OVERVIEW

Metodi Matematici della Fisica 2 (code 61843) has credit value 7 and it is taught in the first semester of the first year of the LM.

Lectures are usually given in Italian. Should it be requested, they can be given in english.

AIMS AND CONTENT

LEARNING OUTCOMES

Boundary and/or initial values problems for second order, linear, partial differential operators of Mathematical Physics (wave operator, diffusion operator, Laplace and Helmholtz operators.

AIMS AND LEARNING OUTCOMES

• ·         Methods of solution for boundary values and initial/boundary values for second order equations of Mathematical Physics.

• ·         Particular emphasis is given to methods based on the use of Green’s functions and Fundamental Solutions. This is why the first part of the course contains an exposition of the Theory of Distributions and Fourier Analysis of Tempered Distributions

PREREQUISITES

• Differential calculus in several variables
• Integral calculus in several variables. The classical yheorems: Stokes, Gauss, Green
• Elementary theory of holomorphic functions of one complex variable
• Linear ordinary differential equations

TEACHING METHODS

Teaching is done in the traditional (chalk and blackboard) way. Students are (very) strongly suggested to attend the class.

SYLLABUS/CONTENT

1.       THEORY OF DISTRIBUTIONS

1.1.    Test functions and Distributions

1.1.1. The space of test functions

1.1.2. The space of distributions

1.1.3. Support of a distribution

1.1.4. Distributions defined by locally integrable functions

1.1.5. Examples of distributions not defined by locally integrable functions

1.1.6. Multiplication by smooth functions

1.1.7. Pullback and image of a distribution

1.2.    Derivation of Distributions

1.2.1. Derivative of a Distribution

1.2.2. Properties of the derivative of a Distribution

1.2.3. Primitive of a Distribution on R

1.2.4. Examples n=1

1.2.5. Examples

1.3.    Tensor product and convolution of distributions

1.3.1. Definition and main properties of the tensor product of Distributions

1.3.2. The convolution of distributions

1.3.3. Properties of convolution

1.3.4. Examples of convolutions

1.4.    Temperate distributions and Fourier Transforms

1.4.1. The Schwartz space of rapidly decreasing test functions

1.4.2. The space of temperate distributions

1.4.3. Examples of temperate distributions

1.4.4. The Fourier transform of temperate distributions

1.4.5. Properties of the Fourier Transform

1.4.6. Fourier transform of convolution

1.4.7. The Poisson summation formula and the Fourier Transform of periodic distributions

1.4.8. Examples n=1

1.4.9. Examples

2.       FUNDAMENTAL SOLUTIONS AND THE CAUCHY PROBLEM

2.1.    Introduction

2.1.1. Generalized solutions of linear differential equations

2.1.2. Fundamental solutions

2.1.3. Non homogeneous  linear equations

2.1.4. Fundamental solutions for linear ordinary differential operators

2.1.5. The Cauchy problem for linear, constant coefficients, ordinary differential equations

2.2.    The Cauchy problem for the wave equation

2.2.1. Fourier analysis of the wave equation

2.2.2. Fourier analysis of the fundamental solutions of the wave equation

2.2.3. The generalized Cauchy problem for the wave equation

2.2.4. Solution of the generalized Cauchy problem

2.2.5. Solution of the classical Cauchy problem (retarded potentials)

2.2.6. Wave propagation in 2 and 3 space dimensions

2.3.    The Cauchy problem for the diffusion equation (heath equation)

2.3.1. Fundamental solutions of the diffusion equation

2.3.2. The generalized Cauchy problem for the diffusion equation

2.3.3. Solution of the Cauchy problem

3.       MIXED PROBLEMS FOR THE WAVE AND DIFFUSION EQUATIONS

3.1.    Separation of variables

3.2.    Mixed problems for the wave equation

3.3.    Mixed problems for the diffusion equation

4.       BOUNDARY VALUES PROBLEMS FOR ELLIPTIC EQUATIONS

4.1.    Introduction to the eigenvalue problem

4.2.    The Sturm-Liuville problem

4.2.1. The Green’s function

4.2.2. Properties of the eigenvalues and eigenfunctions

4.2.3. Explicit calculations of eigenvalues and eigenfunctions

4.3.    Problems related to the Laplacian

4.3.1. Properties of harmonic functions

4.3.2. Separation of

4.3.3. Variables

4.4.    ExamplesFundamental solutions of the Laplacian

4.4.1. Newtonian potential

4.4.2. Volume potential

4.4.3. Single and double layer potential

4.4.4. Properties of single and double layer potentials

4.5.    Boundary values for the Laplace and Poisson equations

4.5.1. Separation of variables for the Laplacian

4.5.2. Examples of boundary values problems for the Laplace equation

4.5.3. Definition and properties of the Green’s function

4.5.4. Solutions of some boundary values problems for the Poisson equation using the Green’s function

4.5.5. The Poisson formula

RECOMMENDED READING/BIBLIOGRAPHY

Suggested textbooks

F.G. Friedlander, Introduction to the theory of distributions, Cambridge UP, 1982
I. Stakgold, Green's functions and boundary value problems, Wiley 1979
L.Schwartz, Methodes mathematiques pour les sciences physiques, Hermann 1965

Some notes by the teacher are distributed to the students.

TEACHERS AND EXAM BOARD

Exam Board

GIOVANNI CASSINELLI (President)

PIERO TRUINI

PIERANTONIO ZANGHI'

LESSONS

LESSONS START

From September 25, 2017

Class schedule

MATHEMATICAL METHODS IN PHYSICS

EXAMS

EXAM DESCRIPTION

Compulsory written examination.Optional oral examination.

ASSESSMENT METHODS

Methodology of ranking

The explicit aim of the course is to  carry the students to the level of making calculations and solving problems. This is why the essential part of the exam is the written one, where  the student is asked  to make calculations and  to solve explicitly problems.

The text is divided into three levels, each level corresponds to a range of marks. (These ranges of marks are clearly indicated in the text given to each student). The first range is for marks up to 24, the second for marks up to 28, the third for marks over 28.

The first corresponds to an easy exercise that do not contain either conceptual or computational difficulties. The second corresponds to an exercise without difficulties, but containing some challenging calculation (as an example, a non elementary integral). Marks over 28 correspond to an exercise whose solution requires some skill, and it is not the immediate application of something seen during the course. In particular it contains a “difficult” part or question that is required to get the “LAUDE”.

It is my firm belief, that comes from many years of teaching, that the oral part of the exam can just be a small correction to the written part. It has to be stressed that this correction is not necessarily positive. This is why the student is allowed to retain the mark of the written part as the final one.

Exam schedule