Mathematical methods in physics: differential equations 1

A.Y. 2018/2019
6
Max ECTS
42
Overall hours
SSD
FIS/02
Language
Italian
Learning objectives
Dare agli studenti gli strumenti essenziali per la soluzione di alcune tra le equazioni differenziali piu` importanti della Fisica e per l'uso delle distribuzioni.
Expected learning outcomes
Undefined
Course syllabus and organization

Single session

Responsible
Lesson period
First semester
Course syllabus
Partial differential equations.
Revisitation of old facts from Physicis and Mathematical Analysis about differential equations (e. g. Frobenius theorem , Helmholtz decomposition and the likes). Comments on the deduction of the vibrating string. The wave equation, homogeneous and inhomogeneous. Uniqueness of the solutions with the energy integral. Construction of the solutions with the separation of variables technique. The eigen-value problem. Series expansion in orthogonal functions.
Poisson equation, uniqueness of the solutions with the minimum-maximum theorems, construction of the solutions by sepearation of variables.
Heat equation, uniqueness and construction of the solutions, convolution of functions, heat and Schroedinger kernels.
Fourier transform applied to the solution of said equations; integral as principal value and Jordan Lemma.
Distributions.
Test functions, space K, distributions, elementary operations, derivation, regularization of non-locally summable functions, Plemelj formulae.
Convolution of distributions and the elementary solutions of the Cauchy problem.
Fourier transform of distributions and applications to the solutions of partial differential equations with initial data. Advaced and retarded Green's functions for the wave equation
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6
Lessons: 42 hours
Professor: Klemm Silke
Professor(s)