MATHEMATICAL PHYSICS
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
Basics of differential and integral calculus of functions of one and several variables and ordinary differential equations.
Oral exam. The exam consists in a discussion on the partial differential equations presented during the course and on the techniques to solve them. The aim of the exam is to evaluate the level of the understanding of the notions and of the solution methods presented during the lectures. The grade of the exam will also take into account the student's ability to express himself in a rigorous mathematical language.
The aim of the course is to provide the basics of the theory of partial differential equations (PDEs), with applications to three fundamental equations of mathematical physics: the wave equation, the heat equation, and the Poisson equation. The students will be able to classify partial differential equations and will be familiar with the concepts of classical and weak solution. The students will also be able to solve the Poisson equation and the Cauchy problem for the heat and wave equation in R^n. Students will learn how to use the Fourier series to solve the one-dimensional heat and wave equation on a bounded domain and will be familiar with the notions of fundamental solution of a PDE, of Green's function, and of harmonic function. The students will be able to use Duhamel's principle to find the solution of a non-homogeneous equation, will learn the basics of the theory of distributions and Fourier transform, and their applications. Students will be able to state and prove, with rigorous mathematical arguments, several theorems concerning the fundamental properties of the solutions of the equations discussed during the course. Additionally, the students will be able to address simple problems within the theory of PDEs, adapting the techniques learned during the course.
Introduction to partial differential equations and their classification. Classical solutions and well-posedness. Deduction of the wave and diffusion equations from physical models. The wave equation on the real line: d’Alembert formula. The heat equation on the real line: the fundamental solution. Wave and heat equation on an interval: maximum principle for the heat equation, boundary conditions, separation of variables, autofunctions and eigenvalues of the Laplacian on an interval, the Fourier series. Laplace equation and harmonic functions: divergence theorem, Green identities, maximum principle for harmonic functions, representation formula for the harmonic functions, Green function. Variational properties of the solutions of the Dirichlet and Neumann problem. Wave equation in several dimensions: Kirchhoff's formula, method of descent, Huygens principle. Introduction to the theory of distributions: convolution of distributions, Fourier transform of distributions, fundamental solutions for the Laplace, heat, and wave equation.
- L.C. Evans, Partial Differential Equations, American Mathematical Society.
- M. Renardy, R.C. Rogers, An introduction to partial differential equations, Springer.
- R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, World Scientific Publishing Company, 2003.
- W. Strauss, Partial differential equations: An Introduction, Wiley&Sons.
The course consists of 64 hours of frontal lessons. Class attendance is not mandatory but is highly recommended.
Office hours: by appointment (email the instructor)