Mathematics

Wave equation 1. From a chain of harmonic oscillators to the wave equation. 2. Density of Lagrangian and equations of motion; boundary conditions. 3. Derivation of the D'Alembert formula. 4. Distributions, the delta function. The Green function for the wave equation. 5. The Duhamel formula. 6. Standing waves and Fourier’s series. 7. Bessel inequality and convergence conditions. 8. Fourier method applied to the homogeneous and inhomogeneous Cauchy-Dirichlet problem. Fourier transform and wave packet 1. Fourier transform on the Schwartz space; the Plancherel-Parseval theorem. 2. Fourier transform of the gaussian. 3. Wave packet; phase and group velocity. Wave equation in Rn 1. Vibration of a membrane, boundary conditions and their interpretations. 2. Density of lagrangian and energy; uniqueness of the solution via the energy method. 3. The Green's function, the Kirchhoff's formula and the Poisson's formula 4. Cone of influence and cone of dependence; Huygens principle. 5. From the Maxwell equations to the wave equation Heat equation 1. Conservation laws and continuity equations; Fourier law. 2. The Green function for the heat equation; the fundamental solution in R^n. 3. Maximum principle; uniqueness. 4. The heat equation with boundary conditions and surces; solutions with Fourier method and Green funztion. 5. From the random walk to the heat equation. 6. Probabilistic interpretation of the Laplace equation in the discrete case Potential theory in Rn 1. Laplace and Poisson equations 2. The Green's function and the electric potential due to a point charge. 3. Regularity an asymptotic behaviour of the electric potential due to a charge distribution. 4. Green's identities. 5. Harmonic functions, mean value theorems, maximum principle; uniqueness theorems. 6. Regularity. Poisson equation in a domain 1. The Laplace-Dirichlet problem for the disk; derivation of the Poisson formula using the Fourier method. 2. The Green function and is properties. 3. The Poisson's formula for the ball in Rn. 4. Inverse of the mean value theorem. Harnack inequality. Liouville theorem. 5. The Poisson problem in Rn: uniqueness.

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.

S. Salsa, G. Verzini, Partial Differential Equations in Action. From Modelling to Theory. Springer, 2022

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 simple exercised proposed during the course. To pass the exam the student needs to achieve a grade not less than 18/30. The student must demonstrate that he has acquired a sufficient knowledge of the topics and and that he is able to apply the methods learned in the course to the simplest of the 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.

V.I. Smirnov "10. A. N Tichonov "Equazioni della fisica matematica". Mosca, MIR, 1981 A. N: Kolmogorov, S. V. Fomin "Elementi di teoria delle funzioni e di analisi funzionale". Mosca, MIR, 1980.

Lectures (2/3), examples and exercises (1/3).

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%).