One Inverse Problem for the Diffusion-Wave Equation in Bounded Domain

Abstract

We prove the theorems on the existence and unique determination of a pair of functions: a(t) >0, t ∈ [0,T], and the solution u(x, t) of the first boundary-value problem for the equation $$ \begin{array}{ll}{D}_t^{\beta }u-a(t){u}_{xx}={F}_0\left(x,t\right),\hfill & \left(x,t\right)\in \left(0,l\right)\times \left(0,T\right],\hfill \end{array} $$

with regularized derivative D _t ^β u of the fractional order β ∈ (0, 2) under the additional condition a(t)u _x (0, t) = F(t), t ∈ [0,T].