1 semester


Title of the discipline

Differential equations in applications.


Training course, specialty

1, 1-31 03 09 Computer mathematics and systems analysis


Semester of  training



The number of credits



Full name of the  lecturer

Gromak Valery Ivanovich., doctor of physical and mathematical sciences.  Professor.


Objectives of study of the discipline

The purpose of studying the discipline is to train specialists who have abilities

and skills in mathematical research on the basis of methods of the analytic theory of differential equations and symmetry analysis.

As a result of the training, the undergraduate must know:

  • the basic concepts of the analytic theory of differential equations and the theory of continuous Lie groups;
  • the main analytical and group methods for investigating of differential equations;
  • the main types of applications that can be effectively investigated by the analytic theory of differential equations and the theory of continuous Lie groups.

be able to:

  • apply the basic methods of the analytic theory of differential equations and the theory of continuous Lie groups to study the properties of dynamic models;
  • construct Lie algebras and universal invariants,
  • find exact solutions or to lower the order of differential equations, that admit one-parameter Lie groups;



Teaching materials of the courses “Algebra and Number Theory”, “Geometry”, “Mathematical Analysis”. “Differential equations”, “Computer mathematics”, “Mathematical modeling of dynamic processes” 


Contents of the discipline

Symmetry analysis of differential equations and applications. Multidimensional mathematical

models. The Boussinesq and Korteweg-De Vries equations. One-parameter Lie groups of transformations. Point and contact transformations.

Local r-parametric Lie group. The tangent vector field and the infinitesimal operator of the Lie group of transformations. Invariants of the Lie group of transformations. Criterion of invariance. Invariant varieties of Lie transformation groups. Criterion for the invariance of a manifold. The theory of the continuation of a group and an infinitesimal operator. Groups of transformations and differential equations. Differential invariants.

Lie algebras of symmetries. Commutator of operators. Calculation of Lie algebras of symmetries. Generalization of Lie groups of point transformations to the multidimensional case.

Invariants of multiparametric Lie groups and complete systems of operators.

The basis of invariants of multiparametric Lie groups. Invariant and partially invariant manifolds.

Symmetric classification of equations of mathematical physics: Burgers, Korteweg-de Vries, heat conduction, Boussinesq, sine-Gordon, and other equations. Invariant and partially invariant solutions. The notions about of the higher symmetries, Bäcklund transformations, integrable equations, boundary conditions compatible with the admitted symmetry groups.       

The elements of the analytic theory of differential equations. Introduction to the theory of special functions and polynomials. The Cauchy theorem of existence and uniqueness. Painleve property.

The  Painlevé small parameter method. The Poincare theorem. The application of the

Painlevé method to equations of the first and second orders. Method of resonances. The Fuchs indices.

Linear differential equations in the complex domain. The Cauchy theorem.

Introduction to the  Frobenius theory. Regular and irregular singular points.

The Fuchs class of linear differential equations. Classification of equations of the Fuchs class.  Confluence of regular singular points. The Airy and Bessel equations.

The Riemann equation. Scheme of Riemann. Invariance of the form of the Riemann equation with respect to the group of Möbius transformations. The Gauss equation. Hypergeometric series. Hypergeometric integrals.

Monodromy group. Definition of the monodromy group and Gauss equation.

The Legendre equation. Periods of the elliptic function as a solution of the Legendre equation. Polynomials of Legendre and Jacobi.

The Riemann problem. Izomodromic deformation of linear systems.  Bolibrukh counterexample.

Introduction to the theory of Painlevé equations. Necessary conditions for the existence of the Painlevé property for the second order equations. Painlevé equations.  Meromorphicity and transcendence of solutions of the Painlevé equations. Sufficiency of conditions for the Painlevé property. Bäcklund transformations of Painlevé equations.

Classification of solutions of the Painleve equations. Special classes of solutions. Asymptotic properties of solutions of the Painlevé equations.

The Painlevé equations of the higher orders. Different approaches: the Painlevé direct method, nonlinear chains, the symmetry approach. Systems of differential equations with  Painleve property. Hamiltonian systems with Painleve property.


Recommended literature

Basic literature:

  1. Ovsyannikov L. Group analysis of differential equations, M., Nauka, 1978.(in Russian)
  2. Olver P. Applications of Lie groups to differential equations, Mir, 1983.(in Russian)
  3. Egorov A.I. Ordinary differential equations with applications, M. Fizmatlit, 2005. (in Russian)
  4. Ibragimov N.Kh. Transformation groups in mathematical physics, M., Nauka, 1983.(in Russian)
  5. Golubev V.V. Lectures on the analytic theory of differential equations, Moscow: Glavpoligraizdat, 1950, 436 pp. .(in Russian)
  6. Ince E.L. Ordinary differential equations, Dover Publications, New York, 1944.
  7. Kudryashov N.A. Analytic Theory of Nonlinear Differential Equations, Moscow-Izhevsk, Institute for Computer Research, 2004, 360 pp. (in Russian)
  8. Smirnov V.I. A course in higher mathematics, vol. III, part 2, Moscow, 1974, 672 p. (in Russian).

Additional literature

  1. Arnold V.I. Mathematical methods of classical mechanics, M., Nauka, 1974.
  2. Blumann G.W., Cole J.D. Similarity methods for Differential Equations, Springer-Verlag, N-Y, 1974.й
  3. Ibragimov N.Kh. ABC of group analysis, Mathematics, Cybernetics, Izd. Znaniya, Moscow, N8, 1989.
  4. Iwasaki K., Kimura H., Shimomura Sh., Yoshida M. From Gauss to Painlevé. A modern theory of special functions: Aspects of Mathematics, E16. Braunschweig, 1991.
  5. Gromak V. I., Laine I., Shimomura S. Painlevé Differential Equations in the Complex Plane, De Gruyter Studies in Mathematics 28, Berlin — New-York, 2002.
  6. Noumi M. Painleve equations through Symmetry, AMS, 2000, 158 p. 


Methods of teaching

Lecture with laboratory works, using elements of distance learning and electronic materials.


Language of teaching



Conditions (requirements), current control

Test work, reports on laboratory works and protection.


Form of current attestation