Inverse problem of unsteady incompressible fluid flow in a pipe with a permeable wall

The process of unsteady flow of viscous incompressible fluid in a pipe with a permeable wall described by a nonlinear system of partial differential equations in the velocity-pressure variables is considered. This system of equations is reduced to a single nonlinear equation of parabolic type with respect to velocity. Within the framework of the obtained model, the inverse problem is posed to determine the permeability coefficient of the pipe wall, which depends only on the time variable. In this case, an additional condition relative to the fluid pressure is set at the outlet of the pipe. A difference analogue of the coefficient inverse problem is built using finite-difference approximations. For the solution of the received difference problem, the special representation is offered allowing to split problems into two mutually independent linear difference problems of the second order on each discrete value of a time variable. As a result, an explicit formula is obtained to determine the approximate value of the wall permeability coefficient for each discrete value of the time variable. On the basis of the proposed computational algorithm, numerical experiments were carried out for model problems.