Non-local variables in group analysis of differential equations

V.F.Zaitsev, L.V.Linchuk

Russian State Pedagogical University

Moyka, 48, 191186, Saint-Petersburg, Russia

valentin_zaitsev@mail.ru

 

The modern step of development of mathematical modeling is characterized by increased interest to higher-order nonlinear differential equations and elaboration of methods of their solutions in closed analytical form.љ This is because the simplest linear models didn't match the requirements of researches to adequacy of model and known integrable classes of equations aren't sufficient to the practical demands. Classical group analysis enabled these classes to be extended and proved their maximality provided that only point and tangent symmetries were considered. Therefore we need to extend the notion of symmetry in order to find new classes of integrable equations. In particular, it is possible if we shall introduce the nonlocal variables. This paper is devoted to different types of operators: exponential nonlocal operators for the 2^nd order ordinary differential equations and several forms of nonlocal (nonexponential) operators for the 3^rd order ordinary differential equations. We consider possibility of solution of inverse problems.