Course detail

# Partial Differential Equations

FSI-SPDAcad. year: 2020/2021

The course deals with the following topics: Ordinary differential equations - a brief survay of material studied within the 3rd semester subject and extending of the subject matter (stability of the solution, autonomous equations and systems, trajectories, boundary value problems).

Partial differential equations - basic concepts. The first-order equations. The Cauchy problem for the k-th order equation. Transformation, classification and canonical form of the second-order equations.

Derivation of selected equations of mathematical physics (heat conduction, wave equation, variational prinsiple), formulation of initial and boundary value problems.

The classical methods: method of characteristics, The Fourier series method, integral transform method, the Green function method. Maximum principles. Properties of the solutions to the elliptic, parabolic and hyperbolic equations.

Supervisor

Department

Learning outcomes of the course unit

Revision and deepening of the knowledge of Ordinary Differential Equations. Elements of the theory of Partial Differential Equations and survey of their application to the mathematical modelling. Ability to formulate mathematical model of the selected problems of mathematical physics and to compute the solution or propose an algorithm for numerical solution.

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.

Recommended or required reading

V. J. Arsenin: Matematická fyzika, Alfa, Bratislava 1977

J. Franců: Parciální diferenciální rovnice, skripta FSI VUT, CERM 2011 (CS)

J. Franců: Obyčejné diferenciální rovnice a Příklady z ODR, http://www.mat.fme.vutbr.cz/home/francu (CS)

L. C. Evans: Partial Differential Equations, AMS, Providence 1998

W. E. Williams: Partial differential equations,

V. J. Arsenin: Matematická fyzika, Alfa, Bratislava 1977. (SK)

K. Rektorys: Přehled užité matematiky II., Prometheus 1995 (CS)

J. Škrášek, Z. Tichý: Základy aplikované matematiky II, SNTL, Praha 1986 (CS)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Course-unit credit is awarded on condition of having attended the seminars actively and passed two control tests:

Control test 1: O.D.E.: (a) solution of the 1st order equation, (b) solution of the 2nd order linear equation, (c) solution of a system of linear equations - stability, trajectories.

Control test 2: P.D.E.: (a) solution of the 1st order equation, (b) classification, and transformation of the 2nd order equation to its canonical form, (c) formulation of an initial boundary value problem related to the physical setting and finding its solution by means of the Fourier series method.

The examination consists of a practical and a theoretical part. Practical part: solving examples of P.D.E., see Control test 2. Theoretical part: theory of O.D.E. and P.D.E. (1 + 3 questions).

Language of instruction

Czech

Work placements

Not applicable.

Aims

After completing knowledge of ordinary differential equations the aim of the subject is to provide students with the basic knowledge of the partial differential equations, their basic properties, methods of solving them, and their application in mathematical modelling. Another goal is to teach the students to formulate and solve simple problems for mathematical physics equations.

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

Absence has to be made up by self-study using lecture notes. Passing the control tests is required, in cases of bad result or absence in additional term.

#### Type of course unit

Lecture

26 hours, compulsory

Teacher / Lecturer

Syllabus

1 Revision of O.D.E. - 1st order equations and higher order linear equations.

2 Systems of linear O.D.E., stability, existence and uniqueness of the solution.

3 Autonomous systems, trajectories and classification of singular trajectories.

4 Elements of P.D.E., 1st order equations.

5 The Cauchy problem, classification of 2nd order equations.

6 Derivation of selected equations of mathematical physics: heat equation.

7 Derivation of the equation of string vibration, wave equations.

8 Derivation of membrane equation via variational principle.

9 Method of characteristics for 1D wave equation.

10 Fourier series method.

11 Integral transform method.

12 Green function method and the maximum principles.

13 Properties of the solutions, reserve.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

1 O.D.E., solution of the 1st order equations and higher order linear equations.

2 Solution of systems of linear O.D.E., stability of the solution.

3 The phase portrait of solutions to autonomous system.

4 P.D.E., solving of the 1st order equations.

5 Written test 1, classification of 2nd order equations.

6 Formulation of problems related to the heat equation.

7 Formulation of problems related to the wave equation.

8 Derivation of membrane equation via variational principle.

9 Solving problems by the method of characteristics.

10 Solving problems by the Fourier series method.

11 Written test 2.

12 Using the Green function method, harmonic functions.

13 Properties of the solutions, course-credits.

eLearning

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