Master methods of solution for linear constant coefficients PDE of first and second order, in particular those relevant in Mathematical Physics (e.g. waves and heat): Fourier analysis and Green function.
Expected learning outcomes
The student will learn the method of characteristics, the basic aspects of Fourier analysis and the method of Green function (propagator). This tools, or paramount relevance for the continuation of his/her studies, will be here applied to the solution of some fundamental equations for the Mathematical Physics of continuous media.
Lesson period: First semester
(In case of multiple editions, please check the period, as it may vary)
1. Quasi-linear PDEs and the method of characteristics 2. Wave equation. Solution by characteristics 3. Function spaces. Fourier series and transform. 4. Wave equation: solution by Fourier method 5. The heat or diffusion equation. Propagator 6. Laplace equation (depending on available time) 7. Some non-linear equations of Mathematical Physics (depending on available time)
V.I. Smirnov, Corso di Matematica Superiore vol. 2 (Editori Riuniti) G. Cicogna, Metodi Matematici della Fisica (Springer Italia) Byron & Fuller, Mathematical Methods of Physics (Dover) Lecture Notes (in Italian)
Assessement methods and criteria
The final examination consists of three parts: a written exam, an oral exam and a lab exam.
- During the written exam, the student must solve some exercises in the format of open-ended and/or short answer questions, with the aim of assessing the student's ability to solve problems in XXX. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take N midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal. - The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding XXX in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them. -The lab exam consists in developing a project, which will be assigned in advance by the professor. The project will be presented by the student during the oral exam. The lab portion of the final examination serves to assess the capability of the student to put a problem of XXX into context, find a solution and to give a report on the results obtained.
The complete final examination is passed if all three parts (written, oral, lab) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.