The Equations of Mathematical Physics

Course guarantor:

Lecturer:

Tutor:

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:

Basic course of Calculus, Linear Algebra and selected topics in mathematical analysis (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2, 01VYMA held at the FNSPE

CTU in Prague).

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:

1. Hilbert space

2. Linear operators on Hilbert spaces

3. Integral equations

4. Partial differential equations

5. Theory of generalized functions

6. Laplace transform

7. Fourier transform

8. Fundamental solution of operators

9. Equations of mathematical physics

10. Eliptic differential equations

11. Mixed boundary problem

Study Objective:

Get acquainted with theory of generalized functions and its application to solving partial differential equations including mixed boundary problem.

Study materials:

Key References:

P. Stovícek: Methods of Mathematical Physics : Theory of generalized functions, CVUT, Praha, 2004. (in czech),

P. Stovícek: Methods of Mathematical Physics II: Theory of generalized functions II. Integral equations, elliptic operators, CVUT, Praha, 2017. (in czech),

V.S. Vladimirov : Equations of Mathematical Physics, Marcel Dekker, New York, 1971

Č. Burdík, O. Navrátil : Rovnice matematické fyziky, Česká technika - nakladatelství ČVUT, 2008

Recommended literature:

L. Schwartz - Mathematics for the Physical Sciences, Dover Publication, 2008

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

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: