A source identification problem in a time-fractional wave equation with a dynamical boundary condition

We focus on a inverse source problem in a partial differential equation containing a fractional derivative in time of the order 1<β<2. The equation is accompanied with a non-standard boundary condition which consists of the classic flux term and the dynamical time-fractional derivative term on the one part of the boundary. To determine both the solution of the equation and the source term, a measurement in a form of a integral over space domain is considered. Using a time-discretization and Rothe's method, we prove an existence of the strong solution in the suitable functional spaces, and the error estimate is established; moreover, the uniqueness is addressed. The results are supported by some numerical experiments.