On the Lie problem in PDEs and effective classification of the differential equations with algebraic coefficients (Q669576)

In this paper the author studies the classification problem for the partial differential equations \[ u_{xy}=f(x,y,u) \] with respect to transformation of variables $(x,y,u)$. Such problem is the natural generalization of the classical problem for ordinary differential equation $y''=f(x,y)$ set by S. Lie. The main result is the computation of the number of independent differential invariants of pure order $k$. The author also studies the classification of above partial differential equations with respect to the action of some symmetry groups.