Course detail

# Differential Equations in Electrical Engineering

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. The method of matrix exponential is applied to solutions of systems with constant coefficients. From the point of utilization, a large family 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, for systems with constant coefficients, e.g. Hurwitz's criterion and Michailov's criterion. Well-known method of Lyapunov functions, being the main method in stability theory, is discussed as well. Full classification of planar linear systems with constant coefficients is given in phase space. In the course is frequently used the matrix method and a lot of results are given in terms of matrices. Partial differential equations serve very often as mathematical models of technical and engineering phenomena. Except others applications of basic methods of solutions (Fourier method with utilization of Fourier series, D'Alembert method) will be applied to solving wave equation, heat equation and Laplace equation. Computer exercises focuse attention to master mathematical software for solving various classes of differential equations.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

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 (related to ordinary and partial differential) equations by use of these methods. Solving these problems by use of modern mathematical software. Main outcomes are:
1) Explicitely solution of basic types of ordinary differential equation of the first order (separated, linear, exact, Bernoulli, Cleiro).
2) Analysis of initial value problems and determining their solvability.
3) Construction of solution using the method of successive approximations.
4) Modeling of electrical curcuits by linear equations of higher-order and their solution.
5) Solution of systems of linear ordinary differential equations, if the fundamentakl system of solutions is known.
6) Solution of homogeneous linear systems of ordinary differential equations by method of eigenvectors and by method of exponential of a matrix.
7) Construction of particular solutions of non-homogeneous linear differential systems.
8) Determining stability of linear systems of differential equations with variable coefficients and with constant coefficients (correct application of stability criterions).
9) Solving of simple partial differential equatioons of the first order.
10) Applicatin of the method of characteristic and first integrals to solve partial differential equations of the first order.
11) Using D’Alembert method to solve linear partial differential equations of the second order.
12) Application of Fourier method to solve linear partial differential equations of the second-order.
13) Detailed construction of wave equation and heat equation.
14) Laplace partial differential equation and their solution.
15) Formulation of Dirichlet’s problem for linear partial second-order differential equations and its solution with utilization of Fourier series.

Prerequisites

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

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods include lectures and computer laboratories. On lectures and computer laboratories are demonstrated applications of differential equations in electrical engineering, related with the annotation of the course. Accent is put as well on self-study of students during semester, e.g. using of available literature as preparation for lectures and activity during lectures and computer laboratories

Assesment methods and criteria linked to learning outcomes

Abilities leading to successful solution of some typical classes of differential equations as well as necessary theoretical knowledge and its application will be positively estimated.
The final evaluation (examination) depends on assigned points (0 points is minimum, 100 points is maximum), 30 points is maximum points which can be assigned during exercises. Final examination is in written form and is estimated
as follows: 0- points is minimum, 70 points is maximum.

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. 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 with utilization of Fourier series.

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 (such as method of solution of linear systems with constant coefficients by the exponential of a matrix, methods for solution of some classes of partial differential equations - Fourier's method with utilization of Fourier series, D'Alembert's method), 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 successful final written test.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Kuben, J., Obyčejné dferenciální rovnice, VA Brno, 2004

Aramanovič, I.G., Lunc, G.L., Elsgolc, L.E., Funkcie komlexnej premennej, operátorový počet, teória stability, Alfa, Bratislava, SNTL Praha, 1973
Myslík, J., Elektrické obvody, BEN - Technická literatura, Praha 1997

Classification of course in study plans

• Programme AUDIO-P Master's

branch P-AUD , 1. year of study, winter semester, optional interdisciplinary

• Programme EEKR-M Master's

branch M-TIT , 1. year of study, winter semester, theoretical subject

• Programme EEKR-M1 Master's

branch M1-TIT , 1. year of study, winter semester, theoretical subject
branch M1-KAM , 1. year of study, winter semester, theoretical subject

• Programme EEKR-M Master's

branch M-KAM , 1. year of study, winter semester, theoretical subject
branch M-EVM , 1. year of study, winter semester, theoretical subject

• Programme EEKR-M1 Master's

branch M1-EVM , 1. year of study, winter semester, theoretical subject

• Programme EEKR-M Master's

branch M-EST , 1. year of study, winter semester, theoretical subject

• Programme EEKR-M1 Master's

branch M1-EST , 1. year of study, winter semester, theoretical subject
branch M1-MEL , 1. year of study, winter semester, theoretical subject

• Programme EEKR-M Master's

branch M-MEL , 1. year of study, winter semester, theoretical subject
branch M-SVE , 1. year of study, winter semester, theoretical subject

• Programme EEKR-M1 Master's

branch M1-SVE , 1. year of study, winter semester, theoretical subject
branch M1-EEN , 1. year of study, winter semester, theoretical subject

• Programme EEKR-M Master's

branch M-EEN , 1. year of study, winter semester, theoretical subject

• Programme EEKR-CZV lifelong learning

branch ET-CZV , 1. year of study, winter semester, theoretical subject

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

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 hours, compulsory

Teacher / Lecturer

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.