A. V. Flegontov
St.-Petersburg, Russia,
Institute for Informatics and Automation of the
Russian Academy of Science
V. F. Zaitsev
St.Petersburg State University, Russia
The symmetry is a fundamental property of any phenomenon or any process. This refers equally to an equation describing a phenomenon or a process. Research methods based on symmetry are effective for nearly all (from algebraic up to integro-differential) types of equations. Main problems of the practical group analysis of differential equations are the following: Elaboration of regular (algorithmic) methods for a search of the equation symmetries of all kinds; Solution of inverse problems, i.e. a search of classes of equations or classes of models having a given symmetry and other a priori properties; Determination of general principles the symmetry has to be used in practical problems. If the inverse problem relates to tangent groups or Lie-Backlund operators, a form of an auxiliary, as a rule, partial differential equation, its order and degree of nonlinearity are predicted. Solving of this auxiliary equation yields a general form of an ordinary differential equation admitting a given operator. This permits to choose among the set of possible models the ones possessing the symmetry required a priori. A computer bank of models makes possible to evolve from the set of admissible models the one with consideration for a priori information: conservation laws, principle of mobile balance, qualitative characteristics of the system behavior and others. Based on group concepts, the model classes unite equations of linear and nonlinear mechanics and mathematical physics and their new exact solutions. Additional information (first integrals, symmetries, invariants, group structures etc.) provides diversified search and analysis of mathematical models in the analytic form.