Module page
MATHEMATICAL PHYSICS
| academic year: | 2013/2014 |
| instructor: | Dario Benedetto |
| degree course: | Mathematics - DM 270/04 (triennale) |
| type of training activity: | caratterizzante |
| credits: | 9 (72 class hours) |
| scientific sector: | MAT/07 Fisica matematica |
| teaching language: | italiano |
| period: | I sem (30/09/2013 - 17/01/2014) |
Lecture meeting time and location
Presence: highly recommended
Module subject:
- Wave equation.
Wave propagation. Heuristic and microscopic derivations of the vibrating string equation. Well posed problems and energy method. The Cauchy problem for the wave equation: D'Alambert, Kirchhoff, and Poisson solutions. Fundamental solution. Theory of Distributions, a short introduction. Solutions of the vibrating string equation in a segment. The Fourier method.
- Introduction to potential theory.
Poisson problem and the fundamental solution for the Laplace operator. Harmonic functions and their properties. Laplace-Dirichlet problem in the disk and in the sphere: the Poisson integral formula. Harnack inequality and consequences. Poisson equation in three and two dimension. Poisson-Dirichlet problem in bounded domains. Examples of exactly solvable problems. Variational formulation of the Dirichlet problem for the Laplace equation.
- Heat equation. Heuristic derivation of the heat equation. Well posed problems and energy method. Microscopic derivation and random walks. Maximum Principle. The Cauchy problem. Problems solved through the Fourier series.
Suggested reading: Notes of Prof. P. Buttà http://www.mat.uniroma1.it/people/butta/didattica, Supplementary bibliographic suggestions can be found in these notes. Exercises and test results will be available on the site See also http://brazil.mat.uniroma1.it/dario/FM.
Type of course: standard
Prerequisites: The courses of Calculus I, Mathematical Analysis I and II, and Mechanics.
Knowledge and understanding: Successful students will know the heuristic arguments and the microscopic models that lead to the formulation of the fundamental equations of Mathematical Physics. They will be able to understand the qualitative aspects of the solutions of the second order PDE (elliptic, parabolic, hyperbolic) and to analyze them through the use of Green's functions and the spectral decomposition.
Skills and attributes: Successful students will have recognized the origin and necessity of mathematical structures like Banach and Hilbert Spaces, Theory of Distributions, Fourier Series, in view of the problems arising in Physics and Applied Sciences. Moreover, they will be able to solve simple (linear) parabolic, elliptic, and hyperbolic problems by means of the separation of variables and the Fourier method.
Personal study: the percentage of personal study required by this course is the 65% of the total.