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 Scheda Riassuntiva
 Anno Accademico 2018/2019 Scuola Scuola di Ingegneria Civile, Ambientale e Territoriale Insegnamento 095853 - MATHEMATICAL PHYSICS Docente Vivarelli Maria Dina Cfu 8.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento
Ing - Civ (Mag.)(ord. 270) - MI (440) INGEGNERIA PER L'AMBIENTE E IL TERRITORIO*AZZZZ095853 - MATHEMATICAL PHYSICS
Ing - Civ (Mag.)(ord. 270) - MI (489) INGEGNERIA PER L'AMBIENTE E IL TERRITORIO - ENVIRONMENTAL AND LAND PLANNING ENGINEERING*AZZZZ095853 - MATHEMATICAL PHYSICS

 Obiettivi dell'insegnamento
 The main object of the course is to provide an overview of some of the most important mathematical models found in the literature to describe several physical and engineering phenomena. Thus, after an introduction to the use of the ordinary differentail equations, it is shown how the partial differential equations are crucial in giving a complete description of pyhsical systems. In particular, this is done by showing how to formulate a partial differential equation starting from the physical problem (thus constructing the mathematical model) and how to solve the equation (along with the initial and boundary conditions). The physical phenomena considered are essentially wave propagation, diffusion-convection (heat flow) and stationary problems. Furthermore, fluid dynamics is studied through a keen study of the Navier-Stokes equations (including the Reynold number and classification of viscous fluids) and through the recent views which regard the transition from laminar to turbolent flows and the application to the fractal theory. Finally, beyond the use of pure analytical methods and laboratory experiments, since the advent of computers allows a good understanding of fluid behaviour, the basic elements of Computational Fluid Dynamics are introduced together with physical applications to related physical and engineering areas.

 Risultati di apprendimento attesi
 The student will have - The knowledge and the comprehension of the fundamental role played by the partial differential equations in physical and engineering problems - The ability to solve the problems being conscious of the appropriate choices to be taken and aware of the adopted procedures.

 Argomenti trattati
 1. Structural analogies in Mathematical Physics 2. Mathematical models described by ordinary (ODEs) and partial differential equations (PDEs): Lagrange and Hamilton ODE-equations. Small oscillations. Hamilton-Jacobi method. 3. Classification of PDE equations: Cauchy Problem. Dirichlet Problem. Vibrating string, vibrating membrane, waves.  Solution of PDEs. Characteristic curves. Separation of variables. Laplace's equation. Potential theory. Heat equation. Rotating fluids. 4. Continuum Mechanics: Navier-Stokes equations. Hierarchy of Reynolds number. Transition from a laminar flux to a turbolent one. Prandtl boundary layer. Brief introduction to fractals and multifractals. Waves around an obstacle.  Enviromental problems. Difficult solution of the Navier-Stokes equations. 5. Brief introduction to Computational Fluid Dynamics (CFD). Physical applications.

 Prerequisiti
 A good knowledge of trigonometry, derivatives, integrals, ordinary differential equations and basic elements of Euclidean and analytical geometry is required

 Modalità di valutazione
 Besides the five official examination sessions (January, February, June, July, September) it is possible to take two partial proofs (one at the middle of the teaching and one at the end).  The two partial proofs are not compulsory. So there are many options. A student can decide to take: 1) an official examination (including the whole program) in one of the official dates  (i.e. without taking both the two partial proofs). 2) both of the two partial proofs (and if he passes both of them there is no need to take any official examination). 3) the first/second partial proof and not  the other one, in this case he will recover only the remaining proof during one of the official exams. If a student does not pass one of the two partial proofs he will recover that proof in one of the official examinations (the mark of the other proof is kept fixed). A  failed official exam can be recovered in a subsequent one. Marks. Total marks available: 30/30 or 30/30 with honours.  18/30 marks are required to pass. In the case of the partial proofs the final mark is given by the arithmetic average of the two proofs. Subject of the exams. The exams are written exams and consist in solving problems according to the following scheme: A) solve a problem by using  the Lagrangian or the Hamiltonian systems of equations;     solve a problem by using the generating functions;     classify the given partial differential equation. B) solve a problem by using a partial differential equation (with the method of separation of variables);     solve a problem on fluid dynamics ( by using the Navier-Stokes equations);     solve a problem by using Computation Fluid Dynamics. The part A) is the argument of the first proof. The part B) of the second proof. Both the two parts are the argument of an official examination session. In solving the problems it is required to specify the theorems or methods adopted.

 Bibliografia
 Notes on Mathematical Physics Beep portal S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Editore: Dover, Anno edizione: 1993

 Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
(hh:mm)
Ore di studio autonome
(hh:mm)
Lezione
52:00
78:00
Esercitazione
28:00
42:00
Laboratorio Informatico
0:00
0:00
Laboratorio Sperimentale
0:00
0:00
Laboratorio Di Progetto
0:00
0:00
Totale 80:00 120:00

 Informazioni in lingua inglese a supporto dell'internazionalizzazione
 Insegnamento erogato in lingua Inglese Disponibilità di materiale didattico/slides in lingua inglese Disponibilità di libri di testo/bibliografia in lingua inglese Possibilità di sostenere l'esame in lingua inglese Disponibilità di supporto didattico in lingua inglese

 Note Docente
 schedaincarico v. 1.6.5 / 1.6.5 Area Servizi ICT 03/12/2020