Non-local and Inverse Problems for the Rayleigh-Stokes Equation

This work consists of two parts. In the first part, a non-local boundary value problem is studied for the fractional version of the Rayleigh-Stokes equation known in hydrodynamics. Namely, instead of the initial condition, the condition \(u(x,T)=\beta u(x,0)+\varphi (x)\) is proposed, where \(\beta \) is an arbitrary real number. Note that if \(\beta =0\) , then we get the backward problem, which is ill-posed. The main goal of this part of the work is to study the influence of parameter \(\beta \) on the correctness of the problem. The second part considers the homogeneous Rayleigh-Stokes equation and investigates the behavior of the solution of the Rayleigh-Stokes problem depending on the order of the fractional derivative \(\alpha \) . In particular, it is proved that for sufficiently large t, the norm \(||u(x,t)||{ }_{L_2(\Omega )}\) of the solution decreases in \(\alpha \) . Moreover, assuming the parameter \(\alpha \) to be unknown, an additional condition on the solution of the problem is found that uniquely determines the order of the derivative \(\alpha \) .