A. Tsitskishvili
abstract:
In the present paper, using linearly independent solutions of the Fuchs class
linear differential equation which contains a term with the first order
derivative of the unknown function, we propose effective methods for solving
both the Schwarz nonlinear equation, whose right-hand side is a doubled
invariant of the Fuchs class linear differential equation, and the plane
problems of filtration with partially unknown boundaries. The modulus of the
difference of the characteristic numbers of the Fuchs class linear differential
equation for every singular point is equal to the corresponding (divided by
$\pi$) angle at the vertex of a circular polygon. For the first time it is shown
that the coefficients at the poles of second order of the doubled invariant of
the Fuchs class linear differential equation and those on the right-hand side of
the Schwarz equation coincide completely.
Relying on the property mentioned above, we suggest simpler methods of solving
the problems of the theory of stationary motion of incompressible liquid in a
porous medium with partially unknown boundaries than those described by us
earlier for the solution of the same problems.
Mathematics Subject Classification: 34A20, 34B15.
Key words and phrases: Filtration, analytic functions, conformal mapping, differential equation.