On the solution of an inverse problem simulating two-dimensional motion of a viscous fluid

An inverse initial boundary value problem for a linear parabolic equation that arises as a result of mathematical modelling of 2D creeping motion of viscous liquid in a flat channel is considered. The unknown function of time is added in the right part of equation and can be found from additional condition of integral overdetermination. This problem has two different integral identities, permitting to obtain a priori estimates of solutions in uniform metric and to proof the uniqueness theorem. Under some restrictions on input data the solution is constructed as a series in the special basis. For this purpose the problem is reduced by differentiation with respect to the spatial variable to a direct non-classic problem with two integral conditions instead of ordinary ones. The new problem is solved by separation of variables, which allows one to find the unknown functions in the form of rapidly converging series. Another method for solving the initial problem is to reduce the problem to the loaded equation and to state the first initial boundary value problem for this equation. In its turn, this problem is reduced to one-dimensional in time Volterra operator equation with a special kernel. It is proved that it has a series solution. Some auxiliary formulas which are useful for the numerical solution of this equation by the Laplace transform are obtained. Sufficient conditions under which the solution with increasing time converges to steady regime by exponential law are established.