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

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 ρt 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 ( ξ 1 ) = 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.