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, D'Alembert method) will be applied to solving wave equation, heat equation and Laplace equation. Computer exercises focuse attention to master modern mathematical software for solving various classes of differential equations.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Entry knowledge

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

Rules for evaluation and completion of the course

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.
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.

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.
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-

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

DIBLÍK, J., BAŠTINEC, J., HLAVIČKOVÁ, I. Diferenciální rovnice a jejich použití v elektrotechnice. 1 vyd. Brno: FEKT VUT, 2005. s. 1 - 174 . ISBN MAT502 (CS)
DIBLÍK, J., PŘIBYL, O., Obyčejné diferenciální rovnice, Akademické vydavatelství Cerm, Brno, 150 str., 2004 (CS)
DIBLÍK, J., RŮŽIČKOVÁ, M., Obyčajné diferenciálne rovnice, EDIS Žilina, ŽU, 2008 (CS)
DIBLÍK, J., BAŠTINEC, J., HLAVIČKOVÁ, I., ŠMARDA, Z. Diferenciální rovnice a jejich použití v elektrotechnice, ESF, OPVK, FEKT, 2013, 1-225 (CS)

ANGOT, A., Užitá matematika pro elektrotechnické inženýry, SNTL, SVTL, 1972. (CS)
MAYER, D., Úvod do teorie elektrických obvodů, SNTL, ALFA, 1978. (CS)
AMARANATH, T., An Elementary Course in Partial Differential Equations, Narosa Publ. House, 1997. (EN)
EPSTEIN, M., Partial Differential Equations, Mathematical Techniques for Engineers, Springer, 2017 (k dispozici ve fakultní knihovně) (EN)
KESKIN, A. Ümit, Ordinary Differential Equations for Engineers, Problems with MATLAB Solutions, Springer, 2019 (k dispozici ve fakultní knihovně) (EN)
Wei-Chau Xie, Differential Equations for Engineers, Cambridge Univerzity Press, 2014 (k dispozici ve fakultní knihovně) (EN)

eLearning

Classification of course in study plans

• Programme MKC-TIT Master's, 1. year of study, winter semester, compulsory-optional
• Programme MKC-EEN Master's, 1. year of study, winter semester, compulsory-optional
• Programme MKC-EKT Master's, 1. year of study, winter semester, compulsory-optional
• Programme MKC-SVE Master's, 1. year of study, winter semester, compulsory-optional

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1) Solution of basic types of ordinary differential equations of the first order (equations with separable variables, linear equations, exact equations, Bernoulli equation, Cleiro equation).
2) Analysis of the initial problem and conditionms for its solvability.
3) Constructions of solutions by the method of successive approximations.
4) Modeling of circuits by linear differential equations of higher order and their solution.
5) Solution of systems of linear ordinary differential Equations if a fundamental system is given.

6) Solution of homogeneous linear ordinary differential systems with constant coefficients by methods of eigenvectors and an exponential of a matrix.
7) Constructions of particular solutions of nonhomogeneous linear differential systems.
8) Stability of linear differential systems with variable coefficients and with constant coefficients (an application of stability criteria).
9) Solution of basic partial differential equations of the first order.
10) Solution of partial differential equations of first order by the method of characteristic and first integrals.
11) Solution of linear partial differential equations of the second order by the method od D’Alembert.
12) Solution of linear partial differential equations of the second order by Fourier method.
13) Solution of the wave equation and the heat equation.
14) Laplace partial differential equation and its solution.
15) Dirichlet problem for linear partial differential equations of the second order and its solution.

Exercise in computer lab

13 hours, optionally

Teacher / Lecturer

Syllabus

1) Solution of basic types of ordinary differential equations of the first order (equations with separable variables, linear equations, exact equations, Bernoulli equation, Cleiro equation).
2) Analysis of the initial problem and conditionms for its solvability.
3) Constructions of solutions by the method of successive approximations.
4) Modeling of circuits by linear differential equations of higher order and their solution.
5) Solution of systems of linear ordinary differential Equations if a fundamental system is given.

6) Solution of homogeneous linear ordinary differential systems with constant coefficients by methods of eigenvectors and an exponential of a matrix.
7) Constructions of particular solutions of nonhomogeneous linear differential systems.
8) Stability of linear differential systems with variable coefficients and with constant coefficients (an application of stability criteria).
9) Solution of basic partial differential equations of the first order.
10) Solution of partial differential equations of first order by the method of characteristic and first integrals.
11) Solution of linear partial differential equations of the second order by the method od D’Alembert.
12) Solution of linear partial differential equations of the second order by Fourier method.
13) Solution of the wave equation and the heat equation.
14) Laplace partial differential equation and its solution.
15) Dirichlet problem for linear partial differential equations of the second order and its solution.

E-learning texts

Diblík J., Baštinec J. Hlavičková I., Šmarda Z.,Diferenciální rovnice a jejich použití v~elektrotechnice (cs)

eLearning