Gevrey Problem for a Loaded Mixed Parabolic Equation with a Fractional Derivative
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GEVREY PROBLEM FOR A LOADED MIXED PARABOLIC EQUATION WITH A FRACTIONAL DERIVATIVE S. Kh. Gekkieva
UDC 517.95
Abstract. In this paper, we consider the Gevrey problem for a loaded parabolic equation with the direct and reverse time in an unbounded domain. The problem on the solvability of this problem is reduced to the problem on the solvability of a generalized Abel equation in the class of function satisfying the H¨ older condition. Keywords and phrases: Gevrey problem, loaded equation, Riemann–Liouville fractional differentiation operator, Wright-type function, Abel equation, H¨ older condition. AMS Subject Classification: 35M12
1. Introduction. Parabolic equations with alternating direction of time have many applications; for example, they describe the processes of heat propagation in inhomogeneous media, the interaction of filtration flows, mass transfer near the surface of an aircraft, the description of the complex flows of viscous fluids, etc. We also indicate another possible application, namely, the problem of calculation of heat exchangers based on the counterflow principle. The pioneer papers devoted to parabolic equations with alternating direction of evolution were the works of M. Gevrey (see, e.g., [4]). Boundary-value problems for equations of mixed parabolic type were considered, for example, in [5–8, 11]. The monograph of S. A. Tersenov [15] is also devoted to the theory of linear parabolic equations with alternating direction of time; the main method of the study of such problems is the theory of singular equations, which allows us to consider problems under general linear sewing conditions. The problem on the uniqueness of a solutio to the Gevrey problem for the fractional mixed diffusion equation was considered by E. A. Zarubin in [16]. We also note that mathematical models of nonlocal physical and biological fractal processes are based, as a rule, on loaded fractional partial differential equations (see [9]). The monograph of A. M. Nakhushev [9] contains a detailed bibliography on loaded equations, including various applications of loaded equations in mathematical biology, mathematical modeling, mathematical modeling of nonlocal processes and phenomena, and mechanics of continuous media with memory. Mixed boundary-value problems for the loaded fractional diffusion-wave equation containing the trace of the solution were studied in [3] The existence of a solution to the Gevrey problem for a fractional parabolic equation with direct and reverse time in a rectangular domain was studied in [2]. In the present paper, we consider the Gevrey problem for the loaded fractional diffusion equation with direct and reverse time in an unbounded domain. 2.
Statement of the problem and the results obtained. Consider the equation α u(x, y), x > 0, D0y uxx (x, y) + λu(0, t) = α u(x, y), x < 0. Dhy
(1)
ν is the Riemann–Liouville fractional differentiation operator of where λ = const, 0 < α < 1, and Dst order ν (see [10]).
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Temati
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