A. Tsitskishvili

Extension of the Class of Effectively Solvable Two-Dimensional Problems with Partially Unknown Boundaries in the Theory of Filtration

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