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MU:MU25014 Solution Meth. for Nonlin. PDE - Course Information

## MU25014 Solution Methods for Nonlinear Partial Differential Equations

**Mathematical Institute in Opava**

Winter 2021

**Extent and Intensity**- 2/2/0. 6 credit(s). Type of Completion: zk (examination).
**Teacher(s)**- RNDr. Hynek Baran, Ph.D. (lecturer)

RNDr. Petr Vojčák, Ph.D. (seminar tutor) **Guaranteed by**- doc. RNDr. Michal Marvan, CSc.

Mathematical Institute in Opava **Timetable**- Tue 14:45–16:20 205
- Timetable of Seminar Groups:

*P. Vojčák* **Prerequisites**(in Czech)- TYP_STUDIA ( N )
**Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Geometry and Global Analysis (programme MU, NMgr-M)
- Mathematical Analysis (programme MU, NMgr-M)
- Mathematical Modelling (programme MU, NMgr-M)

**Course objectives**- An overview of classical and modern methods to find exact solutions of nonlinear partial diferential equations and their systems.
**Syllabus**- 1. Transdformations of variables: point and contact transformations. Jet spaces.

2. Partial differential equations of first order. Complete solution, general solution, singular solution, characteristics, Mayer brackets.

3. Systems of equations and equations of higher order. Compatibility, power series solutions, convergence.

4. Ampere method.

5. Intermediate integrals. Darboux method.

6. Baecklund transformation, coverings. Permutability and nonlinear superposition.

7. Basic soliton equations and phenomenology of their solitons.

8. Zero curvature representations, Lax pairs, solution methods for soliton equations.

- 1. Transdformations of variables: point and contact transformations. Jet spaces.
**Literature**- D. Hilbert a R. Courant.
*Methods of Mathematical Physics, Vol. 2*. Wiley, 1989. info

*required literature*- E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its a V. B.
*Algebro-geometrical approach to nonlinear integrable equations*. info - C. Rogers a W. F. Shadwick.
*Bäcklund transformations and Their Applications*. Academic Press, New York, 1982. info - A. R. Forsyth.
*Theory of Differential Equations, Vol. 5, 6*. Cambridge Univ. Press, 1906. info

*recommended literature*- D. Hilbert a R. Courant.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- Study Materials

The course can also be completed outside the examination period. **Teacher's information**- The course comprises lectures and tutorials. To pass the course, the first step is to earn credit for tutorials by earning 70% on a written test.

The final exam comprises a written part and an oral part. The written part tests the ability to select and apply a suitable solution method. The written part is followed by the oral part, which tests the theoretical knowledge of the subject, including proofs.

- Enrolment Statistics (recent)

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