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
Constant coefficient systems. Higher order ODE with constant coefficients. First integrals. Singular points. Stability.

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

Calculus of variations
Examples. Euler-Lagrange equation. Free boundary 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
Vibration of a string. Wave equation and D'Alembert solution. Laplace equation. Heat equation, heat kernel and random walks. Separation of variables. Black-Sholes equation. Classification of second order PDE in two variables.

1. M. Braun: Differential equations and their applications : an introduction to applied mathematics, 4th ed., New York : Springer, cop. 1993.
2. G. Gandolfo: Economic dynamics : study edition, 4th ed., Berlin ; Heidelberg : Springer, 2010, cop. 2009.
3. F. Križanič: Navadne diferencialne enačbe in variacijski račun, Ljubljana : Mladinska knjiga, 1974.
4. F. Križanič: Parcialne diferencialne enačbe, Ljubljana : DMFA - založništvo, 2004.
5. L. Perko: Differential equations and dynamical systems, 3rd ed., New York : Springer, 2001.
6. Y. Pinchover, J. Rubinstein: An introduction to partial differential equations, Cambridge : Cambridge University Press, 2006.
7. E. Zakrajšek: Analiza III, 3. popravljena izd. - Ljubljana : DMFA - založništvo, 2002.

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

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, Jasna, VLACCI, Fabio. An interpolation theorem for slice-regular functions with application to very tame sets and slice Fatou–Bieberbach domains in H2, AMPA, ISSN 0373-3114, 2022, vol. 201, no. 5, str. 2137-2159 [COBISS-SI-ID - 106389763]
GENTILI, Graziano, PREZELJ, Jasna, VLACCI, Fabio, Slice conformality and Riemann manifolds on quaternions and octonions,: Mathematische Zeitschrift. - ISSN 0025-5874, 2022, vol. 302, no. 2, str. 971-994 [COBISS-SI-ID - 117983235]
GENTILI, Graziano, PREZELJ, Jasna, VLACCI, Fabio, On a definition of logarithm of quaternionic functions, JCNG 2023, vol. 17, no.3, str. 1099-1128 [COBISS-SI-ID - 162763779]

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]