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Course, academic year 2023/2024
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Introduction to Partial Differential Equations - NMMA339
Title: Úvod do parciálních diferenciálních rovnic
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2021
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. Mgr. Petr Kaplický, Ph.D.
Class: M Bc. OM
M Bc. OM > Zaměření MA
M Bc. OM > Zaměření NUMMOD
M Bc. OM > Povinně volitelné
Classification: Mathematics > Differential Equations, Potential Theory
Incompatibility : NMMA334
Interchangeability : NMMA334
Is pre-requisite for: NMMA351, NMNM351
In complex interchangeability with: NMMA334
In complex incompatibility with: NMMA334
Annotation -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (29.05.2019)
An introductory course in partial differential equations for bachelor's program in General Mathematics. Recommended for specializations Mathematical Analysis and Mathematical Modelling and Numerical Analysis.
Course completion requirements -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (03.10.2023)

Credit for the exercise is granted for elaboration of two homeworks and passing successfully a written test.

Literature - Czech
Last update: doc. Mgr. Petr Kaplický, Ph.D. (29.05.2019)
Základní studijní literatura a studijní pomůcky

L. C. Evans: Partial Differential Equations, AMS 1999

M. Renardy, R. C. Rogers: An introduction to partial differential equations, Springer 1993

Další studijní literatura a studijní pomůcky

O. John, J. Nečas: Rovnice matematické fyziky, SPN 1972

S. J. Farlow: PDE for Scientists and Engineers, Dover, 1993

F. Sauvigny: Partial Differential Equations 1, Foundations and Integral Representations, Springer, 2006

Requirements to the exam - Czech
Last update: doc. Mgr. Petr Kaplický, Ph.D. (29.05.2019)

Zkouška se skládá z písemné a ústní části. V písemné části bude testována početní technika v rozsahu probraném na cvičeních. Po úspěšném napsání písemky student postupuje k ústní části zkoušky. Zkoušena bude teorie v rozsahu vyložené látky včetně důkazů.

Syllabus -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (29.05.2019)

Basic examples of PDE's and their numerical solution by the finite difference method. Cauchy problem for a quasilinear PDE of the first order, transport equation, characteristics.

Von Neumann stability analysis of numerical schemes for Cauchy problems. Numerical solution of transport equation: CFL condition, upwinding, maximum principle, truncation error and approximation error, dissipation and dispersion.

Real analytic functions, Cauchy-Kowalevska Theorem, characteristic surfaces, classification of semilinear PDE's of the second order, transformation to canonical form.

Heat equation (fundamental solution, Cauchy problem, problem in bounded domain), wave equation (fundamental solution, Cauchy problem, energy methods).

Numerical solution of the mixed problem for heat equation: implicit and explicit schemes, theta-scheme, Fourier error analysis, maximum principle and convergence.

Relation between consistence, convergence and stability: general scheme for equations of the first order in time, Lax equivalence theorem.

Elliptic equations of the second order: fundamental solution of Laplace equation, Green's representation formula, Dirichlet problem for Laplace equation, mean value theorems, maximum principles.

Numerical solution of elliptic equations of the second order: approximation of general diffusion equation, derivation of schemes in irregular nodes, maximum principle and convergence.

Entry requirements -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (29.05.2019)

Knowledge of mathematical analysis on the level of obligatory courses recommended for the first two years of the study branch General Mathematics is expected.

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