Equations of Mathematical Physics

Course guarantor:

Václav Klika, Matěj Tušek

Lecturer:

Václav Klika, Matěj Tušek

Tutor:

Václav Klika, 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:

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

Examination requirements: Knowledge and understanding of introduced terms, concepts, and results from the theory of generalized functions, integral transforms, integral equations, and selected partial differential equations.

Syllabus of lectures:

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

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

2. Theory of integral transformations - classical and generalized Fourier transformation, classical and generalized Laplace transform, applications.

3. Integral equations - integral operator and its properties, separable kernel of operator, sequential approximation method, iterated degenerate kernel method, Fredholm integral equations, Volterra integral equations.

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

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

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

6. Boundary value and mixed boundary problem for elliptic 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:

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

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

Note:

Time-table for winter semester 2025/2026:

Time-table is not available yet

Time-table for summer semester 2025/2026:

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)