This model describes the Poiseuille type solution in the nonstationary case of the Navier–Stokes problem. An equivalent form of PDE problem is defined as the first-kind Volterra integral equation. The main aim is to analyze a possible ill-posedness of the given problem. For some problems the first-kind Volterra integral equation can be modified to the integral equation of the second kind and the letter equation is well-posed. Different regularization techniques also can be used to control the influence of error pollution with not equal efficiency. Thus we made an extensive analysis and compared classical discretization schemes for PDE and integral Navier–Stokes models and regularization algorithms. The regularization methods are applied to control the influence of the noise in data. The numerical experiment was aimed at obtaining new information about the stability of schemes for the inverse problems. Different approximations methods are used to solve PDE and integral versions of the equation. Results of computational experiments are presented, they confirm the theoretical error analysis and stability estimates.