**Avtandil Tsitskishvili**

## Solution of Some Plane Filtration Problems with Partially Unknown Boundaries

**abstract:**

Plane problems of the stationary filtration theory with partially
unknown boundaries are considered. The porous medium is assumed
to be homogeneous, isotropic and non-deformable. The motion of the fluid
obeys the Darcy law. The simply connected domain occupied by the moving
fluid is bounded by a simple sectionally analytic contour consisting of
unknown depression curves, line segments, half-lines and straight lines.
The paper describes mathematical methods of finding the unknown parts of
the boundary of the fluid motion domain, as well as of determining
geometric, cinematic and physical characteristics of the
moving fluid. In solving the corresponding
mathematical problem, the use is made of the general solution of the
non-linear Schwarz differential equation. The general solution is
constructed in the paper.

**Mathematics Subject Classification:**
34A20, 34B15

**Key words and phrases:**
Filtration, analytic functions, conformal
mapping, differential equation.