Analysis 4

2022/2023
Programme:
Mathematics, First Cycle
Year:
3 year
Semester:
second
Kind:
mandatory
ECTS:
6
Language:
slovenian
Hours per week – 2. semester:
Lectures
3
Seminar
0
Tutorial
3
Lab
0
Prerequisites

Completed courses Analysis 2a and Analysis 2b.

Content (Syllabus outline)

Fundamental notions of the theory of partial differential equations. The order of the equation. Linear, semilinear, quasilinear and completely nonlinear equations. Classical solution. Cauchy problem.
First-order equations for functions of two variables.
Geometric interpretation. Characteristic curves. Pfaff equation. Existence of solutions.
Applications of Fourier and Laplace transforms.
Classification of second-order partial differential equations.
Canonical forms of elliptic, parabolic and hyperbolic equations.
Laplace equation. Harmonic functions.
Newton potential. Green identities.
Elementary properties of harmonic functions. The mean value principle. The maximum principle.
Liouville theorem. Fundamental solutions. Solutions of non-homogeneous Laplace equation. Dirichlet and Neumann problems. Green function. Poisson kernel.
Heat equation. Gauss kernel. Heat equations on a finite and on an infinite rod.
Wave equation. Wave equation on an infinite string. D'Alembert solution. Wave equation on a finite string.

Readings

Y. Pinchover, J. Rubinstein: An Introduction to Partial Differential Equations, CUP, Cambridge, 2005
L. C. Evans: Partial Differential Equations, AMS, Providence, 1998.
G. B. Folland: Introduction to Partial Differential Equations, 2nd edition, Princeton Univ. Press, Princeton, 1995.
F. John: Partial Differential Equations, 4th edition, Springer, New York, 1991.
F. Križanič: Parcialne diferencialne enačbe, DMFA-založništvo, Ljubljana, 2004.
N. Sneddon: Elements of Partial Differential Equations
Suhadolc: Integralske transformacije/Integralske enačbe, DMFA-založništvo, Ljubljana, 1994
A. Suhadolc: Metrični prostor, Hilbertov prostor, Fouriereva analiza, Laplaceova transformacija, DMFA-založništvo, Ljubljana, 1998.

Objectives and competences

Students get acquainted with the theory of partial differential equations in its connections to physics. First and second order equations for the functions of two independent variables are studied. In particular the Laplace, the heat and the wave equations are considered and elementary properties of their solutions are considered. The methods for solving these equations using Fourier series, Fourier transform and Green functions are presented.

Intended learning outcomes

Knowledge and understanding: Understanding of the notion of a partial differential equation and its solution. Skills for analytical solution of certain types of partial differential equations. Proficiency in the use of the Fourier and the Laplace transforms. Understanding of various properties of solutions of the second-order equations.
Application: Formulation of certain mathematical and non-mathematical problems in the form of partial differential equations. Solving these equations.
Reflection: Understanding of the theory by means of studying various examples and applications.
Transferable skills: Identification, formulation and solution of mathematical and non-mathematical problems by means of the theory of partial differential equations. Skills related to the use of literature in different languages.

Learning and teaching methods

Lectures, exercises, homework, 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

ČERNE, Miran. Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces. American journal of mathematics, ISSN 0002-9327, 2004, vol. 126, no. 1, str. 65-87. [COBISS-SI-ID 12895577]
ČERNE, Miran, FORSTNERIČ, Franc. Embedding some bordered Riemann surfaces in the affine plane. Mathematical research letters, ISSN 1073-2780, 2002, vol. 9, no. 5-6, str. 683-696. [COBISS-SI-ID 12391257]
ČERNE, Miran. Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration. Arkiv för matematik, ISSN 0004-2080, 2002, vol. 40, no. 1, str. 27-45. [COBISS-SI-ID 11623513]
SAKSIDA, Pavle. Lattices of Neumann oscillators and Maxwell-Bloch equations. Nonlinearity, ISSN 0951-7715, 2006, vol. 19, no. 3, str. 747-768. [COBISS-SI-ID 13932377]
SAKSIDA, Pavle. Maxwell-Bloch equations, C Neumann system and Kaluza-Klein theory. Journal of physics. A, Mathematical and general, ISSN 0305-4470, 2005, vol. 38, no. 48, str. 10321-10344. [COBISS-SI-ID 13802073]
SAKSIDA, Pavle. Nahm's equations and generalizations Neumann system. Proceedings of the London Mathematical Society, ISSN 0024-6115, 1999, let. 78, št. 3, str. 701-720. [COBISS-SI-ID 8853849]