Mathematics

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

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.

- 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.

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.

- 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%).

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

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.

- 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.

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.

- 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%).

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

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.

- 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.

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.

- 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%).

Basic notions of the theory of distributions, Dirac delta. Elements of Fourier series. Introduction to Fourier transform. Laplace and Poisson equations, properties of the potential, harmonic functions, maximum principle and uniqueness. Solutions of boundary value problems by Green's function and Fourier method. Other physical problems described by Laplace and Poisson equations. Derivation of the equation of the vibrating string. From Maxwell's equations in vacuum to the wave equation. D'Alembert solution, Duhamel formula. Solution of one dimensional wave equation in bounded domains by Fourier method. Solution of the wave equation in dimension two and three in the whole space. Maxwell equations in vacuum, solution of the Cauchy problem, conservation of energy, radiation fields. Fourier law and heat equation. Solution of the Cauchy problem in the whole space. Maximum principle, uniqueness. Solution of heat equation in bounded domains by Fourier method.

It is required a good knowledge of the arguments covered in the courses of Analisi Matematica II and Analisi Reale.

P. Butta', Note del corso di Fisica Matematica, available on the personal website of P. Buttá S. Salsa, Equazioni a derivate parziali, Springer, 2010 A.N. Tichonov, A.A. Samarkij, Equazioni della fisica matematica, Mir, 1981 L.C. Evans, Partial Differential Equations, A.M.S., 2004 A.N. Kolmogorov, S.V. Fomin, Elementi di teoria delle funzioni e di analisi funzionale, Mir, 1981 V.I. Smirnov, Corso di matematica superiore II, Ed. Riuniti, 1977 A. Teta, Appunti di Fisica Matematica, available on the personal website of A. Teta

Attendance at lessons is essential for a good understanding of the course

During the exam the student is required to solve an exercises of the type solved during the course and to discuss some theoretical topics illustrated in the course. To pass the exam the student need to achieve a grade not less than 18/30. The student must demonstrate that he has acquired a sufficient knowledge of the topics and that he is able to apply the methods learned in the course to the simplest examples treated. To achieve a grade of 30/30 cum laude, the student must demonstrate that he has acquired an excellent knowledge of all the topics covered during the course and that he is able to expose them in a logical and coherent way.

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