Vaneeva O. Equivalence groupoids in group classification problems

The thesis is devoted to the development of the theory of equivalence groupoids in classes of differential equations, to the creation of new methods and algorithms of the group analysis of differential equations and to solving a wide range of group analysis problems, including classifications of Lie and nonclassical reduction operators and conservation laws. The main attention is paid to the application of equivalence groupoids to group classification problems and finding solutions of equations of mathematical physics, mathematical biology and financial mathematics. In particular, the notion of regular and singular cases of Lie symmetry extensions is introduced. A modification of the algebraic method of solving group classification problems for non-normalized classes is suggested. The application of contractions to group classification problems is demonstrated. An algorithm of the construction of all cases of Lie symmetry extensions from the list of non-equivalent ones is developed. Equivalence groupoids are also used effectively in the study of integrability and classification of nonclassical reduction operators. Extended group analysis is performed, in particular, for classes of K(m,n) equations, Kawahara equations, third- and fifth-order Korteweg-de Vries equations, Fisher equations, Newell-Whitehead-Segel equations, reaction-diffusion equations, Burgers equations, Benjamin-Bond-Mahony equations and (1+1)-dimensional nonlinear wave and elliptic equations.