Time-Dependent Source Identification Problem for Fractional Schrodinger Type Equations
Abstract
The time-dependent source identication problem for the Schrödinger equation of fractional order $$iD_{t}^{\rho}u(t)+Au(t)=p(t)q+f(t)$$ ( $$0<t\leq T$$ , $$0<\rho<1$$ ), $$u(0)=\varphi$$ , in a Hilbert space $$H$$ is investigated. Here $$A$$ is a self-adjoint positive operator, $$D_{t}$$ is the Caputo derivative. An inverse problem is considered in which, along with $$u(t)$$ , also a time varying factor $$p(t)$$ of the source function is unknown. To solve this inverse problem, we take the additional condition $$B[u(t)]=\psi(t)$$ with an arbitrary bounded linear functional $$B$$ . Existence and uniqueness theorem for the solution to the problem under consideration is proved. Inequalities of stability are obtained. A list of examples of operator $$A$$ and functional $$B$$ is discussed, including linear systems of fractional differential equations, differential models with involution, fractional Sturm–Liouville operators, and others.