This paper presents the method of separation of variables to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Further, in the Rayleigh-Stokes problem, instead of the initial condition, the non-local condition is considered: u(x,T)=βu(x,0)+φ(x), where β is equal to zero or one. It is well known that if β=0, then the corresponding problem, called the backward problem, is ill-posed in the sense of Hadamard, i.e., a small change in u(x,T) leads to large changes in the initial data. Nevertheless, we will show that if we consider sufficiently smooth current information, then the solution exists, it is unique and stable. It will also be shown that if β=1, then the corresponding non-local problem is well-posed and inequalities of coercive type are satisfied.

In this work, we present the advanced fractional model described with the loaded parabolic-hyperbolic-type equation. Also, for this equation, we investigate a Cauchy-type problem with a line of changing type. The existence and uniqueness of the solutions to the problem are proven using the theory of integral equations.

The Fourier method is used to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Then, in the Rayleigh-Stokes problem, instead of the initial condition, consider the non-local condition: u(x, T ) = βu(x, 0) + ϕ(x), where β is either zero or one. It is well known that if β = 0, then the corresponding problem, called the backward problem, is ill-posed in the sense of Hadamard, i.e. a small change in u(x, T ) leads to large changes in the initial data. Nevertheless, we will show that if we consider sufficiently smooth current information, then the solution exists and it is unique and stable. It will also be shown that if β = 1, then the corresponding non-local problem is well-posed and coercive type inequalities are valid.