##### (LIT JINR), (RUDN, MSU), (RUDN, JINR)

Finite differences method and integration of differential equations in finite terms

For ordinary differential equations of a certain class introduced by the French mathematician Painleve, one can construct finite difference schemes that preserve algebraic properties of exact solutions. In terms of Cauchy problem, a differential equation of this class defines an algebraic correspondence between the initial and terminal values. For example, the Riccati equation y’ = p y2 + q y + r defines the one-to-one correspondence between the initial and terminal values of y on a projective straight line. However, standard finite difference schemes do not preserve this algebraic property of the exact solution. Furthermore, the scheme which defines one-to-one correspondence between layers correctly describes behavior of solution not only before but also after movable singularities and preserves such algebraic property of the differential equation as the anharmonic ratio.