The problem of determining the coefficient of thermal expansion of the equation of thermoviscoelasticity

We consider the problem of finding the thermal expansion coefficient $\alpha(z),\ z\in [0,Z],$ occurring in the system of integro-differential termoviscoelasticity equations. The medium density and the Lame parameters are assumed to be function of one variable. The integrand $h(t),\ t\in [0; T]$ is known. The inverse problem is replaced by the equivalent integral equation for unknown functions. The theorem of unique solvability is proved and the stability estimate of solving the inverse problem is obtained.