On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

Abstract

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

On the uniqueness of solutions of two inverse problems for the subdiffusion equation

“…Recall (see, e.g. [12]), E ρ (−t) decreases strictly monotonically as t > 0 and, moreover, has the following estimate…”

“…Let us return to the non-local problem (1.1). The authors of this paper in their previous work [12] studied in detail the influence of parameter α = 0 on the correctness of problem (1.1). It turned out that the critical values of parameter α are in the interval (0, 1).…”

“…It turned out that the critical values of parameter α are in the interval (0, 1). In order to formulate the main result of work [12], we recall the definition of the Mittag-Leffler function E ρ,µ (z) with two parameters (see, e.g. [13], Chapter 1):…”

“…In work [12] it is proved that if α ∈ (0, 1) and E ρ (−λ k t ρ ) = α for all k, then the solution of problem (1.1) exists and is unique. But it may turn out that for some eigenvalue λ k0 of operator A, with multiplicity p 0 (obviously, p 0 is a finite number), equality (1.4) E ρ (−λ k0 t ρ ) = α will hold.…”

Time-dependent source identification problem for a fractional Schrodinger equation with the Riemann-Liouville derivative

“…It should be noted that in all of the listed works, the Cauchy conditions in time are considered (an exception is work [31], where the integral condition is set with respect to the variable t). In the recent paper [32], for the best of our knowledge, an inverse problem for subdiffusion equation with a nonlocal condition in time is considered for the first time.…”