Journal of Mathematical Physics, Analysis, Geometry 2021, vol. 17, No 1, pp. 30-53   https://doi.org/10.15407/mag17.01.030     ( to contents , go back )

### Fractional Derivatives with Respect to Time for Non-Classical Heat Problem

Fatima Berrabah

Oran-1 University Algeria, Lyon University, UJM, Institut Camille Jordan CNRS UMR 5208, 23 Dr Paul Michelon 42023 Saint-Etienne, France
E-mail: berrabah f@yahoo.fr

Mahdi Boukrouche

Lyon University, UJM, Institut Camille Jordan CNRS UMR 5208, 23 Dr Paul Michelon 42023 Saint-Etienne, France
E-mail: Mahdi.Boukrouche@univ-st-etienne.fr

Benaouda Hedia

Laboratory of informatics and mathematics, PO BOX 78 University of Tiaret, Algeria
E-mail: b hedia@univ-tiaret.dz

Received August 20, 2019, revised January 25, 2020.

Abstract

We consider the non-classical heat equation with Caputo fractional derivative with respect to the time variable in a bounded domain $\Omega\subset \mathbb{R}^{+}\times\mathbb{R}^{d-1}$ for which the energy supply depends on the heat flux on a part of the boundary $S=\{0\}\times\mathbb{R}^{d-1}$ with homogeneous Dirichlet boundary condition on $S$, the periodicity on the other parts of the boundary and an initial condition. The problem is motivated by the modeling of the temperature regulation in the medium. The existence of the solution to the problem is based on a Volterra integral of second kind in the time variable $t$ with a parameter in $\mathbb{R}^{d-1}$, its solution is the heat flux $(y,\tau)\mapsto V(y, t)=u_{x}(0, y, t)$ on $S$, which is also an additional unknown of the considered problem. We establish that a unique local solution exists and can be extended globally in time.

Mathematics Subject Classification 2010: 26A33, 34A08, 34A12, 34B15, 35K05, 45D05
Key words: non-classical d-dimensional heat equation, Caputo fractional derivative, Volterra integral equation, existence and uniqueness solution, integral representation of solution