Equations of Mathematical Physics B

Course guarantor:

Václav Klika

Lecturer:

Václav Klika, Jan Novák

Tutor:

Václav Klika, Jan Novák

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.

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. Classical Integral Transforms Laplace and Fourier transforms, their properties, the „Ten Commandments“ of Fourier and Laplace transforms, and simple applications.

2. Introduction to Functional Analysis Function factor spaces, Hilbert spaces, properties of the inner product, orthonormal bases, Fourier series expansions, orthogonal polynomials, Hermitian operators.

3. Integral Equations Integral operators and their properties, separable kernel of the operator, method of successive approximations, method of iterated kernels, Fredholm integral equations, Volterra integral equations.

4. Second-Order Linear Partial Differential Equations (PDEs) Definitions, types of PDE eccentricity, transformation to canonical forms, PDE classification, Cauchy problem, classical problems of mathematical physics.

5. Theory of Generalized Functions Class of test functions, ultrafilter convergence, class of generalized functions, elementary operations on distributions, generalized functions with positive support, elementary introduction to tensor product and convolution.

6. Generalized Versions of Integral Transforms General overview of extending transforms to spaces of generalized functions.

7. Solving PDEs Fundamental solutions of operators, basic theorem on solving PDEs, derivation of general solutions to classical problems in mathematical physics.

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 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:

• Physical Engineering - Computational physics (PS)
• Quantum Technologies (compulsory course in the program)
• Nuclear and Particle Physics (compulsory course in the program)
• Mathematical Engineering - Mathematical Physics (PS)
• Physical Engineering - Plasma Physics and Thermonuclear Fusion (PS)
• Physical Engineering - Solid State Engineering (PS)
• Physical Engineering - Laser Technology and Photonics (PS)