Course detail

Differential equations in electrical engineering

FEKT-NDREAcad. year: 2010/2011

This course is devoted to some important parts of differential equations - ordinary differential equations and partial differential equations which were not explained in the previous bachelor course. From the area of ordinary differential equations we mean e.g. so called exact equation which is a general type of equations representing large family of equations. Attention will be paid to extension of knowledge concerning linear systems including autonomous systems. From the point of utilization, a series of differential equations is important. Let us mention e.g. so called Bessel's or Laplace equations. One of the main notions in applications of differential equations is the notion of stability, which is included in the course. Several methods for detection of stability are discussed, e.g., the method of Lyapunov functions, being the main methid in stability theory. In the course is frequently used the matrix method and many results are given in terms of matrices. Partial equations serve very often as models of technical phenomena. Except other basic methods of solutions of so called wave equation, heat equation and Laplace equation will be presented. Computer exercises focuse attention to master modern mathematical software for solving various classes of differential equations.

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Diblík, DrSc.

Department

Department of Mathematics (UMAT)

Learning outcomes of the course unit

The ability to orientate in the basic notions and problems
of differential equations.
Solving problems in the areas cited in the annotation above
by use of these methods. Solving these problems by use of
modern mathematical software.

Prerequisites

The subject knowledge on the Bachelor´s degree level is requested.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Assesment methods and criteria linked to learning outcomes

Abilities leading to successful solution of some classes of differential equations as well as necessary theoretical knowledge will be positively estimated.

Course curriculum

I. Differential equations of the first order. Basic notions. Existence of solutions. Successive approximations. A summary of basic classes of analytically solvable differential equations of the first order. Higher-order equations. Solution of linear equations of the second-order with power series. Bessel’s equation and Bessel‘s functions.
II. Existence and unicity of solutions of systems differential equations of the first order. Linear systems of ordinary differential equations. General properties of solutions and the structure of family of all solutions. The transient matrix. Solving of initial problem with transient matrix. Linears systems with constant coefficients (homogeneous systems – eliminative method, method of characteristic values, application of the matrix exponential, Putzer’s algorithm, nonhomogeneous systems – method of undetermined coefficients, method of variation of constants). Characterization of circuits by linear systems.
III. Stability of solutions of systems of differential equations. Autonomous systems. Lyapunov direct method for autonomous systems. Lyapunov‘ functions. Lyapunov direct method for nonautonomous systems. Stability of linear systems. Hurwitz‘s criterion. Michailov‘s criterion. Stability by linear approximation. Phase analysis of linear two-dimensional autonomous system with constant coefficients, cases of stability.
IV. Limit cycles and periodic solutions. Poincaré-Bendixson’s theorem. Liénard-Levinson-Smith’s theorem. Aplication to nonlinear equations describing periodic state of electrical processes.
IV. Partial differential equations of the first-order. Initial problem. Simplest classes of equations. Characteristic system. Existence of solutions. General solution. First integrals. Pfaff’s equation.
V. Partial differential equations of the second-order. Classification of equations. Transformatin of variables. Wave equation, D’Alembert’s formula. Heat equation, Dirichlet’s problem. Laplace‘s equation. Fourier’s method of separated variables.

Work placements

Not applicable.

Aims

Differential equations are the base of many fields of engineering science. The purpose of this course is to develop the basic notions concerning the properties of solutions of differential equations and to give the basic techniques for solution of differential equations. In this course not only several exact solution methods are explained, but attention is focused also on possibilities for getting information concerning properties of solutions. Methods are illustrated on concrete electrical-engineering examples.

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

The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year. Necessary conditions for course-unit credit are - regular attendance, nonzero assessment of half-semester written test and final written test.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Amaranath, T, An Elementary Course in Partial Differential Equations, Narosa Publ. House, 1997.
DIBLÍK, J., BAŠTINEC, J., HLAVIČKOVÁ, I. Diferenciální rovnice a jejich použití v elektrotechnice. 1 vyd. Brno: FEKT VUT, 2005. s. 1 - 174 . ISBN MAT502
Farlow, J. Stanley, An Introduction to Differential Equations and Their Applications, McGraw-Hill, Inc., 609 pp., 1994
I.P.Stavroulakis, S.A. Tersian, Partial Differential Equations, An Introduction with Mathematica and Maple, World Scientific, 2004, ISBN 981-238-815-X
Zill, Dennis, G., A first course in differential equations, 5. ed., PWS-Kent Publishing Company, 598 pp., 1993

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme EECC-MN Master's

    branch MN-SVE , 1 year of study, winter semester, theoretical subject
    branch MN-KAM , 1 year of study, winter semester, theoretical subject
    branch MN-EEN , 1 year of study, winter semester, theoretical subject
    branch MN-TIT , 1 year of study, winter semester, theoretical subject
    branch MN-EVM , 1 year of study, winter semester, theoretical subject
    branch MN-EST , 1 year of study, winter semester, theoretical subject
    branch MN-MEL , 1 year of study, winter semester, theoretical subject

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

Syllabus

1. Differential equations of the first ordrer.
2. A summary of basic classes of analytically solvable differential equations of the first order.
3. Higher-order equations. Solution of linear equations of the second-order with power series. Bessel’s equation and Bessel‘s functions.
4. Systems of ordinary differential equations. Linear systems of ordinary differential equations. The transient matrix.
5. Linears systems with constant coefficients (homogeneous systems – eliminative method, method of characteristic values, application of the matrix exponential, Putzer’s algorithm, nonhomogeneous systems – method of undetermined coefficients, method of variation of constants). Characterization of circuits by linear systems.
6. Stability. Autonomous systems. Lyapunov‘ functions. Lyapunov direct method.
7. Stability of linear systems. Criteria of stability. Stability by linear approximation.
8. Phase analysis of linear two-dimensional autonomous system with constant coefficients.
9. Limit cycles and periodic solutions. Criteria of periodicity. Aplications.
10. Partial differential equations of the first-order.
11. Initial problem. Characteristic system. Existence of solutions. General solution. First integrals. Pfaff’s equation.
12. Partial differential equations of the second-order. Classification of equations. Transformatin of variables. Wave equation, D’Alembert’s formula. Heat equation, Dirichlet’s problem.
13. Laplace‘s equation. Fourier’s method of separated variables.
Demonstrating of notions and methods with modern mathematical software.

Exercise in computer lab

13 hod., compulsory

Teacher / Lecturer

prof. RNDr. Josef Diblík, DrSc.

Syllabus

Analytical solutions of differential equations of the firt-order equations and of the higher-order equations.
Directional fields of differential equations. Approximative solution of differential equations of the first and higher order. Characterization of circuits by differential equations.
Van der Pool's equation. Solution in the form of infinite series. Bessel's equation, Bessel's functions. Discussion of advantages and disadvantages of mathematical software. Phase trajectories of two dimensional dynamical system. Algorithms of solutions of linear systems with constant coefficients. Criteria of stability, software determination of stability. Partial differential equation of the first order. Using mathematical software for solution of basic classes of partial equations of the second-order.