


Politecnico di Torino  
Academic Year 2017/18  
02CYTNG, 02CYTMQ Equations of Mathematical Physics 

Master of sciencelevel of the Bologna process in Mathematical Engineering  Torino 1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
The main goal of this course is to present the basic principles of the formulation and of the qualitative analysis of mathematical models appearing in engineering and applied sciences.
The main aim is to treat the complete study path: modeling, classification, qualitative analysis, validation of models. 
Expected learning outcomes
To acquire skills in modeling methods at different scales (micromacro) and in the qualitative analysis of problems of mathematical physics and applied sciences.

Prerequisites / Assumed knowledge
Knowledge of the topics covered in Mathematical Analysis I and Mathematical Analysis II, Physics I and Physics II, Rational Mechanics.

Contents
First part (4 credits)
First order differential systems: first order systems in normal form and equations of order n in normal form. Linear systems, homogeneous and nonhomogeneous, with constant coefficients. Dimensional analysis. Definition of the Cauchy problem. Existence anduniqueness. Maximal solutions. Qualitative analysis. First integrals. Regularity of solutions. Gronwall Lemma. Continuous dependence on initial data. Autonomous systems, phase portrait and classification of trajectories. Equilibrium points and their stability. Liapunov function and Liapunov theorem. Basin of attraction. Linearization. Hyperbolic Points. Bifurcation from the equilibrium points. Transcritical, supercritical, subcritical bifurcations. Hopf bifurcation (sketch), Hopf theorem, limit cycles. Models of classical mechanics and dynamics of populations. Second part (4 credits) Partial differential equations: classical examples of mathematical physics. Balance equations. First order equations. Second order equations: classification and related boundary value and Cauchy problems. Hyperbolic, elliptic, parabolic equations: heuristic and microscopic derivation. Qualitative aspects of the solutions. 
Delivery modes
Theoretical lessons: 5 credits, exercises: 3 credits. The exercises aim at verifying the learning level of the concepts outlined in class lessons, through model analysis and exercises.

Texts, readings, handouts and other learning resources
N.Bellomo, E. De Angelis, M. Delitala, Lecture Notes on Mathematical Modelling From Applied Sciences to Complex Systems, SIMAI eLecture Notes, Vol. 8, 2010, http://cab.unime.it/journals/index.php/lecture/article/view/576
S. Salsa, Equazioni a derivate parziali: metodi, modelli e applicazioni, Springer, seconda edizione L. C Evans. Partial differential equations, AMS 1998 M. Pulvirenti, Appunti per il corso di Fisica Matematica, http://www1.mat.uniroma1.it/people/pulvirenti/didattica/onde_e_calore.pdf Further information about textbooks covering the topics of the course will be provided during the course. Further teaching material will be illustrated in the classroom and will be made available to all students through the teaching portal. 
Assessment and grading criteria
The goal of the exam is to test the knowledge of the subjects listed in the course's official program and the autonomous ability to apply the theory and the related qualitative analysis methods to the mathematical problems dealt with.
The aim is to verify both the level of understanding and the ability to organize a dissertation on the topics. The exam is written and consists of theoretical questions and exercises. Each part of the exam corresponds to a score that is indicated in the text of the question/exercise and the sum of the scores is 30/30. A cum laude mark is assigned in the case of a work scoring 30/30, which in addition exhibits rigor and notational clarity. The duration of the exam is two hours, during which students electronic devices are not allowed. The results of the exam are published and communicated to the students through the didactic portal, along with the date and place for viewing the works. 
