Partial differential equations

2022/2023
Programme:
Practical Mathematics
Year:
3 year
Semester:
second
Kind:
mandatory
ECTS:
5
Language:
slovenian
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed courses Differential equations and Mathematics 2.

Content (Syllabus outline)

Partial differential equations
First order partial differential equations and method of characterisctics.
One and two-dimensional wave equation . Fourier method. D'Alembert solution.
Heat equation.
Laplace equation on two dimensions.
Classification of second order partial differential equations.
Laplace transform
Definition and properties. Convolution. Applications in PDE.
Calculus of Variations
The basic problem.
Euler-Lagrange equations.
Isoperimetric problem.

Readings

E. Kreyszig: Advanced Engineering Mathematics, deveta izdaja, Wiley Publ. Inc., New York, 2006.
F. Križanič, I. Vidav: Navadne diferencialne enačbe, parcialne diferencialne enačbe, variacijski račun. DMFA Slovenije, 1991.
E. Zakrajšek: Analiza III, 3. izdaja, DMFA založništvo, 2002.
J. Cimprič: Rešene naloge iz Analize III, DMFA založništvo, 2001

Objectives and competences

Acquiring knowledge of partial differential equations and calculus of variations. Ability to solve simple partial differential equations in mathematical physics. Solving
Optimization problems by using calculus of variations.

Intended learning outcomes

Knowledge and understanding
Understanding the concept of a partial differential equation. Solving specific types of partial differential equations analytically. Understanding the concept of calculus of variations.
Application
Formulation and sloving of certain mathematical and physical problems by using partial differential equations or calculus of variations.
Reflection
Undersanding the theory through applications.
Transferable skills
Identification of the problems and problem solving. Formulation of some nonmathematical problems in mathematical language.
Knowledge and use of literature.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Jasna Prezelj:
FORSTNERIČ, Franc, IVARSSON, Björn, KUTZSCHEBAUCH, Frank, PREZELJ-PERMAN, Jasna. An interpolation theorem for proper holomorphic embeddings. Mathematische Annalen, ISSN 0025-5831, 2007, bd. 338, hft. 3, str. 545-554. [COBISS-SI-ID 14335065]
PREZELJ-PERMAN, Jasna. A relative Oka-Grauert principle for holomorphic submersions over 1-convex spaces. Transactions of the American Mathematical Society, ISSN 0002-9947, 2010, vol. 362, no. 8, str. 4213-4228. [COBISS-SI-ID 15641433]
PREZELJ-PERMAN, Jasna, SLAPAR, Marko. The generalized Oka-Grauert principle for 1-convex manifolds. Michigan mathematical journal, ISSN 0026-2285, 2011, vol. 60, iss. 3, str. 495-506. [COBISS-SI-ID 16134745]
ČERNE, Miran, ZAJEC, Matej. Boundary differential relations for holomorphic functions on the disc. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 2, str. 473-484. [COBISS-SI-ID 15710553]
ČERNE, Miran, FLORES, Manuel. Generalized Ahlfors functions. Transactions of the American Mathematical Society, ISSN 0002-9947, 2007, vol. 359, no. 2, str. 671-686. [COBISS-SI-ID 14227801]
ČERNE, Miran, FLORES, Manuel. Quasilinear [overline{partial}]-equation on bordered Riemann surfaces. Mathematische Annalen, ISSN 0025-5831, 2006, vol. 335, no. 2, str. 379-403. [COBISS-SI-ID 13970777]