616-3001/01 – Applied Mathematics (AM)
| Gurantor department | Department of Environmental Protection in Industry | Credits | 7 |
| Subject guarantor | Ing. Jan Bitta, Ph.D. | Subject version guarantor | Ing. Jan Bitta, Ph.D. |
| Study level | undergraduate or graduate | Requirement | Compulsory |
| Year | 1 | Semester | winter |
| | Study language | Czech |
| Year of introduction | 2014/2015 | Year of cancellation | 2019/2020 |
| Intended for the faculties | FMT, HGF | Intended for study types | Follow-up Master |
Subject aims expressed by acquired skills and competences
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
Teaching methods
Lectures
Tutorials
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.
Compulsory literature:
Stoer J., Bullirsch R.: Introduction to Numerical Analysis, Springer, l978
Recommended literature:
Stoer J., Bullirsch R.: Introduction to Numerical Analysis, Springer, l978
Way of continuous check of knowledge in the course of semester
E-learning
Other requirements
Students prepare 6 separate tasks covering the theoretical content of the subject.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
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.
8th 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.
9th 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.
11th 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).
12th 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.
13th 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.
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction