Course detail

# Differential Equations in Electrical Engineering

FEKT-MDREAcad. year: 2016/2017

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

Number of ECTS credits

Mode of study

Guarantor

Department

Learning outcomes of the course unit

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

Co-requisites

Planned learning activities and teaching methods

Assesment methods and criteria linked to learning outcomes

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

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

Aims

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

Recommended optional programme components

Prerequisites and corequisites

Basic literature

Recommended reading

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

Teacher / Lecturer

Syllabus

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

Teacher / Lecturer

Syllabus

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.