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Partial Differential Equations

General data

Course ID: 0600-MS2-2RRC2 Erasmus code / ISCED: 11.105 / (0541) Mathematics
Course title: Partial Differential Equations Name in Polish: Równania różniczkowe cząstkowe
Department: (in Polish) Instytut Matematyki
Course groups: (in Polish) 2 rok 2 stopnia sem. zimowy Matematyka spec. Teoretyczna
(in Polish) 2L stac. II st. studia matematyki - przedmioty obowiązkowe
ECTS credit allocation (and other scores): 5.00
view allocation of credits
Language: Polish
Type of course:

obligatory courses

Short description:

Course objectives: Knowledge of fundamenta notions and theorems: 1. Cauchy-Kowalewska theorem. 2. Integration of first order quasi-linear and linear PDEs. First integrals. Hamiltonian systems. 3. Classification of second order PDEs. 4. Boundary value problems of different kinds. Well-posed boundary value problems. 5. Hiperbolic equations . Cauchy problem for wave equations. mixed boundary problem for wave equation. 6. Elliptic equations. Properties of harmonic functions. Green function and its properties. Solution of Dirichlet problem. 7. Parabolic equations. Heat equation. Maximum and minimum principles. existence and uniqueness theorem for solutions of Cauchy problem for heat equation. 8. Applications of knowledge to solving theoretical problems as well as to practical ones.

Full description:

Course profile: academic

Form of study: stationary

Course type: obligatory

Academic discipline: Mathematics, field of study in the arts and science: mathematics

Year: 2, semester: 3

Prerequisities: Differential Geometry, Functional Analysis

lecture 30 h. exercise class 30 h.

Verification methods: lectures, exercises, consultations, studying literature, home works, discussions in groups.

ECTS credits: 5

Balance of student workload:

attending lectures15x2h = 30h

attending exercise classes 15x2h = 30h

preparation for classes 7x3h = 21h

completing notes after exercises and lectures 7x2h = 14h

consultations 12x1h = 12h

the final examination: preparation.and take 12h + 3h = 15h

control works: repeating the material and preparation 3x4h = 12h

Quantitative description

Direct interaction with the teacher: 75 h., 3 ECTS

Practical exercises: 77 h., 3 ECTS

Learning outcomes:

Learning outcomes:

Student knows classification of first order partial differential equations (PDE), understands theorem of existence and uniqueness of solutions of Cauchy problem for quasilinear first order PDE. Student knows a notion of first integral; is able to construct general solution of problem for quasilinear first order PDEs by using characteristics. KA7_WG02, KA7_WG03, KA7_UW02, KA7_UW06

Student knows classification of first order PDEs and boudary value problems of different kinds; knows a notion of well-posed problem for mathematical physics equations and understands connection between equations and physical processes, described by them. Student is able to determine the type of PDE with two independent variables.KA7_WG02, KA7_WG04, KA7_WG06, KA7_UW06, KA7_UW10

Student knows a canonical form of hiperbolic PDE, methods of wave propagation, d'Alembert formula and understand Kirchoff formula. Student is able to use these formulas in simple examples. KA7_WG02, KA7_WG06, KA7_UW06

Student knows fundamental solution of Laplace equation, properties of harmonic functions, notion of Green function and its applications.KA7_WG02, KA7_WG03, KA7_UW01, KA7_UW06

Student understands maximum pronciple and uniqueness of solution in boundary value problem for heat equation with two independent variables. Student knows fundamental solution and formula for solutions in Cauchy problem for heat equation.KA7_WG02, KA7_WG04, KA7_WG05, KA7_UW06, KA7_UW10

Student obtains basic practice in creative development of theory of differential equations. KA7_KK01, KA7_KK02, KA7_KK07, KA7_UU01

Assessment methods and assessment criteria:

The overall form of credit for the course: final exam

Classes in period "Academic year 2019/2020" (in progress)

Time span: 2019-10-01 - 2020-06-30
Choosen plan division:


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Type of class: Class, 30 hours more information
Lecture, 30 hours more information
Coordinators: Tomasz Czyżycki
Group instructors: Tomasz Czyżycki
Students list: (inaccessible to you)
Examination: Course - Examination
Class - Grading
Type of course:

obligatory courses

Mode:

(in Polish) w sali

Short description: (in Polish)

Założenia i cele przedmiotu: Znajomość podstawowych pojęć i twierdzeń: 1. Twierdzenie Cauchy'ego-Kowalewskiej. 2. Całkowanie liniowych i quasi-liniowych równań I rzędu. Całki pierwsze. Układy Hamiltonowskie. 3. Klasyfikacja równań różniczkowych cząstkowych II rzędu. 4. Zagadnienia graniczne i ich rodzaje. Zagadnienia graniczne poprawnie postawione. 5. Równanie typu hiperbolicznego. Zagadnienie Cauchy'ego dla równania falowego. Zagadnienie mieszane dla równania falowego. 6. Równania typu eliptycznego. Własności funkcji harmonicznych. Funkcja Greena i jej własności. Rozwiązanie zagadnienia Dirichleta. 7. Równania typu parabolicznego. Równanie przewodnictwa ciepła. Zasada ekstremum. Twierdzenie o istnieniu Cauchy'ego równania przewodnictwa ciepła. 8. Stosowanie zdobytej wiedzy zarówno do rozwiązywania zagadnień teoretycznych, jak i zagadnień praktycznych.

Bibliography: (in Polish)

1. J. Wolska–Bochenek, A. Borzymowski, J.Chmaj, M.Tryjarska „Zarys teorii równań całkowych i równań różniczkowych cząstkowych” PWN Warszawa 1973

2. M. Krzyżański „Równania różniczkowe cząstkowe rzędu II” cz. I i II PWN Warszawa 1979

3. M.M. Smirnow „Zadania z równań różniczkowych cząstkowych” PWN Warszawa 1987

4. K. Bieńkowska–Lipińska „Wybrane zagadnienia równań fizyki matematycznej” PWN Warszawa 1992

5. Budak, Samarski „Równania fizyki matematycznej” PWN Warszawa 1983

6. F. Bierski „Równania różniczkowe cząstkowe” Wyd. AGH 1986

7. L. Evans „Równania różniczkowe cząstkowe” American Mathematical Soc., 2010, tłum. PWN Warszawa 2012

8. Bicadze „Zadania z równań fizyki matematycznej” PWN Warszawa 1980

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