Course detail

# Differential equations in electrical engineering

FEKT-NDREAcad. year: 2019/2020

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, 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.

Supervisor

Department

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.

Prerequisites

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

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Zill, Dennis, G., A first course in differential equations, 5. ed., PWS-Kent Publishing Company, 598 pp., 1993

Farlow, J. Stanley, An Introduction to Differential Equations and Their Applications,
McGraw-Hill, Inc., 609 pp., 1994

Amaranath, T, An Elementary Course in Partial Differential Equations, Narosa Publ. House, 1997.

I.P.Stavroulakis, S.A. Tersian, Partial Differential Equations, An Introduction with Mathematica and Maple, World Scientific, 2004, ISBN 981-238-815-X

Evans, G., Blackledge, J., Yardley, P., Analytic Methods for Partial Differential Equations, Springer, Inc., 1999.

DIBLÍK, J., BAŠTINEC, J., HLAVIČKOVÁ, I. Diferenciální rovnice a jejich použití v elektrotechnice. 1 vyd. Brno: FEKT VUT, 2005. s. 1 - 174 . ISBN MAT502

Haberman, R., Elementary Applied Partial Differential Equations, Prentice Hall, Inc., 1998.

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 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.

Language of instruction

English

Work placements

Not applicable.

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.

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, 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.

Classification of course in study plans

- Programme EEKR-MN Master's
branch MN-TIT , 1. year of study, winter semester, 5 credits, theoretical subject

branch MN-KAM , 1. year of study, winter semester, 5 credits, theoretical subject

branch MN-EVM , 1. year of study, winter semester, 5 credits, theoretical subject

branch MN-EST , 1. year of study, winter semester, 5 credits, theoretical subject

branch MN-MEL , 1. year of study, winter semester, 5 credits, theoretical subject

branch MN-SVE , 1. year of study, winter semester, 5 credits, theoretical subject

branch MN-EEN , 1. year of study, winter semester, 5 credits, theoretical subject

#### Type of course unit

eLearning

**eLearning:** currently opened course