The main tasks of group analysis of differential equations of mechanics. Multiphase Systems. 17 (2022) 1–2. 51–62 (in Russian).
The main tasks of group analysis of differential equations of mechanics
Mavlyutov Institute of Mechanics UFRC RAS, Ufa, Russia
Group analysis of differential equations uses the Lie theory of correspondence between continuous transformation
groups and Lie algebras of first-order differentiation operators. Differential equations of mechanics necessarily admit
an extensive group of transformations. Lie theory studies the structure of the algebra of this group. The group
analysis of the equations of mechanics uses the structure of the admissible algebra to produce submodels and exact
solutions, to study the boundary value problems of submodels and the behavior of the mechanical medium for exact
solutions. The main tasks of group analysis are formulated and simple examples of their solutions are given.
equations of mechanics,
Objective: to describe the main modern problems of group analysis of differential equations of mechanics. All tasks are divided into 4 large sections: permissible group and group classification, structure of permissible groups, stratification and submodels, and other group analysis tasks.
In the first section, the first three main sequential problems are formulated and solutions to these problems are given by the example of equations of one-dimensional gas dynamics with plane waves.
In the second section, two more problems are presented and an example of the kernel of admissible algebras for one-dimensional gas dynamics is considered.
In the third section, three more tasks with examples are formulated.
In the fourth section, other tasks of group analysis are formulated. Methods for solving some problems are well developed, but for some there may be no algorithmic approaches.
Result: the article lists 13 new problems of group analysis of differential equations of mechanics. For the first 8 tasks, the simplest examples of the implementation of these tasks are given. The other five tasks are announced with links to ways to solve them.
Conclusions: the group analysis of differential equations of mechanics contains many more unsolved problems.
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