Multipoint (in Time) Problem for One Class of Evolutionary Pseudodifferential Equations

Inverse problems for a time fractional diffusion equation in the Schwartz‐type distributions

“…It satisfies the condition (5) if R 0 is determined by (18). Indeed, let F * , H, H * be define as in the proof of Lemma 2, u * is the solution of Equation (19) in which u * 1 is defined by (20) with F * instead of F. Then from (18), we obtain…”

“…Theorem 3. It exists T 1 > 0 such that for each T ∈ (0, T 1 ), under the assumption (B) and if 0 < g min ≤ |g(t)| ≤ g max , t ∈ [0, T], g(t) and T (t) keep the same sign on [0, T], g max ∕g min is a monotonically nondecreasing function of T, the inverse problems (3) to (5) has the unique solution (R 0 , u) ∈S ′ ,(a) (R n ) ×S ′ ,(a),C (Q), R 0 is defined by (18), u is the solution of Equation (19) with u 1 ∈S ′ ,(a),C (Q), defined by (20).…”

Inverse problem with a time‐integral condition for a fractional diffusion equation

“…We get F * = F in  ′ ,(a) (R n ) if F 0 is determined by (18) and that the pair (u, F 0 ), defined by (17) and (18) is the solution of the problem (2) to (4).…”

“…Note that such type of overdetermination condition for an abstract parabolic equation was studied for the first time in Prilepko and Kostin, 21 the problems with the multipoint and an integral conditions for differential equations with partial derivatives of entire orders were studied, for example, in other works. 4,[22][23][24][25][26][27] For the fractional diffusion equation, by a time-integral overdetermination condition, the uniqueness of restoration the equation's minor coefficient (in Janno and Kasemets 28 ) and the unique solvability of the inverse problem of restoration the solution's initial data from the space of periodic distributions (in Lopushanska et al 29 ) were obtained.…”