Partial Differential Equations (FSI-SPD)

Academic year 2024/2025
Supervisor: doc. Mgr. Zdeněk Opluštil, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:

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.
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.
Learning outcomes and competences:
 
Prerequisites:
 
Course contents:

The course deals with the following topics: 
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.
Teaching methods and criteria:
 
Assesment methods and criteria linked to learning outcomes:
 
Controlled participation in lessons:
 
Type of course unit:
    Lecture  13 × 2 hrs. compulsory                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture  
    Exercise 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.
Literature - fundamental:
1. V. J. Arsenin: Matematická fyzika, Alfa, Bratislava 1977
2. L. C. Evans: Partial Differential Equations, AMS, Providence 1998
3. W. E. Williams: Partial differential equations,
Literature - recommended:
1. J. Franců: Parciální diferenciální rovnice, skripta FSI VUT, CERM 2011
2. J. Franců: Obyčejné diferenciální rovnice a Příklady z ODR, http://www.mat.fme.vutbr.cz/home/francu
3. V. J. Arsenin: Matematická fyzika, Alfa, Bratislava 1977.
4. K. Rektorys: Přehled užité matematiky II., Prometheus 1995
5. J. Škrášek, Z. Tichý: Základy aplikované matematiky II, SNTL, Praha 1986
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-MAI-P full-time study --- no specialisation -- Cr,Ex 5 Compulsory 1 3 W