#
Mathematical Physics 2

A.Y. 2017/2018

Learning objectives

To give an introduction to partial differential equations.

Expected learning outcomes

To be able to study the properties of the solutions of the main partial differential equations of Mathematical Physics.

**Lesson period:**
First semester

**Assessment methods:**

**Assessment result:**

Course syllabus and organization

### Single session

Responsible

Lesson period

First semester

**Course syllabus**

This is an introductory course on partial differential equations of mathematical physics. It is divided into several chapters.

1. Transport equation. Shock in Burgers equation (non-linear transport). Classification of second-order equations with constant coefficients in two independent variables (reduction to the heat equation, the Laplace equation and Wave equation)

2. Wave equation in one space dimension. Deduction. general solution in R. Solution of the Cauchy problem. Main properties. Domains of dependence and domain of influence. Conservation of energy. Solution of the equation with Dirichlet boundary conditions on a segment: separation of variables, series representation of the solution.

3. Heat equation in one space dimension. Deduction. the maximum principle for the problem of a rectangle. Use of the maximum principle to demonstrate uniqueness' of solutions.

Equation in R. Symmetries of the equation. Use for the construction of the general solution of the Cauchy problem. Properties. Solution of the equation with Dirichlet boundary conditions on a segment: separation of variables, the representation of the solution through series. Analysis of the solution. solution of the equation on the half-line employees with conditions periodically by time (the winery). Duhamel principle.

4. Fourier series. Definition. Theorem of pointwise, uniform and elle2 convergence.

5. Elliptic Equations. physical systems described by the equation of Laplace and Poisson from that. 1d Laplace equation solution for some simple domains. Poisson formula. Green formulae. Variational property of the laplace equation. Green's functions and properties 'regularity' of solutions.

6. Wave equation in 3-d. Kirchoff formula. Conservation of energy and the principle of causality. Formula of variation of arbitrary constants. Non-homogeneous equation. Formula of retarded potentials.

7. Electromagnetism. Maxwell equations for the electromagnetic field in interaction with a particle. Conservation of energy. Electromagnetic potentials, wave equations in Lorentz gauge. Solution in the case of motion of the particle assigned. Hertz formula for the radiation of a moving charge in dipole approximation.

8. Introduction to distribution.

1. Transport equation. Shock in Burgers equation (non-linear transport). Classification of second-order equations with constant coefficients in two independent variables (reduction to the heat equation, the Laplace equation and Wave equation)

2. Wave equation in one space dimension. Deduction. general solution in R. Solution of the Cauchy problem. Main properties. Domains of dependence and domain of influence. Conservation of energy. Solution of the equation with Dirichlet boundary conditions on a segment: separation of variables, series representation of the solution.

3. Heat equation in one space dimension. Deduction. the maximum principle for the problem of a rectangle. Use of the maximum principle to demonstrate uniqueness' of solutions.

Equation in R. Symmetries of the equation. Use for the construction of the general solution of the Cauchy problem. Properties. Solution of the equation with Dirichlet boundary conditions on a segment: separation of variables, the representation of the solution through series. Analysis of the solution. solution of the equation on the half-line employees with conditions periodically by time (the winery). Duhamel principle.

4. Fourier series. Definition. Theorem of pointwise, uniform and elle2 convergence.

5. Elliptic Equations. physical systems described by the equation of Laplace and Poisson from that. 1d Laplace equation solution for some simple domains. Poisson formula. Green formulae. Variational property of the laplace equation. Green's functions and properties 'regularity' of solutions.

6. Wave equation in 3-d. Kirchoff formula. Conservation of energy and the principle of causality. Formula of variation of arbitrary constants. Non-homogeneous equation. Formula of retarded potentials.

7. Electromagnetism. Maxwell equations for the electromagnetic field in interaction with a particle. Conservation of energy. Electromagnetic potentials, wave equations in Lorentz gauge. Solution in the case of motion of the particle assigned. Hertz formula for the radiation of a moving charge in dipole approximation.

8. Introduction to distribution.

MAT/07 - MATHEMATICAL PHYSICS - University credits: 6

Practicals: 22 hours

Lessons: 36 hours

Lessons: 36 hours

Professors:
Bambusi Dario Paolo, Penati Tiziano

Professor(s)

Reception:

to be fixed by email

office num. 1039, first floor, Dep. Mathematics, via Saldini 50