ID:
SCC0238
Duration (hours):
72
CFU:
8
SSD:
FISICA MATEMATICA
Year:
2025
Overview
Date/time interval
Primo Semestre (22/09/2025 - 16/01/2026)
Syllabus
Course Objectives
Partial differential equations describe a wide range of natural phenomena, among which: vibrations of solids; propagation of acoustic and electromagnetic waves, heat diffusion, dynamics of quantum particles. The goal of the course is to give an introduction to the theory of partial differential equations. Three fundamental equations of mathematical physics will be analyzed: the wave equation, the heat equation, and the Poisson equation. Physical motivations for the study of these equations will be provided. The description of natural phenomena leads to the study of several typical problems associated to the equations: problems with initial and/or boundary data. Using methods from functional analysis, these problems will be solved at a rigorous mathematical level. At the end of the course the students will be able to: - obtain representation formulae for the solutions of the problems associated to the three partial differential equations; - state and prove theorems concerning the existence, uniqueness, and stability of the solutions; - discuss the fundamental properties of the solutions; - give examples of physical applications.
Course Prerequisites
Basis of integral and differential calculus in one or several variables. The notions of the courses of Linear Algebra and Geometry, Mathematical Analysis I and II will be taken for granted. It is recommended to have followed the course Mathematical Analysis III.
Teaching Methods
The course consists of a total of 72 hours of lectures, divided as follows: 48 hours devoted to the presentation and discussion of the general theory of partial differential equations; 24 hours dedicated to problem-solving and the analysis of the fundamental properties of explicit solutions. Class attendance is not mandatory, but it is strongly recommended. Lectures will be in presence with the support of a blackboard.
Assessment Methods
Oral examination. The exam consists of a discussion on the partial differential equations presented during the course, and on the methods used to solve the associated problems. The purpose of the exam is to verify: the level of knowledge and deepening of the topics addressed; the full understanding of the solving techniques and of the properties of the solutions; the ability to state theorems and expose proofs in a mathematically rigorous way; the ability to discuss the examples and solve exercises similar to the ones presented in the lectures. The student who aspires to excellence must be able to: justify all the assumptions in the statements of the theorems and each individual step of the proofs; discuss several particular cases that are not part of the general theory; use all the solving techniques presented during the course. The mark is expressed in thirtieths
Contents
The course is essentially divided in four, mutually connected parts: 1. Heat, wave, and Poisson equation in the whole space; 2. Harmonic functions and Poisson equation in bounded domains; 3. Distributions; 4. Heat and wave equation in bounded domains. Eigenvalues and eigenvectors.
Course Language
Italian
More information
Office hours: by appointment. e-mail: claudio.cacciapuoti@uninsubria.it
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MATHEMATICS
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3 years
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