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Lecturer(s)
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Černá Dana, doc. RNDr. Ph.D.
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Course content
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Lectures: 1. Solution of nonlinear equations - Brower's fixed point theorem, the convergence of iterative methods, fixed-point iterations. 2. Bisection method. Newton's method - derivation, geometric interpretation, convergence theorem, and error estimation. Newton's method for systems of nonlinear equations. 3. Secant method - derivation, geometric interpretation, convergence theorem, and error estimation. 4. Numerical methods for integrals. Quadrature formula, order, Newton-Cotes formulas. Rectangular rule. 5. Trapezoidal and Simpson's rule, error estimation by half step method. 6. One-step methods for initial problems. Problem formulation, solution existence, conversion of the m-th order equation to a system, order, convergent and consistent methods, the relationship between total and relative discretization error. 7. Euler's method - derivation, error order, stability, error estimation by half step method. 8. Methods based on direct application of Taylor expansion, Runge-Kutta method. 9. Approximation of derivatives using differences - derivation, geometric interpretation, error estimates. 10. Finite difference method for ordinary differential equations - algorithm, properties of discretization matrices, operator stability, convergence theorem. 11. Classification of linear partial differential equations of the second order in two variables. Basic tasks of mathematical physics. 12. Finite difference method for elliptic differential equations. Poisson's equation, mesh construction, node types, algorithm derivation, discrete maximum principle, convergence, error. 13. Solving initial boundary value problems for parabolic differential equations. Finite difference method - mesh construction, explicit and implicit Euler scheme, Crank-Nicolson scheme, stability, error estimates. 14. Reserve. Exercises: The exercise is focused on the implementation of the discussed numerical methods in Matlab.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Class attendance
- 56 hours per semester
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Learning outcomes
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The subject is concerned with the analysis and implementation of numerical methods for interpolation, computation of integrals, solution of nonlinear equations, solution of ordinary differential equations with initial and boundary conditions, and solution of elliptic and parabolic partial differential equations.
Numerical methods for interpolation, computation of integrals, and solution of nonlinear equations. The solution of ordinary differential equations - initial value problems, boundary value problems. The solution of elliptic and parabolic partial differential equations.
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Prerequisites
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Condition of registration: Calculus 1, Calculus 2.
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Assessment methods and criteria
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Written exam
Carrying out two given semester works on computer. Written, consists of a theroetical and computational parts. Exam: Written, composed of the theoretical and computational part.
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Recommended literature
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Grossmann, Ch., Roos, H., Stynes, M.:. Numerical Treatment of Partial Differential Equations. Berlin, Springer, 2007. ISBN 978-3-540-71584-9.
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PRESS, W.H.:. Numerical Recipes 3rd Edition, The Art of Scientific Computing. 2007. ISBN 9780521880688.
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Stoer J., Bulirsch R.:. Introduction to Numerical Analysis. Springer. ISBN 0-387-95452-X.
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