# MATHEMATICAL PHYSICS

Degree course:
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree:
2015/2016
Year:
3
Academic year in which the course will be held:
2017/2018
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 variable and ordinary differential equations.

Final Examination:
Orale

Oral exam

Assessment:
Voto Finale

The aim of the course is to provide the basics of the theory of partial differential equations. The students will be able to classify partial differential equations and will be familiar with the concepts of classical solution and weak solution.
Three basic equations of mathematical physics will be studied with greater detail: the wave equation, the heat equation, and the Laplace equation. The most relevant properties of their solutions will be discussed and the most common solving methods will be presented.

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.
- 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.