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# Course detail

## Differential Equations in Electrical Engineering

Course unit code: FEKT-LDRE
Type of course unit: theoretical subject
Level of course unit: Master's (2nd cycle)
Year of study: 1
Semester: winter
Number of ECTS credits:
 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.
 Mode of delivery: 20 % face-to-face, 80 % distance learning
 Prerequisites: The subject knowledge on the Bachelor´s degree level is requested.
 Co-requisites: Not applicable.
 Recommended optional programme components: Not applicable.
 Course contents (annotation): 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.
 Recommended or required reading: Diblík, J. a kolektiv, Diferenciální rovnice a jejich použití v elektrotechniceAramanovič, I.G., Lunc, G.L., Elsgolc, L.E., Funkcie komlexnej premennej, operátorový počet, teória stability, Alfa, Bratislava, SNTL Praha, 1973Kuben, J., Obyčejné dferenciální rovnice, VA Brno, 2004Myslík, J., Elektrické obvody, BEN - Technická literatura, Praha 1997Evans, G., Blackledge, J., Yardley, P., Analytic Methods for Partial Differential Equations, Springer, Inc., 1999.
 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: Czech
 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.

Type of course unit:

Lecture: 39 hours, optionally RNDr. Zdeněk Svoboda, CSc. 13 hours, compulsory RNDr. Zdeněk Svoboda, CSc.

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