Module also offered within study programmes:
General information:
Annual:
2017/2018
Code:
AMA-2-042-MZ-s
Name:
Group Analysis of Differential Equations
Faculty of:
Applied Mathematics
Study level:
Second-cycle studies
Specialty:
Matematyka w zarządzaniu
Field of study:
Mathematics
Semester:
0
Profile of education:
Academic (A)
Lecture language:
English
Form and type of study:
Full-time studies
Course homepage:
 
Responsible teacher:
dr hab. Vladimirov Vsevolod (vladimir@mat.agh.edu.pl)
Academic teachers:
dr hab. Vladimirov Vsevolod (vladimir@mat.agh.edu.pl)
Module summary
Theory of Lie groups of transformations. Jet space. Symmetry.
Description of learning outcomes for module
MLO code Student after module completion has the knowledge/ knows how to/is able to Connections with FLO Method of learning outcomes verification (form of completion)
Social competence
M_K001 Student can notice symmetry of surrounding world and understands the role of symmetry in modeling the nonlinear phenomena. MA2A_K05, MA2A_K01 Activity during classes
Skills
M_U001 Student can implement the criterium of invariance of the differential equations (DE). Student knows how to find out the elementary symmetries of a given DE (such as scalings and shifts) without implementing the Sophus Lie machinery. MA2A_U01, MA2A_U05, MA2A_U02 Activity during classes,
Examination,
Test
M_U002 Student can use the symmetry of a nonlinear PDE in finding out particular solutions and constructing new solutions from the already known ones. Student is able to formulate the problem of the group-theory classification of a family of DE containing the parameters and unknown elements. Student possesses the practical skill of solving the ODEs of the first and the second order, using their symmetries. MA2A_U06, MA2A_U10 Activity during classes,
Examination,
Test
Knowledge
M_W001 Student knows the foundations of theory of Lie groups of transformations. Student understands links between the local Lie groups and their generators. Student knows the foundations of the theory of invariants and is familiar with the concept of prolongation of the Lie group of transformation onto the jet space. Student knows the criterium of invariance of the system of differential equations and is familiar with its practical implementation. Student is able to apply the Lie algorithm for studying the symmetry of both ordinary and partial differential equations (ODEs and PDEs to abbreviate) and knows how (and when) it is possible to implement the symmetry of nonlinear differential equations. MA2A_W01, MA2A_W02 Activity during classes,
Examination,
Test
M_W002 Student is able to write down a system of defining equations. Student can implement the procedure of splitting the system of defining equations. Student knows the procedure enabling to find out the invariants of the symmetry group and is able to construct an ansatz leading to the group theory reduction in the case of PDEs. MA2A_W03, MA2A_W07 Activity during classes,
Examination,
Test
FLO matrix in relation to forms of classes
MLO code Student after module completion has the knowledge/ knows how to/is able to Form of classes
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Zaj. terenowe
Zaj. warsztatowe
Others
E-learning
Social competence
M_K001 Student can notice symmetry of surrounding world and understands the role of symmetry in modeling the nonlinear phenomena. + + - - - - - - - - -
Skills
M_U001 Student can implement the criterium of invariance of the differential equations (DE). Student knows how to find out the elementary symmetries of a given DE (such as scalings and shifts) without implementing the Sophus Lie machinery. + + - - - - - - - - -
M_U002 Student can use the symmetry of a nonlinear PDE in finding out particular solutions and constructing new solutions from the already known ones. Student is able to formulate the problem of the group-theory classification of a family of DE containing the parameters and unknown elements. Student possesses the practical skill of solving the ODEs of the first and the second order, using their symmetries. + + - - - - - - - - -
Knowledge
M_W001 Student knows the foundations of theory of Lie groups of transformations. Student understands links between the local Lie groups and their generators. Student knows the foundations of the theory of invariants and is familiar with the concept of prolongation of the Lie group of transformation onto the jet space. Student knows the criterium of invariance of the system of differential equations and is familiar with its practical implementation. Student is able to apply the Lie algorithm for studying the symmetry of both ordinary and partial differential equations (ODEs and PDEs to abbreviate) and knows how (and when) it is possible to implement the symmetry of nonlinear differential equations. + + - - - - - - - - -
M_W002 Student is able to write down a system of defining equations. Student can implement the procedure of splitting the system of defining equations. Student knows the procedure enabling to find out the invariants of the symmetry group and is able to construct an ansatz leading to the group theory reduction in the case of PDEs. + + - - - - - - - - -
Module content
Lectures:

1. Local Lie group. One-parametric local Lie group of transformations.
Canonical parameter.

2. Generator of a Lie group of transformation (IFG) and its geometric interpretation. The first
fundamental Lie theorem.

3. Exponential map. Lie group induced by a diffeomorphism. Transformation of coordinates of IFG
induced by the change of variables. Canonical coordinates.

4. Invariance of a function. Algebraic manifold and its invariance. One-parematric Lie group defined
on a set of dependent and independent variables.

5. Jet space. Theory of prolongation. Criterium of invariance of a differential equation. Defining
equations.

6. Symmetry of the linear heat transport equation.

7. Symmetry of the potential Burgers equation. Symmetry-based reconstruction of the Cole-Hopf
transformation.

8. Multi-parameter Lie group and its Lie algebra. The problem of group theory classification.

9. Symmetries and invariant solutions. Particular solutions of some nonlinear equations of
mathematical physics.

10. Dissemination of solutions. Inverse problem of the group analysis.

11. Symmetry and integrability of the first-order ODE.

12. Symmetry of the second-order ODE: lowering of the order. A class of the second-order ODEs,
admitting two symmetry generators. The strategies of integration.

13. Canonical IFO of a one-parameter Lie group of transformations. Generalized symmetries.

14. Algorithm for searching the generalized symmetries of DE. Examples.

Auditorium classes:
Group analysis of differential equations (classes)

Tresci ćwiczeń audyrotyjnych pokrywają sie z tresciami prowadzonego wykładu. Pod czas zajęc praktycznych studenci rozwiązuja zadania praktyczne dotyczące analizy grupowej równań różniczkowych

Student workload (ECTS credits balance)
Student activity form Student workload
Summary student workload 160 h
Module ECTS credits 6 ECTS
Participation in lectures 30 h
Participation in auditorium classes 30 h
Preparation for classes 28 h
Realization of independently performed tasks 60 h
Contact hours 10 h
Examination or Final test 2 h
Additional information
Method of calculating the final grade:

Ocena końcowa jest średnią oceny z egzaminu i zaliczenia ćwiczeń audytoryjnych

Prerequisites and additional requirements:

Prerequisites and additional requirements not specified

Recommended literature and teaching resources:

1. P. Olver, Application of lie Groups to Differential Equations, Springer, NY, 1993.

2. G. Bluman , S. Kumei, Symmetries and Differential Equations,
Springer, NY, 1989.

3. H. Stephani, Differential Equations: Their Solutions Using
Symmetries, Cambridge Univ. Press, NY, 1989.

4. G. Baumann, Symmetry Analysis of Differential Equations with
Mathematica, Springer, NY, 2000.

5. N. Ibragimov, Transformation Groups Applied to Mathematical
physics, Reidel, Boston, 1985.

6. N. Ibragimov (Ed.), CRC Handbook on Group Analysis. Vol.
I-III, Boca Rata, 1994.

Scientific publications of module course instructors related to the topic of the module:

1. Likus, W.; Vladimirov, V.A.
Solitary waves in the model of active media, taking into account effects of relaxation; Rep. Math. Phys. 75, No. 2, 213-230 (2015).

2. Danylenko, V.A.; Danevich, T.B.; Makarenko, O.S.; Moskaliuk, V.S.; Skurativskiy, S.I.; Vladimirov, V.A.
Exact solutions and wave patterns within some non-local hydrodynamic-type models;
Algebras Groups Geom. 31, No. 4, 407-477 (2014).

3. Vladimirov, V.A.; Morgulis, A.B.
Relative equilibria in the Bjerknes problem. (English. Russian original);
Sib. Math. J. 55, No. 1, 35-48 (2014); translation from Sib. Mat. Zh. 55, No. 1, 44-60 (2014).

4. Danylenko, V.A.; Danevich, T.B.; Makarenko, O.S.; Moskaliuk, S.S.jun.; Skurativskiy, S.I.; Vladimirov, V.A.
Group analysis of reaction-diffusion-convection of nonlinear equations;
Algebras Groups Geom. 30, No. 3, 275-365 (2013).

5. Vladimirov, V.A.
Dumbbell micro-robot driven by flow oscillations; J. Fluid Mech. 717, R8, 11 p., electronic only (2013).

6. Vladimirov, V.A.
On the self-propulsion of an N-sphere micro-robot; J. Fluid Mech. 716, R1, 11 p., electronic only (2013).

7. Danylenko, V.A.; Danevich, T.B.; Makarenko, O.S.; Moskaliuk, N.M.; Skurativskiy, S.I.; Vladimirov, V.A.
Algebra-invariant models for nonlinear nonlocal media;
Algebras Groups Geom. 29, No. 3, 309-376 (2012).

8. Vladimirov, V.A.; Ma̧czka, Cz.;
On the stability of kink-like and soliton-like solutions of the generalized convection-reaction-diffusion equation; Rep. Math. Phys. 70, No. 3, 313-329 (2012).

9. Vladimirov, V.A.; Magnetohydrodynamic drift equations: from Langmuir circulations to magnetohydrodynamic dynamo? J. Fluid Mech. 698, 51-61 (2012).

10. Vladimirov, V.A.; Kutafina, E.V.; Zorychta, B.
On the non-local hydrodynamic-type system and its soliton-like solutions;
J. Phys. A, Math. Theor. 45, No. 8, Article ID 085210, 12 p. (2012).

11. Vladimirov, V.A.; Ma̧czka, Cz.
On the stability of some exact solutions to the generalized convection-reaction-diffusion equation; Chaos Solitons Fractals 44, No. 9, 677-684 (2011).

Additional information:

None