Equations of Mathematical Physics

Course guarantor:

Václav Klika

Lecturer:

Václav Klika

Tutor:

Lukáš Heriban, Václav Klika, Filip Konopka, Matěj Tušek

Supervisor:

Department of Mathematics

Synopsis:

The subject of this course is solving integral equations, theory of generalized functions, classification of partial

differential equations, theory of integral transformations, and solution of partial differential equations (boundary value

problem for eliptic PDE, mixed boundary problem for eliptic PDE).

Requirements:

All courses of linear algebra and mathematical analysis (calculus) including Lebesgue integration. Some understanding of functional analysis is beneficial but not crucial.

Syllabus of lectures:

1. Introduction to functional analysis - factor space, Hilbert space, scalar product, orthonormal basis, fourier series,

orthogonal polynoms, hermite operators, operator spectrum and its properties, bounded operators, continuous operators,

eliptic operators

2. Integral equations - integral operator and its properties, separable kernel of operator, sequential approximation

method, iterated degenerate kernel method, Fredholm integral equations, Volterra integral equations.

3. Classification of partial differential equations - definitions, types of PDE, transformations of partial differential

equations into normal form, classification of PDE, equations of mathematical physics.

4. Theory of generalized functions - test functions, generalized functions, elementary operations in distributions,

generalized functions with positive support, tensor product and convolution, temepered distributions.

5. Theory of integral transformations - classical and generalized Fourier transformation, classical and generalized

Laplace transform, applications.

6. Solving differential equations - fundamental solution of operators, solutions of problems of mathematical physics.

7. Boundary value problem for eliptic partial differential equation.

8. Mixed boundary problem for eliptic partial differential equation.

Syllabus of tutorials:

Study Objective:

Study materials:

Key references:

[1] A. G. Webster, Partial Differential Equations of Mathematical Physics, Second Edition, Dover, New York, 2016

[2] A. Tikhonov, A. Samarskii: Equations of Mathematical Physics, Courier Corp., Science, 2013

Recommended literature:

[5] L. Schwartz: Mathematics for the Physical Sciences, Dover Publication, 2008.

[6] I. M. Gel'fand, G. E. Shilov: Generalized Functions. Volume I: Properties and Operations, Birkhäuser Boston, 2004.

Note:

Time-table for winter semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue	roomTR:211 Kováč J. 10:00–12:50 (parallel nr.201) Trojanova 13
Wed	roomTR:201 Klika V. 14:00–15:50 (lecture parallel1) Trojanova 13
Thu	roomTR:201 Klika V. 12:00–13:50 (lecture parallel1) Trojanova 13
Fri	roomTR:208 Tušek M. 08:00–09:50 (parallel nr.101) Trojanova 13 roomBR:10 Konopka F. 12:00–13:50 (parallel nr.103) Břehová 7 roomTR:209 Konopka F. 08:00–09:50 (parallel nr.102) Trojanova 13 roomTR:210 Heriban L. 08:00–09:50 (parallel nr.104) Trojanova 13

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Matematické inženýrství - Matematická fyzika (PS)
• Matematické inženýrství - Matematické modelování (PS)