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Applied Mathematics

* Exchange students do not have to consider this information when selecting suitable courses for an exchange stay.

Course Unit Code616-3031/01
Number of ECTS Credits Allocated6 ECTS credits
Type of Course Unit *Compulsory
Level of Course Unit *Second Cycle
Year of Study *First Year
Semester when the Course Unit is deliveredWinter Semester
Mode of DeliveryFace-to-face
Language of InstructionCzech
Prerequisites and Co-Requisites There are no prerequisites or co-requisites for this course unit
Name of Lecturer(s)Personal IDName
BIT02RNDr. Jan Bitta, Ph.D.
Summary
Applied mathematics is a subject of mathematics concerned with the study of those areas of mathematics, which are used as a convenient tool in a non-mathematical field. Develops mathematical methods used outside mathematics itself, specifies the way in which such methods can be used, and is responsible for the accuracy of their results.
Learning Outcomes of the Course Unit
- Knowledge of capabilities, limitations and errors of mathematical models and numerical calculations
- Knowledge of basic methods for analytical and numerical computation solving initial value problems for ordinary differential equations
- Knowledge of basic methods for analytical and numerical computation solving boundary value problems for ordinary differential equations
- Knowledge of basic methods for analytical and numerical computation solving boundary value problems for partial differential equations
- Ability to apply the acquired knowledge on numerical algorithms
Course Contents
1. Introduction. Modelling. Physical and abstract models. Mathematical modeling, computational mathematics. Errors in mathematical models. Methods of verification models. A priori and a posteriori error estimation. The use of mathematical models in practice.
2. Basic concepts in transmission phenomena 1st - scalars, vectors - Cartesian and geometric representation of vectors, vector spaces, dimension of vector spaces. Scalar, vector and tensor product of vectors and their geometric meanings. Nut - phenomena may be represented by matrices, singular and regular matrix, determinant of a matrix and its geometrical meaning, transposition of matrices, symmetric matrices, eigenvalues and eigenvectors of a matrix.
3. Basic concepts in transmission phenomena 2nd - tensors, tensor relationship to vectors and matrices, fundamentals of tensor calculus, differential operations with tensors. Basics of field theory - scalar and vector fields, gradient of a scalar field, divergence and curl of a vector field and its geometric meaning, scalar potential vector field.
4. Initial problems for ordinary differential equations - Formulation of the initial problem for the equation of the first order. Formulation of the initial value problems for systems of equations of the first order. Formulation of the initial tasks for the equation n-th order and its conversion to a system of equations of the 1st order. The theorem on the existence of solutions to the initial task of the 1st order. Relations between Lipschitz, continuous and continuously diferencovatelnými functions. Properties of solutions of initial value problems of the 1st order.
5. Analytical methods of solving ODE - Method of direct integration method of separation of variables, linear equations of n-th order and systems of linear equations. Distinctive features, fundamental solutions, method of variation of constants.
6. Numerical methods for solving ODE 1 - Discretization tasks. The general scheme of numerical solution of ODE. Explicit and implicit methods. Single- and multi-step methods. Euler method - explicit, implicit, trapezoid. Interpolation of functions - Lagrange interpolation, Hermite interpolation.
7. Numerical methods for solving ODE 2 - Consistency, stability and convergence of numerical schemes. The speed of convergence problems. Conditionality jobs. Computer representation of numbers. Rounding errors. Convergence and stability of processes based on Euler's method. Methods type predictor-corrector. Methods of Runge-Kutta type.
8. Boundary value problems for ordinary differential equations 1 - Formulation of boundary value problems. Boundary conditions - Dirichlet, Neumann boundary condition, Newton boundary condition, samoadjugovaný shape linear 2nd order ODE. Orthogonal functions and their basic properties. Fourier series. Homogeneous boundary value problems. Eigenvalues and functions homogeneous boundary value problems. Analytical solutions through direct integration. Fourier method.
9. Boundary value problems for ordinary differential equations 2 - Numerical solution of boundary value problems using the finite difference method (finite difference method). Numerical methods diagram representation of boundary conditions. Features moment matrices. Special Gaussian elimination for three-band matrix. Convergence Methods networks. Stationary one-dimensional heat conduction in a rod, a plate, a cylinder and a sphere. Limitations networks.
10. Marginal problems for ordinary differential equations 3 - Finite element method for boundary value problems of ODE. Weak formulation of the problem, Galerkinovské approximation, Courantova base. Features stiffness matrix. Convergence of the finite element method. Adaptive refinement grid computing.
11. Boundary value problems for partial differential equations 1 - Formulation of boundary value problems for PDR. Linear partial differential equations of second order and their classification. Boundary conditions for PDR. Method of separation of variables (Fourier method). Method combination of variables in parabolic problems. The method of fundamental solution (Green's function).
12. Boundary value problems for partial differential equations 2 - Numerical solution of boundary value problems for PDR finite difference method (finite difference method). Requirements for computer network to ensure convergence problems. Numerical solution of boundary value problems for PDE finite element method. Time-varying tasks, time cuts method for parabolic problem. Time slices type - explicit, implicit, Crank-Nicholson.
13. Boundary value problems for partial differential equations 3 - Application of analytical and numerical techniques for the job - unsteady conduction in poloomezeném body. Symmetric and asymmetric heating plate limited. Heating plate final thickness. Solving boundary value problems of diffusion equations.
Recommended or Required Reading
Required Reading:
STOER, Josef a Roland BULIRSCH. Introduction to numerical analysis. 3rd ed. New York: Springer, c2002. ISBN 0-387-95452-X.
BIRD, R. Byron, Warren E. STEWART a Edwin N. LIGHTFOOT. Přenosové jevy: sdílení hybnosti, energie a hmoty. Přeložil Štefan ŠALAMON, přeložil Vladimír MÍKA. Praha: Academia, 1968.
DONT, Miroslav. Úvod do parciálních diferenciálních rovnic. Praha: České vysoké učení technické, 1997. ISBN 80-01-01676-5.
FEISTAUER M. Diskrétní metody řešení diferenciálních rovnic. Praha: SPN, 1981.
Recommended Reading:
STOER, Josef a Roland BULIRSCH. Introduction to numerical analysis. 3rd ed. New York: Springer, c2002. ISBN 0-387-95452-X.
JENSON, Victor George a Godfrey Vaughan JEFFREYS. Matematické metódy v chemickom inžinierstve. Bratislava: Alfa, c1973.
KUBÍČEK, Milan. Numerické algoritmy řešení chemickoinženýrských úloh. Praha: SNTL - Nakladatelství technické literatury, 1983.
ŠKRÁŠEK, Josef a Zdeněk TICHÝ. Základy aplikované matematiky II: integrální počet, nekonečné řady, diferenciální geometrie, obyčejné a
parciální diferenciální rovnice, funkce komplexní proměnné, Laplaceova transformace, diferenční rovnice. Praha: SNTL - Nakladatelství
technické literatury, 1986.
REKTORYS, Karel. Přehled užité matematiky II. 6. přeprac. vyd. Praha: Prometheus, 1995. ISBN 80-85849-62-3.
Planned learning activities and teaching methods
Lectures, Tutorials
Assesment methods and criteria
Task TitleTask TypeMaximum Number of Points
(Act. for Subtasks)
Minimum Number of Points for Task Passing
Credit and ExaminationCredit and Examination100 (100)51
        CreditCredit40 10
        ExaminationExamination60 10