Inverse boundary value problems for diffusion-wave equation with generalized functions in right-hand sides

Abstract

We prove the unique solvability of the problem on determination of the solution $u(x,t)$ of the first boundary value problem for equation

$$u^{(\beta)}_t-a(t)\Delta u=F_0(x)\cdot g(t), \;\;\; (x,t) \in (0,l)\times (0,T],$$

with fractional derivative $u^{(\beta)}_t$ of the order $\beta\in (0,2)$, generalized functions in initial conditions, and also determination of unknown continuous coefficient $a(t)>0, \; t\in [0,T]$ (or unknown continuous function $g(t)$) under given the values $(a(t)u_x(\cdot,t),\varphi_0(\cdot))$ ($(u(\cdot,t),\varphi_0(\cdot))$, respectively) of according generalized function onto some test function $\varphi_0(x)$.