Course detail

# Equations of Mathematical Physics I

Partial differential equations - preliminaries. First order equations.
Classification and canonical form of the second order equations Derivation of selected equations of mathematical physics, formulation of initial and boundary value problems.
Classical methods: method of characteristics, Fourier series method, integral transform method, Green function method. Maximum principles.
Properties of the solutions to elliptic, parabolic and hyperbolic equations.

Language of instruction

Czech

Number of ECTS credits

0

Mode of study

Not applicable.

Entry knowledge

Solution of algebraic equations and system of linear equations, differential and integral calculus of functions of one and more variables, ordinary differential equations.

Rules for evaluation and completion of the course

The examination consists of a practical and a theoretical part.
Practical part: solving examples of P.D.E.:
1) solution of the 1st order equation,
2) classification and transformation of the 2nd order equation to its canonical form,
3) formulation of an initial boundary value problem related to the physical setting
and finding its solution by means of the Fourier series method.
Theoretical part: 3 questions from the theory of P.D.E.
Absence has to be made up by self-study using lecture notes.

Aims

The aim of the subject is to provide students with the basic knowledge
of the partial differential equations, particularly equations of
mathematical physics, their basic properties, methods of solving them
and their application in mathematical modelling. Another goal is to teach
the students to formulate and solve the basic problems of mathematical physics.
Elements of the theory of P.D.E. and survey of their application in mathematical modelling. Ability to formulate mathematical model of the selected problems of mathematical physics and to compute the solution in some simple cases.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Arsenin, V. J.: Metody matematičeskoj fyziky i specialnyje funkcii, Nauka, Moskva 1974, překlad do slovenštiny: Matematická fyzika. Základné rovnice a špeciálne funkcie. Alfa, Bratislava, 1977. (RU)
Evans, L. C.: Partial differential equations, American Math. Society Providence 1998. (EN)
Sobolev, S. L.: Partial differential equations of mathematical physics Pergamon Press, Oxford 1964 (EN)
T. A. Bick: Elementary boundary value problems. Marcel Dekker, New York 1993 (EN)
Williams, W. E.: Partial Differential Equations, Clarendon Press, Oxford 1980. (EN)

J. Franců: Parciální diferenciální rovnice. Akad. nakl. CERM, Brno 2011 (CS)
V. J. Arsenin: Matematická fyzika, Alfa, Bratislava 1977 (SK)
Drábek, P., Holubová, G.: Elements of Partial Differential Equations, De Gruyter, Berlin, 2014 (EN)
Renardy, M., Rogers, R., C.: An introduction to partial differential equations, Springer, New York 2004. (EN)

eLearning

Classification of course in study plans

• Programme D-FIN-K Doctoral, 1. year of study, summer semester, recommended
• Programme D-FIN-P Doctoral, 1. year of study, summer semester, recommended

#### Type of course unit

Lecture

20 hours, optionally

Teacher / Lecturer

Syllabus

1 Introduction, 1st order equations.
2 Classification of 2nd order equations.
3-4 Derivation of selected equations of mathematical physics and formulation of initial and boundary value problems.
5 Method of characteristics.
6 Fourier series method.
7 Integral transform method.
8 Green function method.
9 Maximum principles and harmonic functions.
10 Survey of properties of the solutions to hyperbolic, parabolic and elliptic equations.

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