Avtandil Tsitskishvili
abstract:
In the present work we consider a general mathematical method of constructing
the solutions of spatial axisymmetric stationary problems of the jet and
filtration theories with
partially unknown boundaries. The $x$-axis coincides with the symmetry axis, and
the distance to the $x$-axis is denoted by $y$. The use is made of the right
coordinate system. Of infinitely many half-planes we arbitrarily select one
passing through the symmetric axis. But for the sake of effectiveness sometimes
it is more convenient to take two symmetric half-planes lying in one plane. The
boundary of the domain under consideration consists of the known and unknown
parts. The known ones consist of straight lines and their portions,
while the unknown parts consists of curves. Every portion of the boundary is
assigned two boundary conditions. The unknown functions (the velocity potential,
the flow function) and their arguments on every portion of the boundary must
satisfy two inhomogeneous boundary conditions.
The system of differential equations with respect to the velocity potential and
flow function is reduced to a normal equation. Unknown functions are represented
as sums of holomorphic and generalized analytic functions.
One problem of the jet
theory and one problem of the filtration theory are solved.
Mathematics Subject Classification: 34A20, 34B15
Key words and phrases: Filtration, analytic functions, generalized analytic functions, quasi-conformal mappings, differential equation