# MATHEMATICAL PHYSICS

Degree course:
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree:
2017/2018
Year:
3
Academic year in which the course will be held:
2019/2020
Course type:
Compulsory subjects, characteristic of the class
Credits:
8
Period:
First Semester
Standard lectures hours:
80
Detail of lecture’s hours:
Lesson (80 hours)
Requirements:

Basics of differential and integral calculus of functions of one and several variables and ordinary differential equations.

Final Examination:
Orale

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.

Assessment:
Voto Finale

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. At the end of the course the student is expected:
- To be able to classify partial differential equations and to be familiar with the concepts of classical and weak solution.
- To know the representation formulae for the solutions of the Poisson equation and of the Cauchy problem for the heat and wave equation in R^n.
- To be familiar with the notions of fundamental solution of a PDE, of Green's function, and of harmonic function.
- To use the Fourier method to solve the one-dimensional heat and wave equation on a bounded domain.
- To know the basics of the theory of distributions and Fourier transform, and their applications.
- To 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.

Partial differential equations; wave equation; diffusion equation; Laplace equation and harmonic functions; theory of distributions.

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 and in R^n: the fundamental solution. Wave equation in R^n: Kirchhoff's formula, method of descent, Huygens principle. Duhamel's principle. Laplace equation and harmonic functions: divergence theorem, Green identities, maximum principle for harmonic functions, representation formula for the harmonic functions, Green function. Wave and heat equation on an interval: maximum principle for the heat equation, boundary conditions, separation of variables, eigenfunctions and eigenvalues of the Laplacian on an interval, the Fourier series. Variational properties of the solutions of the Dirichlet and Neumann problem. 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 it is strongly recommended.

Office hours: by appointment (email the instructor)