Part I: The spaceof test functions and their duals, generalized functions (distributions) analysis and differential equations of distributions. Green's function method for boundary problems of second order linear differential equations. The dual problem and the solution of the inhomogeneous problem via Green's function.
Part II: Tempered distributions, Fourier transform, convolution and properties. Fourier Transform on a Hilbert space, Parseval and Plancherel identities.
Part III (Applications): Fundamental solution of diffusion equation, Green's function and method of images for boundary value problems. Green's function method for Poisson equation on the plane, 2-dimensional Dirac delta function. Method of images, non-homogeneous Dirichlet problem on the disk.