Analysis 4

Completed courses Analysis 2a and Analysis 2b.

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.
Fourier and Laplace transforms: elementary properties, inverse formulas and applications.
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.

1. L. C. Evans: Partial differential equations, 2nd ed., Providence : American Mathematical Society, cop. 2010.
2. G. B. Folland: Introduction to partial differential equations, 2nd ed., Princeton : Princeton Univ. Press, cop. 1995.
3. F. John: Partial differential equations, 4th ed., New York : Springer, 1982.
4. F. Križanič: Parcialne diferencialne enačbe, Ljubljana : DMFA - založništvo, 2004.
5. Y. Pinchover, J. Rubinstein: An Introduction to Partial Differential Equations, CUP, Cambridge, 2005
6. I. N. Sneddon: Elements of partial differential equations, New York : McGraw-Hill, 1957.
7. A. Suhadolc: Integralske transformacije ; Integralske enačbe, 2. natis. - Ljubljana : Društvo matematikov, fizikov in astronomov Slovenije, 1994.
8. A. Suhadolc: Metrični prostor, Hilbertov prostor, Fourierova analiza, Laplaceova transformacija, Ljubljana : Društvo matematikov, fizikov in astronomov Slovenije, 1998.

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.

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.

Lectures, exercises, homework, consultations

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

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]