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 2024
- Offered
- 2026/27
- Learning activities
- Caratterizzante
- Area
- Formazione modellistico-applicativa
- Sector
- MAT/07
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Partial differential equations and their applications to mathematical modelling. Initial and boundary value problems. First- and second-order equations. Classification of second-order equations into elliptic, parabolic and hyperbolic types. Main models of mathematical physics and solution methods.
- Reference texts
- S. Salsa, "Equazioni a derivate parziali: Metodi, modelli e applicazioni" (Vol. 98). Springer, 2016.
- Educational objectives
- The course aims to provide students with the fundamental tools for the analysis of partial differential equations arising in the mathematical modelling of physical and applied phenomena. At the end of the course, students will be able to classify the main types of partial differential equations, understand the qualitative properties of their solutions, and apply the basic solution techniques to initial and boundary value problems. The course also aims to develop the ability to interpret mathematical models formulated through partial differential equations and to relate their mathematical structure to the underlying phenomena.
- Prerequisites
- Students are expected to have a basic knowledge of mathematical analysis, linear algebra, geometry and ordinary differential equations acquired during the Bachelor's degree programme. In particular, familiarity with matrices, eigenvalues and eigenvectors, multiple integrals, integral theorems of vector calculus, Fourier series and ordinary differential equations is required.
- Teaching methods
- The course consists of classroom lectures devoted to the presentation of the theoretical topics and the main methods for the analysis and solution of partial differential equations. Lectures are complemented by the discussion of relevant examples and applications in mathematical physics. All teaching material used during the course, including lecture notes, is made available to students through the Unistudium platform.
- Other information
- Class attendance is not mandatory, but it is strongly recommended. All teaching materials used during the course are available on the Unistudium platform.
- Learning verification modality
- Learning assessment consists of an oral examination lasting approximately 30–45 minutes. The examination is aimed at evaluating the student's knowledge and understanding of the topics covered in the course, the ability to apply the main theoretical results to problem solving, and the ability to present the subject matter in a rigorous and appropriate manner. Assessment takes into account the correctness of the answers, the command of mathematical language, and the ability to establish connections among the different topics of the course.
For information on support services for students with disabilities and/or specific learning disorders (DSA), please visit http://www.unipg.it/disabilita-e-dsa. - Extended program
- Introduction to partial differential equations: initial and boundary value problems, linear operators, classification of first-order equations and geometric interpretation. Quasilinear first-order equations, method of characteristics, Cauchy theorem and reduction to canonical form.
Separation of variables. Classification of second-order equations into elliptic, parabolic and hyperbolic types. Reduction to canonical forms and examples of constant-coefficient equations.
Heat equation: derivation of the model, well-posed problems, separation of variables, convergence of series solutions, uniqueness, maximum principles, fundamental solution and Dirac distribution. Introduction to random walks and their connection with diffusion processes.
Poisson's equation and harmonic functions. Dirichlet problems, mean value properties, maximum principles, Poisson formula, Harnack inequality, Liouville theorem, fundamental solution and Newtonian potential.
Conservation laws and transport models. Applications to pollutant transport and road traffic flow. Inflow and outflow characteristics, rarefaction waves, shock waves, Rankine–Hugoniot condition, Burgers equation, weak solutions, entropy condition and the Riemann problem.
Wave equation. Derivation of the vibrating string model, mechanical energy, Cauchy and Cauchy–Dirichlet problems, d'Alembert formula, domains of dependence and influence, generalized solutions and propagation of singularities. - Obiettivi Agenda 2030 per lo sviluppo sostenibile
- Goal 4 - Quality Education
The course contributes to the achievement of Sustainable Development Goal 4 by providing advanced skills in mathematical modelling and quantitative analysis of natural phenomena, fostering lifelong learning and supporting further studies in scientific disciplines.