universit¨at, Halle-Wittenberg (2003).

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Krasnoschok, M. V. Solvability in Hölder Space of an Initial Boundary Value Problem for the Time-Fractional Diffusion Equation. Журн. мат. фіз. анал. геом. 2016, 12, 48-77.

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