Inverse initial problem for a time-fractional diffusion-wave equation

Abstract

We find sufficient conditions for a unique classical solvability of the inverse problem of restoration of two functions in initial conditions of the Cauchy problem for a time-fractional diffusion-wave equation with the Caputo-Djrbashian-Nersesian derivative and the right-hand side with values in Schwartz-type spaces of smooth functions rapidly decreasing to zero at infinity. We use two time-integral overdetermination conditions \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_1(t)dt=\Phi_1(x), \quad\frac{1}{T}\int_{0}^{T}u(x,t)\eta_2(t)dt=\Phi_2(x), \quad x\in \mathbb R^n,\] where $u$ is the solution of the Cauchy problem for such equation, $\Phi_1$, $\Phi_2$ are given functions from the Schwartz-type space, $\eta_1$, $\eta_2$ are given functions from $C^2[0,T]$. We use the method of the Green's vector-function. The initial data sought are expressed through the solution of a certain linear Fredholm integral equation of the second kind in the space of continuous functions with values in Schwartz-type spaces.