Analysis III - HS 2011

Analysis III - Herbstsemester 2011

Kurzbeschreibung - Abstract

Diese Lehrveranstaltung konzentriert sich auf drei Grundtypen von partiellen Differentialgleichungen zweiten Grades: Wellengleichung, Wärmeleitungsgleichung und Laplace-Gleichung.

This course focuses on three fundamental types of partial differential equations of second order and their solving methods: the wave equation, the heat equation and the Laplace equation.

Vorlesungsskripten - Lecture notes

1. 26. September (PDF) Wave equation and its derivation.
Separation of variables.
2. 3. Oktober (PDF) Superposition principle. Wave equation.
Fourier series and Fourier theorem.
3. 10. Oktober (PDF) Heat equation.
Nonzero boundary conditions and adiabatic conditions.
4. 17. Oktober (PDF) Heat equation on an infinite bar.
Heat equation on a solid ball.
5. 24. Oktober (PDF) 2D and 3D equations.
Double Fourier series and Fourier transform.
6. 31. Oktober (PDF) Classification of second order PDEs. Types of boundary conditions.
Laplace's equation. Sturm-Liouville problems.
7. 7. November (PDF) Laplace's equation in polar coordinates and in cylindrical coordinates.
Bessel equations and Bessel functions.
8. 14. November (PDF) Laplace's equation in spherical coordinates. Spherical harmonics.
Legendre equations and Legendre polynomials.
9. 21. November (PDF) Application of complex analysis to Laplace's equation in 2D.
Equipotential lines and lines of force. Conformal mapping.
10. 28. November (PDF) Maximum principle for harmonic functions.
Poisson's integral formula.
11. 5. December (PDF) Integral representations. Definition of Green's functions.
Distributions. Dirac delta.
12. 12. December (PDF) Derivative of a distribution. Fundamental solutions.
Use of Green's functions. Reflection principle.
13. 19. December (PDF) Poisson's kernel vs. Green's function on a disk.
Neumann problem for the Poisson equation. Neumann's functions.

Links to relevant materials:
Please let know of any mistakes or imprecisions.

Coordination webpage:

Dozierende: Ana Cannas

Stand: 28. August 2012