Enhancement of the method of A.N. Ivanov for flows with free boundaries (to the 100th anniversary of Ivanov's birth)

The analysis of flows with free boundaries is a non-linear problem, since their forms are not known in advance and must be found. The described method is effective for various problems with various dynamic conditions on free boundaries. This iterative method is based on the quasi-linearization of the boundary conditions for the velocity potential proposed by A.N. Ivanov in 1962. The iterations include the successive solutions of the Fredholm integral equations of the second kind (allowing to determine the pressure over a known boundary) and singular integral equations (to correct the boundary shape). Here both the general formulation of the method and its development for specific types of flows are presented, as well as numerous examples of solutions for them.