Methods of Mathematical Physics

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The course provides an introduction to the theory of distributions with applications to solutions of partial differential equations with constant coefficients, further the Fredholm theorems are discussed for the case of a continuous kernel on a compact set as well as Sturm-Liouville operators on bounded intervals, and applications of the separation of variables method to the solution of some boundary value problems and mixed problems.

Requirements:

Basic courses of Calculus and Linear Algebra (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2 held at the FNSPE CTU in Prague).

Syllabus of lectures:

1. Definition of spaces of distributions and basic operations, periodic distributions, tensor product and convolution. 2. Tempered distributions and the Fourier transformation. 3. The generalized Laplace transformation. 4. The Fredholm theorems for integral operators with continuous kernels on a compact set. 5. Elliptic operators, Sturm-Liouville operators on a bounded interval, the Green function. 6. Solutions of a boundary value problem for the Laplace equation on a symmetric domain. 7. Solutions of a mixed problem by the separation of variables method.

Syllabus of tutorials:

1. Classification of partial differential equations of second order. 2. Exercises focused on the calculus with distributions - the limit, the differential calculus, an expansion into a Fourier series. 3. Derivative of a piece-wise smooth function. 4. Fundamental solutions for the most common partial differential operators with constant coefficients. 5. Examples of convolutions. 6. Applications of the theory of distributions to solutions of ordinary differential equations with constant coefficients, the wave equation and the heat equation. 7. Examples on the generalized Fourier transformation. 8. Examples on the generalized Laplace transformation with applications. 9. Solutions of integral equations with degenerate kernels. 10. Solutions of the Sturm-Liouville equation with a right-hand side. 11. Solutions of a boundary value problem for the Laplace equation on a disk and a rectangle. 12. Solutions of a mixed problem by the separation of variables method.

Study Objective:

Knowledge of the theory of distributions including the Fourier transformation and the Laplace transformation, knowledge of basic results concerning the solvability of integral equations with continuous kernels (Fredholm's theorems), furthermore some basic results about elliptic operators, particularly about Sturm-Liouville operators. Skills to apply this knowledge in solving the most common problems with partial differential operators as well as in solving integral equations.

Study materials:

Key references: [1] V. S. Vladimirov: Equations of Mathematical Physics, (Marcel Dekker, New York, 1971); Recommended references: [2] P. Šťovíček: Methods of mathematical physics I, (in Czech, ČVUT, Praha, 2004), [3] L. Schwartz, Méthodes Mathematiques pour les Sciences Physiques, (Hermann, Paris, 1965)

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• BS Matematické inženýrství - Matematické modelování (compulsory course of the specialization, elective course)
• BS Matematické inženýrství - Matematická fyzika (compulsory course of the specialization, elective course)
• BS Matematické inženýrství - Aplikované matematicko-stochastické metody (elective course)
• BS Informatická fyzika (compulsory course of the specialization, elective course)
• BS Aplikace softwarového inženýrství (elective course)
• BS Aplikovaná informatika (elective course)
• BS jaderné inženýrství B (elective course)
• BS Jaderné inženýrství C (elective course)
• BS Dozimetrie a aplikace ionizujícího záření (elective course)
• BS Experimentální jaderná a částicová fyzika (compulsory course of the specialization, elective course)
• BS Inženýrství pevných látek (elective course)
• BS Diagnostika materiálů (elective course)
• BS Fyzika a technika termojaderné fúze (compulsory course of the specialization, elective course)
• BS Fyzikální elektronika (elective course)
• BS Jaderná chemie (elective course)