Course detail

# Equations of Mathematical Physics I

FSI-9RF1Acad. year: 2016/2017Summer semesterNot applicable.. year of study1  credit

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.

Learning outcomes of the course unit

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.

Prerequisites

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

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

V. J. Arsenin: Matematická fyzika. Základné rovnice a špeciálne funkcie. Alfa, Bratislava, 1977
J. Franců: Parciální diferenciální rovnice. Akad. nakl. CERM, Brno 2011
T. A. Bick: Elementary boundary value problems. Marcel Dekker, New York 1993
V. J. Arsenin: Matematická fyzika, Alfa, Bratislava 1977
I. G. Petrovskij: Parciální diferenciální rovnice. Přir. vydavatelství, Praha 1952

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

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.

Language of instruction

Czech, English

Work placements

Not applicable.

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.

Specification of controlled education, way of implementation and compensation for absences

Absence has to be made up by self-study using lecture notes.

#### 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.