Course details

Differential Equations in Electrotechnics

DRE Acad. year 2005/2006 Winter semester 5 credits

Ordinary and partial differential equations. 1. Ordinary differential equations: typical properites of solutions of differential equations (geometrical meaning, notion of initial problem, unicity of solutions, singular points). Two basic classes of equations (separable and linear). The structure of solutions of linear equations of higher order and systems of linear differential equations. Applications of differential equations (in electrical engineering, second order equations, oscillations, resonance, damping, transient phenomena). Linear differential equations of the second order and special functions. Bessel's functions and Legendreas polynomials. Exponential of a matrix and its computation. Matrix solution (with the aid of matrix exponential) of linear systems with constant coefficients. The vector form of particular solution (as result of variation of constants method). Discussion and classification of solutions of a planar system with constant coefficients 2. Partial differential equations: Method of solution of some classes of firts order equations. Telegraph equation. The geometrical conception of solution. Basic (canonical) forms of second order differential equations. Characteristics. Solution of some classes of second order equations by separation method and by Fourier method (Laplace equation, heat equation, wave equation). The wave equation and D'Alembert method for solution. Illustrations of notions and methods by modern mathematical software. 1.Ordinary differential equations of the first order. (Basic notions. Existence and unicity of solutions. Basic classes of equations). 2. Ordinary differentiál equations of the n-th order. Qualitative theory. Linear equations od n-th order. 3. Systems of ordinary differential equations. Linear homogeneous and nonhomogeneous systems. Systems with constant coefficients. The vector form of solution. 4. Apllication of differential equations. Linear systems with noncontinuous coefficients. Mechanical systems. Van der Pool equations. 5. Partial differential equations. Classes of equations, canonical forms. Partial differential equations of the first order. Some methods of solution of second order partial differential equations.

Guarantor

Diblík Josef, prof. RNDr., DrSc. (MAT)

Language of instruction

Czech

Completion

Examination

Time span

39 hrs lectures
13 hrs pc labs

Department

Department of Mathematics (UMAT)

Subject specific learning outcomes and competences

The ability to orientate in the basic notions and problems of differential equations. Solving problems in the areas cited in the annotation above by use of these methods. Solving these problems by use of modern mathematical software.

Learning objectives

Differential equations are the base of many filds of engineering science. The purpose of this course is to develop the basic notion 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, but attention is focused also on possibilities for getting information concerning properties of solutions. Methods are illustrated on concrete electroengineering examples.

Prerequisite knowledge and skills

Rudiments of higher mathematics.

Study literature

Kuben, J., Obyčejné diferenciální rovnice, VA v Brně, katedra matematiky 1998 (skriptum).
Greguš, M., Švec, M., Šeda, V., Obyčajné diferenciálne rovnice, ALFA, Bratislava, 1985.
Angot, A., Užitá matematika pro elektrotechnické inženýry, SNTL, SVTL, 1972.
Kalas, J., Ráb, M., Obyčejné diferenciální rovnice, Masarykova universita, Brno, 1995.
Mayer, D., Úvod do teorie elektrických obvodů, SNTL, ALFA, 1978.
Amaranath, T, An Elementary Course in Partial Differential Equations, Narosa Publ. House, 1997.
Haberman, R., Elementary Applied Partial Differential Equations, Prentice Hall, Inc., 1998.
Evans, G., Blackledge, J., Yardley, P., Analytic Methods for Partial Differential Equations, Springer, Inc., 1999.

Fundamental literature

Kuben, J., Obyčejné diferenciální rovnice, VA v Brně, katedra matematiky 1998 (skriptum).
Greguš, M., Švec, M., Šeda, V., Obyčajné diferenciálne rovnice, ALFA, Bratislava, 1985.
Angot, A., Užitá matematika pro elektrotechnické inženýry, SNTL, SVTL, 1972.
Kalas, J., Ráb, M., Obyčejné diferenciální rovnice, Masarykova universita, Brno, 1995.
Mayer, D., Úvod do teorie elektrických obvodů, SNTL, ALFA, 1978.
Amaranath, T, An Elementary Course in Partial Differential Equations, Narosa Publ. House, 1997.
Haberman, R., Elementary Applied Partial Differential Equations, Prentice Hall, Inc., 1998.
Evans, G., Blackledge, J., Yardley, P., Analytic Methods for Partial Differential Equations, Springer, Inc., 1999.

Syllabus of lectures

Notion of differential equation. Classification and geometrical sense, directional field.
Existence and unicity of solutions. Illustrative examples.
Separable equations and linear equations of first order.
Higher order equations. Linear highre order equations with constant coefficients.
Variation of constants for higher order equations.
Apllications of highre order equations (oscillations, damping, transient phenomena).
Linear second order equations and special functions (Bessel's equation and Bessel's functions, Legendrea's equation and Legendrea's polynomials)
Systems of differential equations. The structure of solutions of linear systems, variation of constant method. Two-dimensional differential system. Classification of solutions.
Exponential of a matrix, its computation. Solution of linear systems with constant coefficients by exponential of a matrix.
Partial differential equations of the first order. Notion of solution. Telegraph equation.
Partial second order equations. Characteristics. Canonical forms.
Solution of some classes of partial differential equations by separation method and by Fourier method (Laplace equation, heat equation).
Wave equation and its solution by D'Alembert method.

Syllabus of computer exercises

Notion of differential equation. Classification and geometrical sense, directional field.
Existence and unicity of solutions. Illustrative examples.
Separable equations and linear equations of first order.
Higher order equations. Linear highre order equations with constant coefficients.
Variation of constants for higher order equations.
Apllications of highre order equations (oscillations, damping, transient phenomena).
Linear second order equations and special functions (Bessel's equation and Bessel's functions, Legendrea's equation and Legendrea's polynomials)
Systems of differential equations. The structure of solutions of linear systems, variation of constant method. Two-dimensional differential system. Classification of solutions.
Exponential of a matrix, its computation. Solution of linear systems with constant coefficients by exponential of a matrix.
Partial differential equations of the first order. Notion of solution. Telegraph equation.
Partial second order equations. Characteristics. Canonical forms.
Solution of some classes of partial differential equations by separation method and by Fourier method (Laplace equation, heat equation).
Wave equation and its solution by D'Alembert method.

Progress assessment

Study evaluation is based on marks obtained for specified items. Minimimum number of marks to pass is 50.

The activity in computer practices is evaluated by 0-15 points. The evaluation is, besides, performed in the form of test in the middle of semester (0-15 points). The resulting estimation is the sum of previous evaluations and the estimation of final written test (0-70 points).

Controlled instruction

Test in the middle of semester. Activity in computer practices.