Partial differential equations

Completed courses Differential equations and Mathematics 2.

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.

1. J. Cimprič: Rešene naloge iz analize III, Ljubljana : DMFA - založništvo, 2001, 2014.
2. J. Cimprič, J. Prezelj: Rešene naloge iz analize IV, Ljubljana : DMFA - založništvo, 2017.
3. E. Kreyszig: Advanced engineering mathematics, 7th ed. in 9th ed., Hoboken : J. Wiley & Sons, cop. 1993, 2006.
4. F. Križanič: I. Vidav: Navadne diferencialne enačbe ; Parcialne diferencialne enačbe ; Variacijski račun, Ljubljana : Društvo matematikov, fizikov in astronomov Slovenije, 1991.
5. E. Zakrajšek: Analiza III, 3. popravljena izd. - Ljubljana : DMFA - založništvo, 2002.

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.

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.

Lectures, exercises, homeworks, consultations

2 midterm exams instead of written exam, written exam
Oral exam
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]

Miran Černe:
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]

Jakob Cimprič:
CIMPRIČ, Jaka. A nullstellensatz for linear partial differential equations with polynomial coefficients. Multidimensional systems and signal processing. Jan. 2019, vol. 30, iss. 1, str. 363-372. [COBISS-SI-ID 18545497]
CIMPRIČ, Jaka. Local linear dependence of linear partial differential operators. Integral equations and operator theory. June 2018, vol. 90, iss. 3, art. 38 (8 str.). [COBISS-SI-ID 142629635]
CIMPRIČ, Jaka, PREZELJ-PERMAN, Jasna. Rešene naloge iz analize IV. Ljubljana: DMFA - založništvo, 2017. 151 str., ilustr. Izbrana poglavja iz matematike in računalništva, 51. [COBISS-SI-ID 292729600]
CIMPRIČ, Jaka. Rešene naloge iz analize III. 1. ponatis. Ljubljana: DMFA - založništvo, 2014. 130 str., graf. prikazi. Izbrana poglavja iz matematike in računalništva, 41. [COBISS-SI-ID 271410176]