1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. Mathematics - drugiego stopnia z tyt. magistra
5. Partial Differential Equations

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Partial Differential Equations - opis przedmiotu

Cel przedmiotu

The main goal of this course is to acquire by the students a skill of solving initial-boundary value problems (IBVP) for the linear partial differential equations of the first and second order with the use of the the method of characteristics, the method of the separation of variables and the Fourier transform as well as to learn the basics of the theory of Sobolev spaces and the so-called weak formulations of IBVP for partial differential equations.

Wymagania wstępne

Zapisz zmiany

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

Zakres tematyczny

Lectures

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 characteristics. Cauchy-Kowalewski theorem.

3. Equations of the second order. A classification of the second order equations.

Elliptic equations - basic properties of the harmonic functions, the fundamental solution to Laplace's and Poisson's equations, maximum principles, Green's function for an elliptic equation.
Parabolic equations - the fundamental solution of the Cauchy problem for the heat equation, maximum principles, the method of the separation of variables.
Hyperbolic equations - D'Alembert formula, formulas for the solutions of the wave equation in higher dimensions, Duhamel's principle.

4. The 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. An 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.

Classes

Solving assignments related to the content presented in lectures, with a particular emphasis on practical applications of the learned concepts.

Laboratory

Solving assignments related to partial differential equations using a mathematical package.

Metody kształcenia

Traditional lectures; exercises during which students solve assignments; laboratory exercises in the computer lab.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

A final grade from the course is consisted of a grade from laboratory (25%), a grade from classes (25%) and a grade from the final exam (50%). To take a final exam, students must receive a positive grade from classes and to attain a pass in the course students are required to pass the final exam. A grade from both, classes and laboratory will be determined by the sum of points gained from two tests consisted of assignments of different degree of difficulty. A grade from the final exam, which consists of questions testing the student's theoretical knowledge, is determined by the sum of points gained for the answers to these questions.

Literatura podstawowa

L. Evans, Partial Differential Equations, second edition, American Mathematical Society, 2010

Literatura uzupełniająca

Uwagi

Zmodyfikowane przez dr Tomasz Małolepszy (ostatnia modyfikacja: 15-02-2024 18:10)