Unit MATHEMATICAL PHYSICS I
- Course
- Mathematics
- Study-unit Code
- 55109909
- Curriculum
- In all curricula
- Teacher
- Francesca Di Patti
- Teachers
-
- Francesca Di Patti
- Hours
- 42 ore - Francesca Di Patti
- CFU
- 6
- Course Regulation
- Coorte 2020
- Offered
- 2022/23
- 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 differential equations. First and second order equations. Initial and boundary value problems. Equations of hyperbolic, parabolic and elliptic type. Methods of solution and applications.
- Reference texts
- S. Salsa, Equazioni a derivate parziali: Metodi, modelli e applicazioni. Unitext, Springer.
A. Tichonov and A. Samarsky, Equazioni della fisica matematica, Editori riuniti, 2020.
Tyn-Mynt,U. and L. Debnath, Partial Differential Equations for Scientist and Engineer, North Holland. - Educational objectives
- The goals of this course are:
- provide students with the mathematical tools that are essential to the formation of an undergraduate student to tackle problems related to mathematical models implemented by problems for partial differential equations,
- motivating the study of these instruments, indicating the issues that led to their development also showing applications.
- to be able to study and analyze simple mathematical models concerning partial differential equations and to study the classical solutions.
These objectives involve the discussion of problems of classical mathematical physics such as: first order linear equations and their applications, second order linear equations: elliptic, parabolic and hyperbolic types. These describe the main mathematical models regarding population dynamics, potential, heat distribution, diffusion and reaction of interacting elements, vibrating string. - Prerequisites
- A mandatory prerequisite for students, planning to attend the course with profit, is the knowledge and resolution of:
matrices, eigenvalues and eigenvectors; multiple integrals, surface integrals;
divergence and transport theorems;
ordinary differential equations, Cauchy problems;
Fourier series and their respective convergence theorems;
fundamental law of the dynamics, energy of a material system. - Teaching methods
- Lectures on all subjects of the course and respective exercises.
Supporting material and a detailed program will be posted on unistudium. - Other information
- It is recommended to attend the lectures.
- Learning verification modality
- The oral examination consists in an interview about 2/3 arguments treated during the course. This allows to verify the ability of knowledge and understanding, the ability to apply the acquired skills, the ability to display and learn. Operating time up to 30/45 minutes.
For information about services for students with disabilities and/or DSA visit the page http://www.unipg.it/disabilita-e-dsa. - Extended program
- First-Order, Quasi-Linear Equations and Method of Characteristics
Classification of First-Order Equations
Construction of a First-Order Equation
Geometrical Interpretation of a First-Order Equation
Method of Characteristics and General Solutions ¿Canonical Forms of First-Order Linear Equations ¿Method of Separation of Variables
Classification of Second-Order Linear Equations
Second-Order Equations in Two Independent Variables
Canonical Forms¿Equations with Constant Coefficients
GeneralSolutions
The Diffusion Equation
Uniqueness¿The Fundamental Solution ¿Symmetric Random Walk (n=1)
The Global Cauchy Problem (n=1)
The Laplace Equation
Well Posed Problems. Uniqueness
Harmonic Functions
Fundamental Solution
The Green Function
Scalar Conservation Laws
Linear Transport Equation
Traffic Dynamics
Integral (or Weak) Solutions
Waves and Vibrations
Types of waves
The One-dimensional Wave Equation
The d’Alembert Formula
Small vibrations of an elastic membrane
Small amplitude sound waves ¿