Mathematics

PAOLO BUTTA'

General objectives:
To acquire basic knowledge on modeling and solving classical problems of continuum physics.

Specific objectives:

Knowledge and understanding:
At the end of the course the student will know the fundamental equations of mathematical physics (transport, waves, Laplace, heat), their derivation from concrete physical problems and the classical techniques of solution.

Applying knowledge and understanding:
Students who have passed the exam will be able to solve transport and Liouville's equation, simple initial and boundary value problems for wave and heat equations and boundary value problems for Laplace and Poisson equations, using the classical techniques of mathematical physics, like Green's functions and Fourier method.

Making judgments :
Students who have passed the exam will be able to recognize a mathematical physics approach to problems, linking the mathematical properties of the models based on partial differential equations to the concrete description of the problems of continuum physics.

Communication skills:
Students who have passed the exam will have gained the ability to communicate concepts, ideas and methodologies of Mathematical Physics related to continuum physics.

Learning skills:
The acquired knowledge will allow a study, individual or given in other courses, concerning more advanced methods of Mathematical Physics.

General objectives:
To acquire basic knowledge on modeling and solving classical problems of continuum physics.

Specific objectives:

Knowledge and understanding:
At the end of the course the student will know the fundamental equations of mathematical physics (transport, waves, Laplace, heat), their derivation from concrete physical problems and the classical techniques of solution.

Applying knowledge and understanding:
Students who have passed the exam will be able to solve transport and Liouville's equation, simple initial and boundary value problems for wave and heat equations and boundary value problems for Laplace and Poisson equations, using the classical techniques of mathematical physics, like Green's functions and Fourier method.

Making judgments :
Students who have passed the exam will be able to recognize a mathematical physics approach to problems, linking the mathematical properties of the models based on partial differential equations to the concrete description of the problems of continuum physics.

Communication skills:
Students who have passed the exam will have gained the ability to communicate concepts, ideas and methodologies of Mathematical Physics related to continuum physics.

Learning skills:
The acquired knowledge will allow a study, individual or given in other courses, concerning more advanced methods of Mathematical Physics.

It is required to have a knowledge of basic concepts and methods of the courses in Rational Mechanics, Mathematical Analysis, and Linear Algebra, acquired in the first level degree program.

The equation of the vibrating string
1. Microscopic model: chains of oscillators.
2. The Lagrangian for the wave equation; equations of motion; boundary conditions.
3. D'Alembert's formula. Fundamental solution.
3. Forced equation and Duhamel's formula.
4. Fourier series. The Fourier method with applications to different boundary problems.

Distributions and Fourier transform
1. Space of fundamental functions; distributions.
2. Examples, the Dirac function δ, operations on distributions.
3. The fundamental solution as solution in a generalized sense.

The wave equation in dimension two and three
1. The vibrating membrane, boundary conditions and their physical meaning. Maxwell equations.
2. Well posed problems and uniqueness of regular solutions.
3. Fourier series in higher dimension; the wave equation in rectangular domains and solution by series.
4. Green's function and the Kichhoff and Poisson formulas.
5. Cone of influence and domain of dependence; Huygens principle.
6. Wave packet; phase and group velocity.

Introduction to potential theory
1. The equation for the electrostatic/Newtonian potential, Laplace and Poisson equations.
2. Fundamental solution of the Laplace operator. Green's functions in dimension 2 and 3 and their physical interpretation.
3. Green's identities.
4. Harmonic functions and their properties.
5. Separation of variables.
6. Laplace problem in the disk with continuous boundary conditions: derivation of the Poisson formula with the Fourier method.
8. Green's function in bounded domains and its properties. Method of image charges. Poisson's formula for the Laplace problem in the ball in three dimensions with continuous boundary conditions.
9. Potential generated by a charge distribution: the Poisson equation in the whole space in dimension two and three.

Heat equation
1. Conservation laws in divergence form; Fourier's law.
2. Green's function for the heat equation.
3. The principle of the maximum parabolic; the uniqueness of the solution in bounded domains and in the whole space.
4. The Fourier method for the heat equation in bounded domains. Asymptotic behavior of solutions.
5. From the symmetric random walk to the heat equation.

Appendices
1. Introduction to the Fourier transform.
2. Fourier transform in Schwartz space and its properties. Examples.
3. Fourier transform of distributions.
4. Calculating fundamental solutions with the Fourier transform

- Lecture notes available on-line (http://www1.mat.uniroma1.it/~butta/didattica/note_FM.pdf)
- S. Salsa, Equazioni a Derivate Parziali: Metodi, Modelli e Applicazioni. Milano: Springer 2010.

- V.I. Arnold, Lectures on Partial Differential Equations (Coll. Universitext). Berlin: Springer 2004.
- L.C. Evans, Partial Differential Equations. Providence: A.M.S. 2004.
- A. N. Kolmogorov, S. V. Fomin, Elementi di teoria delle funzioni e di analisi funzionale. Mosca: MIR 1980.
- V.I. Smirnov, Corso di Matematica Superiore Vol. II. Roma: Editori Riuniti 1977.
- A.N. Tichonov, A.A. Samarskij, Equazioni della fisica matematica. Mosca: MIR 1981.
- S. Vladimirov, Equazioni della Fisica Matematica. Mosca: MIR 1987.

Lectures (60%), examples and exercises (40%).

Attendance at lessons is strongly recommended for a good understanding of the course content.

The exam aims to evaluate learning through an oral test. This test consists in the discussion of some of the most relevant topics illustrated in the course and the resolution of a simple exercise. To pass the exam the student needs to achieve a grade not less than 18/30. The student should prove to have a sufficient knowledge of the topics covered during the course and to apply the related techniques to basic examples. To achieve a grade of 30/30 cum laude, the student should exhibit an excellent knowledge of all the topics covered during the course and he should be able to expose them in a logical and coherent way.

The theory questions concern one or more of the most relevant topics of the course. The exercises are quite simple and along the lines of those contained in various exercise sheets that are published and subsequently corrected in the classroom during the course.