Equations of Mathematical Physics
Code  Completion  Credits  Range  Language 

01RMFM  Z,ZK  6  4P+2C  Czech 
 Garant předmětu:
 Václav Klika
 Lecturer:
 Václav Klika
 Tutor:
 Václav Klika, Juraj Kováč, 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:
 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:
[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:
 Timetable for winter semester 2023/2024:
 Timetable is not available yet
 Timetable for summer semester 2023/2024:
 Timetable is not available yet
 The course is a part of the following study plans:

 Radiolgická fyzika (compulsory course in the program)