A. Tsitskishvili
abstract:
In the present paper, first we briefly describe effective methods for solving
such two-dimensional problems of the theory of filtration to which on the plane
of complex velocity there correspond circular pentagons of special type and
removable singular points. Of five vertices of the circular pentagon at least
one is a cut end formed by two neighboring sides, with the angle $2\pi$. Then we
present effective methods of solving two
problems of the theory of filtration dealing with the motion of an
incompressible liquid through the plane earth dams of trapezoidal shape with
water nonpermeable bases. To each problem there corresponds one removable
singular point. The first problem deals with the plane earth dam whose lower
slope is vertical and the upper one is inclined to the horizon. The second
problem deals with the plane earth dam whose lower slope is inclined to the
horizon and the upper one is vertical. To these problems on the plane of complex
velocity there correspond circular pentagons one of whose vertices is a cut end
with the angle $2\pi$. To find unknown parameters, we write a system of
equations which is decomposed into three systems. We substitute the solutions of
the first system into the second and third systems, and we substitute the
solution of the second system into the third system.
Mathematics Subject Classification: 35J25, 35L99, 76S05
Key words and phrases: Filtration, analytic functions, conformal mapping, differential equation