Lectures on the Theory of Group Properties of Differential Equations

These lecturers provide a clear introduction to Lie group methods for determining and using symmetries of differential equations, a variety of their applications in gas dynamics and other nonlinear models as well as the author's remarkable contribution to this classical subject. It contains material that is useful for students and teachers but cannot be found in modern texts. For example, the theory of partially invariant solutions developed by Ovsyannikov provides a powerful tool for solving systems of nonlinear differential equations and investigating complicated mathematical models.

Contents:

• One-Parameter Continuous Transformation Groups Admitted by Differential Equations:

  • One-Parameter Continuous Transformation Group
  • Infinitesimal Operator of the Group
  • Invariants and Invariant Manifolds
  • Theory of Prolongation
  • Groups Admitted by Differential Equations
  • Lie Algebra of Operators

• Lie Algebras and Local Lie Groups:

  • Lie Algebra
  • Adjoint Algebra
  • Local Lie Group
  • Subgroup, Normal Subgroup and Factor Group
  • Inner Automorphisms of a Group and of Its Lie Algebra
  • Local Lie Group of Transformations

• Group Invariant Solutions of Differential Equations:

  • Invariants of the Group GNr
  • Invariant Manifolds
  • Invariant Solutions of Differential Equations
  • Classification of Invariant Solutions
  • Partially Invariant Solutions
  • Reduction of Partially Invariant Solutions
  • Some Problems

Readership: Professional mathematics researchers and graduate students.