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FEKT-MKC-DREAcad. year: 2024/2025
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.
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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.
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