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Lecturer(s)
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Calábek Pavel, RNDr. Ph.D.
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Course content
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1. Fundametals of the matrix analysis. Matrix norm, characteristic equation, eigenvalues. Matrix sequences and series, Cayley-Hamilton theorem. 2. Existence and uniqueness of solution for an initial value problem. Lipschitz condition, Picard iterations. Solution to differential equations by means of power series. 3. Homogeneous systems of linear differential equations. Fundamental systems, fundamental matrices, the Jacobi formula, the fundamental matrix theorem. 4. Selected parts from the theory of linear differential equations. Adjoint systems, theorem about fundametal matrix of adjoint system, self-adjoint systems. Nonhomogeneous systems of linear differential equations, method of variation of parameters. 5. Linear systems with constant coefficients. Characteristic equations, canonical form of a matrix. Calculation exp(At), using in solving systems. The normal solution. The Putzer method. 6. Linear differential equations of the n-order, relation to systems of n differential equations of the first order. Wronskian, Lieuvill formula. Operator methods. 7. Phase plains, phase curves. Saddle points, focuses, centers, nodes. Analysis of systems of two differential equations. Stability and asymptotic stability. 1. Complex plane, extended Gauss plane. 2. Functions of a complex variable (limit, continuity). 3. Derivative of functions of a complex variable (Cauchy-Riemann conditions). 4. Holomorphic functions. 5. Conformal mapping. 6. Elementary functions of a complex variable. 7. Sequences and series of functions, power series. 8. Plane curves. 9. Integrals of functions of a complex variable. 10. Cauchy theorem, Cauchy integral formula. 11. Primitive functions. 12. Taylor series. 13. Zero points of holomorphic functions. 14. Isolated singularities. 15. Laurent series. 16. Residue, residue theorem and its application.
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
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Learning outcomes
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Summary of topics of diferential equations and complex calculus
5. Synthesis Students summarise and deepen their knowledges from the theory of systems of differential equations.
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Prerequisites
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unspecified
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Assessment methods and criteria
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Oral exam, Didactic Test
Credit: active participation, homework. Exam: written test, the student has to understand the subject and prove principal results.
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Recommended literature
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J. B. Conway. (1984). Functions of One Complex Variable. Springer New York Inc.
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J. Kalas, M. Ráb. (1995). Obyčejné diferenciální rovnice. Brno.
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M. Greguš, M. Švec, V. Šeda. (1985). Obyčajné diferenciálne rovnice. Alfa, SNTL.
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P. Hartman. (1964). Ordinary Differential Equations. John Wiley and Sons, New York.
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Zeman, J. (1998). Úvod do komplexní analýzy. VUP Olomouc.
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