1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2019/2020
4. Mathematics - drugiego stopnia z tyt. magistra
5. Równania różniczkowe cząstkowe

Wygeneruj PDF dla tej strony

Równania różniczkowe cząstkowe - opis przedmiotu

Cel przedmiotu

The main aim of this course is to acquire by students skills to solve the initial-boundary value problems (IBVP) for linear PDE of first and second orders by the means of the method of the characteristics, the method of the separation of variables and Fourier transform. During that course students also will learn the basics of the theory of Sobolev spaces and so-called weak formulation of IBVP for some PDE.

Wymagania wstępne

Zapisz zmiany

Mathematical Analysis 1 and 2, Functional Analysis, Linear Algebra 1 and 2.

Zakres tematyczny

1. Basic definitions - linear, semilinear and nonlinear equations, Cauchy problems, the types of boundary problems, characteristic surfaces.
2. Equations of the first order. The method of the characteristics. Cauchy-Kowalewski theorem.
3. Equations of the second order. Classification of the second order equations.
a. Elliptic equations - basic properties of the harmonic functions, the fundamental solution to Laplace's and Poisson's equations, the maximum principles, Green's function for elliptic equation.
b. Parabolic equation - the fundamental solution of the Cauchy problem for the heat equation, the maximum principles, the method of the separation of variables.
c. Hyperbolic equations - D'Alembert formula, formulas for the solutions of the wave equation in higher dimensions, Duhamel's principle.
4. Fourier transform and its application in the theory of partial differential equations.
5. Elements of the theory of Sobolev spaces.
a. Weak derivatives.
b. Sobolev spaces.
c. Approximation of the elements of the Sobolev spaces by smooth functions.
d. Trace of the function.
e. Sobolev-type inequalities.
6. Weak solutions of the second order equations - the methods of Ritz and Galerkin.

Metody kształcenia

Traditional lectures; classes with the lists of exercises to solve by students; computer lab.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

Class and Laboratory: learning outcomes will be verified through two tests consisted of exercises of different degree of difficulty. A grade determined by the sum of points from these two tests is a basis of assessment.
Lecture: final exam. A grade determined by the sum of points from that exam is a basis of assessment.
A grade from the course is consisted of the grade from laboratory (25%), the grade from classes (25%) and the grade from the final exam (50%). To take a final exam, students must receive a positive grade from classes. To attain a pass in the course students are required to pass the final exam.

Literatura podstawowa

1. Warsztaty z Równań Różniczkowych Cząstkowych, pod red. naukową prof. dr. hab. P. Bilera, Torun, 2003.
2. Evans, L., Partial differential equations, AMS, 1998.
3. Marcinkowska, H., Dystrybucje, przestrzenie Sobolewa, równania różniczkowe, PWN, 1993.
4. Walter A. Strauss, Partial differential equations: an introduction, Wiley, New York 1992.

Literatura uzupełniająca

1. Strzelecki, P., Krótkie wprowadzenie do równań różniczkowych cząstkowych, Wydawnictwa Uniwersytetu Warszawskiego, 2006.

Uwagi

Zmodyfikowane przez dr Robert Dylewski, prof. UZ (ostatnia modyfikacja: 20-09-2019 11:45)