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

Abstract

UDC 517.9 We consider a Schrödinger equation i ∂ t ρ u ( x , t ) - u x x ( x , t ) = p ( t ) q ( x ) + f ( x , t ) , 0 < t ≤ T , 0 < ρ < 1 , with the Riemann–Liouville derivative. An inverse problem is investigated in which, parallel with u ( x , t ) , a time-dependent factor p ( t ) of the source function is also unknown. To solve this inverse problem, we use an additional condition B [ u ( ⋅ , t ) ] = ψ ( t ) with an arbitrary bounded linear functional B . The existence and uniqueness theorem for the solution to the problem under consideration is proved. The stability inequalities are obtained. The applied method make it possible to study a similar problem by taking, instead of d 2 / d x 2 , an arbitrary elliptic differential operator A ( x , D ) with compact inverse.