Mathematical Methods for Physics II

Time: Παρασκευές 11:00-12:00
Semester: Εαρινό
Venue: Αμφιθέατρο “ΘΑΛΗΣ”
Course Code: 10ΥΚΟ16
eClass: https://eclass.uoa.gr/courses/PHYS244/

Description:

• Introduction to partial differential equations, with examples from physics (wave equation, diffusion equation, Laplace equation, etc.). Classification of partial differential equations. Initial and boundary conditions. Solution methods.
• Inner product spaces: Cauchy-Schwarz inequality, Gram-Schmidt orthogonalization. Complete infinite-dimensional functional spaces: Bessel’s inequality, Parseval’s theorem, basis of an infinite-dimensional space.
• Fourier series. Linear operators in complete spaces: Self-adjoint operators, eigenvalue problems, spectral theorem of self-adjoint operators. Sturm-Liouville systems.
• Study of the wave equation and the diffusion equation on the line, the half line and a finite interval. Fundamental solutions and Green’s functions. Reflections and sources.
• Boundary value problems with homogeneous and inhomogeneous boundary conditions for the wave equation and the diffusion equation. Problems in cartesian, cylindrical and spherical coordinates.
• The Laplace equation. Basic properties of harmonic functions. Solution of Laplace equation in special geometries in two and three dimensions.

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