- ECTS Credits: 4.5
- Semester: 1
- Year: 2
- Campus: Ivan Franko National University of Lviv
- Language: English
- Unit Coordinator: dr hab. inż. Edyta Hetmaniok, prof. PŚ
- ECTS Credits: 5
- Year: 2
- Campus: Silesian University of Technology
- Language: English
- Aims:
The aim of the course is to get knowledge on constructing mathematical models of selected technical and economic problems, verifying them and simulating by using Mathematica software.
- Content:
1. Types of models, types of variables, selection of variables to the model.
2. Construction of the model.
3. Linear model - assumptions, estimation of parameters, verification procedures.
4. Analysis of the selected nonlinear models.
5. Analysis of the selected models described by means of differential equations.
6. Computer simulations of selected models.
- Code: DT0627
- Unit Coordinator: Donatella Donatelli
- ECTS Credits: 3
- Semester: 1
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Aims:
Learning Objectives:
The aim of the course is to give an overview of fluid dynamics from a mathematical viewpoint, and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for the analysis of other partial differential equations.Learning Outcomes:
On successful completion of this course, the student should:- understand the basic principles governing the dynamics of non-viscous fluids;
- be able to derive and deduce the consequences of the equation of conservation of mass;
- be able to apply Bernoulli's theorem and the momentum integral to simple problems including river flows;
- understand the concept of vorticity and the conditions in which it may be assumed to be zero;
- calculate velocity fields and forces on bodies for simple steady and unsteady flows derived from potentials;
- demonstrate skill in mathematical reasoning and ability to conceive proofs for fluid dynamics equations.
- demonstrate capacity for reading and understand other texts on related topics. - Content:
CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical Modelling of Continuum Media (3 ECTS)
- Derivation of the governing equations: Euler and Navier-Stokes
- Eulerian and Lagrangian description of fluid motion; examples of fluid flows
- Fluidi di tipo Poiseulle e Couette
- Vorticity equation in 2D and 3D
CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid dynamics (6 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS),- Dimensional analysis: Reynolds number, Mach Number, Frohde number.
- From compressible to incompressible models
- Existence of solutions for viscid and inviscid fluids
- Fluid dynamic modeling in various fields: mixture of fluids, combustion, astrophysics, geophysical fluids (atmosphere, ocean)CONTENTS FOR: Modelling and analysis of fluids and biofluids (9 ECTS), Mathematical fluid and biofluid dynamics (6 ECTS)
- Modeling for biofluids: hemodynamics, cerebrospinal fluids, cancer modelling, animal locomotion, bioconvection for swimming microorganisms.
- Pre-requisites:
PREREQUISITES for Mathematical Modelling of Continuum Media:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.
PREREQUISITES for Mathematical fluid and biofluid dynamics, Mathematical fluid dynamics, Modelling and analysis of fluids and biofluids:
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations, Sobolev spaces. - Reading list:
- Alexandre Chorin, Jerrold E. Marsden, A Mathematical Introduction to Fluid Mechanics. Springer.
- Roger M. Temam, Alain M. Miranville, Mathematical Modeling in Continum Mechanics. Cambridge University Press.
- Franck Boyer, Pierre Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models. Springer-Verlag Italia.
- Andrea Bertozzi, Andrew Majda, Vorticity and Incompressible Flow. Cambridge University Press.
- Unit Coordinator: Marco Di Francesco, Antonio Esposito
- Programme: InterMaths
- ECTS Credits: 6
- Semester: 1
- Year: 2
- Campus: University of L'Aquila
- Language: English
- Delivery: In-class
- Aims:
At the end of the course, the student will be familiar with multi-agent systems models of discrete type (both deterministic and stochastic), with their meso-scopic formulation, with their continuum formulation, both of first and second order, and with their formulation on a graph. The students will acquire the mathematical techniques to solve those models suitably and will be able to use those models in various interdisciplinary applications and to adapt them to specific problems in contexts of interest in health-care systems such as diagnostics and imaging, neural networks, genetics, epidemiology, dynamic data management, and biological aggregation phenomena in physiology.
- Content:
- Discrete particle systems for interacting agents. Models with external field. Models with nonlocal aggregation/repulsion forces. Models with alignment, self-propulsion and friction. Swarms models (Vicsek) . Opinion models (Sznajd, Krause). Examples of asymptotic behaviour. The stochastic case.
- Control for discrete models. Mean-field games. Application to optimisation problems. Many species models and models with species transitions. Applications in genetics, imaging, and data science.
- Complementary topics of abstract measure theory. Measure topologies. Transport of measures.
- Mesoscopic models of Vlasov type. Derivation as mean field limits from discrete particle models. Formal derivation of continuum second order models. Derivation of first order models in friction dominated regimes.
- Derivation of linear and nonlinear diffusion models from particle systems. Existence of solutions to the nonlinear diffusion equation. Asymptotic self-similar behavior.
- Aggregation-diffusion equations. Existence of solutions with linear diffusion. Existence in the diffusion-less case and formation of clusters in finite time. Stationary states with quadratic diffusion. The case of many species. Application to epidemiology.
- Introduction to graph modelling of nonlocal type. Applications to neural networks.
- Probabilistic label-switching models and applications.
- Pre-requisites:
Ordinary differential equations, real and functional analysis
- Reading list:
Lecture notes will be provided.
