Prystavka Y. Symmetry properties and exact solutions of the reaction-convection-diffusion equations

The thesis is devoted to the study of symmetry properties and the construction of exact solutions of nonlinear reaction-convection-diffusion equations and systems of nonlinear convection-diffusion equations. In the thesis work, for classification of reaction-convection-diffusion equations, we classified Lie symmetries and found some exact solutions of equations with power nonlinearities. The nonlocal ansätze and nonlocal operators corresponding to them are constructed for the system of convection-diffusion equations. With the help of the found ansätze, the data of the equations were reduced and some exact solutions of them were found. A complete group classification of the symmetry properties of the (1+2)-dimensional reaction-convection-diffusion equation was carried out to within equivalence transformations. The exact solutions of the reaction-convection-diffusion equation with power nonlinearities are constructed, the properties of some of the solutions obtained are investigated and their geometrical interpretation is given. Nonlocal equivalence transformations of a system of nonlinear convection-diffusion equations are found. These transformations are used to study the system of the van der Waals equations. Two images of the van der Waals system of equations, which belongs to the class of systems of nonlinear convection-diffusion equations, are obtained, and the symmetry properties of these images are investigated. The nonlocal ansätze were constructed, a reduction was carried out, and some exact solutions of the van der Waals system of equations were found. Nonlocal ansätze were found, and both images of the van der Waals system of equations were reduced. Nonlocal operators of invariance, which correspond to the nonlocal ansätze of this system, are constructed.