Analysis 4

Completed courses Analysis 1, Analysis 2 and Analysis 3.

Difference equations
Linear diference equations. Stability. Examples from economics.

Differential equations
First order differential equations. Phase space. Vector fields and integral curves. Examples. Singular solutions. Population models.

Existence and dependence on parameters
General solution. Existence and uniqueness. Picard's method. Newton's solution.

Systems of first order linear differential equations
Linearization. Constant coefficient systems. Higher order ODE with constant coefficients. First integrals. Singular points. Stability.

Nonlinear systems
Classification of the critical points. Stability and Hartman – Grobman theorem. The Lotka-Volterra model.

Variational calculus
Examples. Euler-Lagrange equation. Free boudary conditions. Examples from economy.

Laplace transform
Elementary properties. Inverse formula. Examples.

First order partial differential equations
Quasilinear PDE. Characteristics method. Characteristic system. Nonlinear PDE and characteristic equations.

Second order PDE
Basics of Sturm–Liouville theory. Vibration of a string. Wave equation and D'Alembert solution. Laplace equation. Heat equation, heat kernel and random walks. Separation of variables. Solving PDE using integral transforms. Black-Sholes equation. Classification of second order PDE in two variables.

M.Braun: Differential equations and their applications, Springer, 1992
L. Perko: Differential Equations and Dynamical Systems, 3rd edition, Springer, New York, 2004.
E. Zakrajšek: Analiza III, DMFA-založništvo, Ljubljana, 2002.
F. Križanič: Navadne diferencialne enačbe in variacijski račun, DZS, Ljubljana, 1974.
Pinchover and Rubinstein: An introduction to partial differential equations,Cambridge University press, 2005
F. Križanič: Parcialne diferencialne enačbe, DMFA-založništvo, Ljubljana, 2004.

The aim is that student understands the concept of differential equation and its solution and is able to solve special types of differential equations with emphasis on linear ones. Basic topics of variational calculus and examples from economy are represented.

Knowledge and understanding:
Understanding the concept of differential equation and its solution. Solving special type of differential equations. Modeling of economic problems. Understanding the concept of variational calculus. Understanding the concept of partial differential equation and especially heat equation.

Lectures, tutorial, homeworks, consultations

Examination
Oral examination
A student gets two grades for the course: for the examination and the oral examination separetely examination oral examination
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Jasna Prezelj:
PREZELJ-PERMAN, Jasna. Weakly regular embeddings of Stein spaces with isolated singularities. Pacific journal of mathematics, ISSN 0030-8730, 2005, vol. 220, no. 1, str. 141-152. [COBISS-SI-ID 13910873]
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]
Pavle Saksida:
SAKSIDA, Pavle. On zero-curvature condition and Fourier analysis. Journal of physics. A, Mathematical and theoretical, ISSN 1751-8113, 2011, vol. 44, no. 8, 085203 (19 str.). [COBISS-SI-ID 15909465]
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]