Geodesic
On solvability of some classes of equations with Hilfer derivative in Banach spaces
Čelâbinskij fiziko-matematičeskij žurnal, Tome 7 (2022) no. 1, pp. 11-19.

Voir la notice de l'article provenant de la source Math-Net.Ru

The issues of unique solvability of a Cauchy-type problem for an evolutionary equation in a Banach space with a Hilfer fractional derivative and with a bounded operator are investigated. The existence of a unique solution to the problem is proved, the solution is presented using Mittag-Leffler operator functions. The abstract result is used when considering a class of initial boundary value problems for partial differential equations
Keywords: Hilfer derivative, fractional order equation, initial problem, theorem on the existence and uniqueness of a solution, linear inhomogeneous equation, initial boundary value problem, Laplace operator. 
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A. R. Volkova; V. E. Fedorov; D. M. Gordievskikh. On solvability of some classes of equations with  Hilfer derivative in Banach spaces. Čelâbinskij fiziko-matematičeskij žurnal, Tome 7 (2022) no. 1, pp. 11-19. http://geodesic.mathdoc.fr/item/CHFMJ_2022_7_1_a1/

[1] Nakhushev A.M., Elements of the fractional calculus and its applications, Kabardino-Balkarian Scientific Center of RAS, Nalchik, 2000 (In Russ.)

[2] Nakhushev A.M., Fractional calculus and its applications, Fizmatlit Publ., Moscow, 2003 (In Russ.)

[3] Uchaykin V.V., The method of fractional derivatives, Artishok Publ., Ul'yanovsk, 2008 (In Russ.)

[4] Tarasov V. E., Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, 2011 | MR

[5] Caputo M., Fabrizio M., “A new definition of fractional derivative without singular kernel”, Progress in Fractional Differentiation and Applications, 1:2 (2015), 1–13

[6] Atangana A., Baleanu D., “New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model”, Thermal Science, 20 (2016), 763–769 | DOI

[7] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives: Theory and Applications, OPA, Amaterdam, 1993 | MR

[8] Podlubny I., Fractional Differential Equations, Academic Press, San Diego; Boston, 1999 | MR | Zbl

[9] Pskhu A.V., Partial differential equations of a fractional order, Nauka Publ., Moscow, 2005 | MR

[10] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, Elsevier Science Publishing, Amsterdam; Boston; Heidelberg, 2006 | MR | Zbl

[11] Hilfer R., Applications of Fractional Calculus in Physics, WSPC, Singapore, 2000 | MR

[12] Dzhrbashyan M.M., Nersesyan A.B., “Fractional derivatives and the Cauchy problem for differential equations of fractional order”, Izvestiya Akademii Nauk Armyanskoy SSR. Matematika, 3 (1968), 3–28 (In Russ.)

[13] Fedorov V. E., Plekhanova M. V., Izhberdeeva E. M., “Initial value problems of linear equations with the Dzhrbashyan — Nersesyan derivative in Banach spaces”, Symmetry, 13 (2021), 1058 | DOI

[14] Volkova A.R., Izhberdeeva E.M., Fedorov V.E., “Initial value problems for equations with a composition of fractional derivatives”, Chelyabinsk Physical and Mathematical Journal, 6:3 (2021), 269–277 (In Russ.) | MR | Zbl

[15] Bogatyreva F.T., “Boundary value problems for first order partial differential equation with the Dzhrbashyan — Nersesyan operators”, Chelyabinsk Physical and Mathematical Journal, 6:4 (2021), 403–416 | MR | Zbl