Serov N. I.
Classification of linear representations of the Galilei, Poincaré, and conformal algebras in the case of a two-dimensional vector field and their applications
Ukr. Mat. Zh. - 2006. - 58, № 8. - pp. 1128–1145
We present the classification of linear representations of the Galilei, Poincaré, and conformal algebras nonequivalent under linear transformations in the case of a two-dimensional vector field. The obtained results are applied to the investigation of the symmetry properties of systems of nonlinear parabolic and hyperbolic equations.
Lie Symmetries, Q-Conditional Symmetries, and Exact Solutions of Nonlinear Systems of Diffusion-Convection Equations
Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1340-1355
A complete description of Lie symmetries is obtained for a class of nonlinear diffusion-convection systems containing two Burgers-type equations with two arbitrary functions. A nonlinear diffusion-convection system with unique symmetry properties that is simultaneously invariant with respect to the generalized Galilei algebra and the operators of Q-conditional symmetries with cubic nonlinearities relative to dependent variables is found. For systems of evolution equations, operators of this sort are found for the first time. For the nonlinear system obtained, a system of Lie and non-Lie ansätze is constructed. Exact solutions, which can be used in solving relevant boundary-value problems, are also found.
Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 846-849
We investigate the $Q$-conditional symmetry of a nonlinear two-dimensional heat-conduction equation. By using ansatzes, we obtain reduced equations.
Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1262–1270
The Lie symmetries of nonlinear diffusion equations with convection term are completely described. The Lie ansatzes and exact solutions of a certain nonlinear generalization of the Murray equation are constructed. An example of the family of non-Lie solutions of the Murray equation is given.
Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 806–813
We study the conditional symmetry of the Navier-Stokes equations and construct multiparameter families of exact solutions of the Navier-Stokes equations.
Ukr. Mat. Zh. - 1993. - 45, № 2. - pp. 293–302
By a nonlocal substitution, a nonlinear system of heat-conduction equations is reduced to a scalar nonlinear heat-conduction equation. The Lie and conditional invariance of the scalar equation is used to find nonlocal ansatze which reduce the original system to systems of ordinary differential equations.
Ukr. Mat. Zh. - 1990. - 42, № 10. - pp. 1370–1376