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Department of Physics

ΥΚΟ16

10ΥΚΟ16 Mathematical Methods in Physics II

Course webpage:  https://eclass.uoa.gr/courses/PHYS244

Course outline

Course content

  • 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.