Geodesic
Inhomogeneous Fractional Evolutionary Equation in the Sectorial Case
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 149 (2018), pp. 103-112.

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In this paper, we prove the unique solvability of the Cauchy problem for a linear inhomogeneous equation in a Banach space that is solved with respect to the Gerasimov–Caputo fractional derivative. We assume that the operator acting on the unknown function in the equation generates a set of resolving operators of the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis, which is in the sector. The general form of solutions to the Cauchy problem is obtained. The general results are applied to the study of the unique solvability of a certain class of initial-boundary-value problems for partial differential equations solvable with respect to the Gerasimov–Caputo fractional derivative with respect to time, containing in the simplest case initial-boundary-value problems for fractional diffusion and diffusion-wave equations.
Keywords: Gerasimov–Caputo fractional derivative, evolutionary equation, analytic in a sector resolving family of operators, initial-boundary-value problem
Mots-clés : diffusion-wave equation. 
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V. E. Fedorov; E. A. Romanova. Inhomogeneous Fractional Evolutionary Equation in the Sectorial Case. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, Tome 149 (2018), pp. 103-112. http://geodesic.mathdoc.fr/item/INTO_2018_149_a11/

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