Unit MATHEMATICAL PHYSICS I

Course
Mathematics
Study-unit Code
55109909
Curriculum
In all curricula
Teacher
Diletta Burini
Teachers
  • Diletta Burini
Hours
  • 42 ore - Diletta Burini
CFU
6
Course Regulation
Coorte 2022
Offered
2024/25
Learning activities
Caratterizzante
Area
Formazione modellistico-applicativa
Academic discipline
MAT/07
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Italian
Contents
Partial derivative equations. Mathematical models. First- and second-order linear equations and related initial and/or boundary value problems. Second-order equations of hyperbolic, parabolic and elliptic type. Solving methods.
Reference texts
S. Salsa, "Equazioni a derivate parziali: Metodi, modelli e applicazioni" (Vol. 98). Springer, 2016.
Educational objectives
The aims of this course are
- provide students with the essential mathematical tools that must be part of a Bachelor's degree student's training to tackle problems related to mathematical models implemented by problems to partial differential equations.

- motivate the study of these tools by pointing out the issues that led to their development and also by showing applications.

- be able to tackle the study and analysis of simple mathematical models involving partial differential equations.

These objectives involve the treatment of classical topics in mathematical physics. Linear equations of the first order and their applications, linear equations of the second order: elliptic, parabolic and hyperbolic equations and a description of the main mathematical models involving population dynamics, potential, heat diffusion, diffusion and reaction of interacting substances, vibrating chords.
Prerequisites
The knowledge and skills prerequisites for students wishing to successfully follow the course are:
matrices, eigenvalues and eigenvectors;
multiple integrals, surface integrals;
divergence and transport theorem;
ordinary differential equations, Cauchy problems;
Fourier series developments and their convergence theorems;
fundamental laws of dynamics, energy of a material system.
Teaching methods
Classroom lectures on all course topics and related exercises.

Students will find the detailed syllabus of the lectures and additional material on unistudium.
Other information
Class attendance: optional but recommended.
Learning verification modality
A single final interview lasting 30/45 minutes.
The oral test consists of a discussion of 2/3 topics aimed at ascertaining the level of knowledge and understanding reached by the student on the theoretical and methodological contents indicated in the programme. This oral test will also be an opportunity to verify the degree of communication achieved by the student with language property and autonomous organisation of the exposition.

For information on support services for students with disabilities and/or DSA visit http://www.unipg.it/disabilita-e-dsa
Extended program
Basic concepts and definitions of PDEs, initial value and boundary problems, linear operators, classifications of first-order PDEs, geometric interpretation of a first-order PDE.
Solving a quasi-linear PDE with the method of characteristics, Cauchy's theorem for a first-order PDE, Cauchy's theorem for a quasi-linear first-order PDE, exercises on solving first-order PDEs with the method of characteristics.
Separation of variables method for a first-order PDE and related exercises, classification of second-order PDEs, reduction to the first canonical form for a hyperbolic equation.
Second canonical form of hyperbolic equations, canonical form of parabolic and elliptic equations, special case of second order equations with constant coefficients, general solutions.
Heat equation, heat conduction, well-placed problems (case n=1), the method of separation of variables.
Issues related to the solution found by the method of separation of variables (convergence of the series, uniqueness, initial conditions), maximum principle in weak form.
Enunciation of the strong maximum principle, the fundamental solution for n=1, the Dirac distribution, introduction to the symmetric random walk.
Calculation of the first and second moment in the symmetric random walk, generating functions, the limit transition probability, hints at the connection with Brownian motion, hints at the global homogeneous Cauchy problem for n=1, Poisson's equation, well-placed problems and uniqueness.
Harmonic functions in the discrete, discrete Dirichlet problem, averaging properties, maximum principles.
Dirichlet's problem in a circle, Poisson's formula.
Harnack's inequality, Liouville's theorem, the fundamental solution, potentials in finite domains.
Green's function for the Dirichlet problem, method of images in the upper half-space and in the sphere, Green's representation formula, conservation laws, pollutant in a river.
Pollutant in a river with distributed source, extinction and localised source, inflow and outflow characteristics, road traffic, solving by the method of characteristics, what at a traffic light.
Rarefaction waves, increasing traffic with x, Rankine-Hugoniot condition, shock wave, breaking time.
Integral solution, Rankine-Hugoniot condition and integral solutions, Burgers equation, entropy condition, entropy inequality.
Entropy condition theorems, Riemann problem, types of waves.
Model for transverse vibrations of a string, wave equation, energy, initial and a conditions at the edge, method of separation of variables.
Uniqueness and dependence on initial data of the solution of the Cauchy-Dirichlet problem, solution of the global Cauchy problem, d'Alembert's formula, introduction to singularity propagation.
Rankine-Hugoniot type condition for the wave equation in which a singularity propagates, the fundamental solution, non-homogeneous equation solved by Duhamel's method, special solutions for n>1 (progressive plane waves, cylindrical waves and spherical waves.
Spherical waves, small vibrations in an elastic membrane, solution for a square membrane, sound waves in gases.
Fundamental solution at n=3, Huygens' principle, Kirchhoff's formula.
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