Non-local ansätze for nonlinear heat and wave equations

Symmetry and Solution of Neutron Transport Equations in Nonhomogeneous Media

“…Integrating this reduced equation one can obtain exact solutions in explicit form, such as one-and many-soliton solutions. Moreover, the existence of Lie-Bäcklund symmetry enables one to construct conservation laws for initial equation [3][4][5][6][7][8][9][10][11]. In this section we use another property of an invariance group, namely, the generation of the new solutions by action of the group of finite transformations on the known one.…”

“…The group-theoretical analysis is known to be used for the construction of exact solutions of a number of linear and nonlinear equations of mathematical physics [1,2]. One of the most efficient methods for the obtaining of explicit solutions is the method of group reduction [1][2][3][4][5][6][7][8][9][10][11]. Finite transformations of the invariance group of differential equations can also be applied to generate new solutions (both exact and approximate).…”

On new ways of group methods for reduction of evolution-type equations

“…In order to construct ansatz reducing given equation to system of two ordinary DEs, we should use two-dimensional subalgebra of invariance algebra of equation. At the same time according to [15], obtained solution will be invariant solution in the Lie sense and can be obtained by classical Lie method. It follows that we should use operators of non-point symmetries to obtain new results.…”