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

2022/05/06 by Ashurov, Ravshan, Fayziev, Yusuf
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2205.03405

Abstract

Let A be an arbitrary positive selfadjoint operator, defined in a separable Hilbert space H. The inverse problems of determining the right-hand side of the equation and the function ϕ in the non-local boundary value problem D_t^ρ u(t) + Au(t) = f(t) (0 < ρ< 1, 0 < t ≤ T), u(ξ) = αu(0) + ϕ, (α is a constant and 0 < ξ≤ T), is considered. Operator D_t on the left-hand side of the equation expresses the Caputo derivative. For both inverse problems u(ξ₁) = V is taken as the over-determination condition. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence and uniqueness of a solution to problems is investigated. An interesting effect was discovered: when solving the forward problem, the uniqueness of the solution u(t) was violated, while when solving the inverse problem for the same values of α, the solution u(t) became unique.

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