ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: I0459

Unit Coordinator: Bruno Rubino

Aims:

The course is intended to introduce and develop an understanding of the concepts in nonlinear dynamical systems and bifurcation theory, and an ability to analyze nonlinear dynamic models of physical systems. The emphasis is to be on understanding the underlying basis of local bifurcation analysis techniques and their applications to structural and mechanical systems.

Content:

Review of: first-order nonlinear ODE, first-order linear systems of autonomous ODE. Local theory for nonlinear dynamical systems: linearization, stable manifold theorem, stability and Liapunov functions, planar non-hyperbolic critical points, center manifold theory, normal form theory. Global theory for nonlinear systems: limit sets and attractors, limit cycles and separatrix cycles, Poincaré map. Hamiltonian systems. Poincaré-Bendixson theory. Bifurcation theory for nonlinear systems: structural stability, bifurcation at non-hyperbolic equilibrium points, Hopf bifurcations, bifurcation at non hyperbolic periodic orbits. Applications.

Pre-requisites:

Ordinary differential equations

Reading list:

Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001

ECTS Credits: 3   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: DT0627

Unit Coordinator: Donatella Donatelli

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.

ECTS Credits: 3   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |      |   Code: DT0807

Unit Coordinator: Roberta Antonetti

Aims:

The aim of this course is to provide students with knowledge of Italian language at level A1 (beginner). The course will concentrate on four aspects of communication: speaking, writing, listening and reading on a basic level. Attention is given to correct pronunciation of Italian.
Furthermore, the goals of the course are to enable students to:
- develop the language proficiency required to communicate effectively in Italian at level A1;
- develop awareness of the nature of language and language learning;
- develop transferable skills;
- form a sound base of the skills, language and attitudes required for progression to work or further study, either in Italian or another subject area.
On successful completion of this module, a student should be able to:
- understand familiar words and very basic phrases concerning himself/herself, his/her family and immediate concrete surroundings when people speak slowly and clearly;
- read familiar names, words and very simple sentences, for example on notices and posters or in catalogues;
- interact in a simple way provided the other person is prepared to repeat or rephrase things at a slower rate of speech;
- ask and answer simple questions in areas of immediate need or on very familiar topics;
- use simple phrases and sentences to describe where he/she lives and people he/she knows;
- write short messages, postcards, fill in forms with personal details or a hotel registration form.

Content:

Students will be required to show knowledge and understanding of the broad topic areas listed below. These provide contexts for the acquisition of vocabulary and the study of grammar and structures. 
VOCABULARY AND TOPIC AREAS 
Clothes and accessories; colors; countries, nationalities and languages; culture, customs and celebrations; education e.g. learning institutions, education and training, learning tools, subjects; family and friends; feelings and emotions; food and drink; interests, sports and activities; jobs; measurements e.g. size, shape, weight; numbers (cardinal, ordinal) and money; rooms and furniture; shops and places; street directions; the human body and health e.g. parts of the body, health and illness; time expressions e.g. telling the time, dates, days of the week, months, seasons; travel and transport; weather.

FUNCTIONAL SYLLABUS
Accepting/ thanking; applying for a job; asking about personal information; asking permission; buying and asking prices; describing people and objects; giving opinions; giving instructions; greeting and introducing; inviting, refusing; making suggestions; requesting, offering; talking about likes and dislikes; talking about routines; telling the time.

GRAMMAR SYLLABUS
Sounds and spelling; nouns: gender, number; irregular plural nouns; definite article; indefinite article; adjectives, first group adjectives and second group adjectives, position of adjectives; formal address; pronouns, subject pronouns and direct and indirect object pronouns; possessive adjectives; prepositions simple and prepositions with articles; interrogatives; adverbs; present tense of regular and irregular verbs; modal verbs and the verb sapere; present perfect and some verbs with an irregular past participle.

Pre-requisites:

There are no prerequisites for attending the course.

Reading list:

"Dieci A1" - Alma edizioni, Firenze 2019.
“New Italian grammar in practice”, Alma edizioni, Firenze 2015.

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: DT0626

Unit Coordinator: Michele Palladino

Aims:

• Introducing basic tools of advanced real analysis such as metric spaces, Banach spaces, Hilbert spaces, bounded operators, weak convergences, compact operators, weak and strong compactness in metric spaces, spectral theory, in order to allow the student to formulate and solve linear ordinary differential equations partial differential equations, classical variational problems, and numerical approximation problems in an "abstract" form.

Content:

• Metric spaces, normed linear spaces. Topology in metric spaces. Compactness.
• Spaces of continuous functions. Convergence of function sequences. Approximation by polynomials. Compactness in spaces of continuous functions. Arzelà's theorem. Contraction mapping theorem.
• Crash course on Lebesgue meausre and integration. Limit exchange theorema. Lp spaces. Completeness of Lp spaces.
• Introduction to the theory of linear bounded operators on Banach spaces. Bounded operators. Dual norm. Examples. Riesz' lemma. Norm convergence for bounded operators.
• Hilbert spaces. Elementary properties. Orthogonality. Orthogonal projections. Bessel's inequality. Orthonormal bases. Examples.
• Bounded operators on Hilbert spaces. Dual of a Hilbert space. Adjoin operator, self-adjoint operators, unitary operators. Applications. Weak convergence on Hilbert spaces. Banach-Alaoglu's theorem.
• Introduction to spectral theory. Compact operators. Spectral theorem for self-adjoint compact operators on Hilbert spaces. Hilbert-Schmidt operators. Functions of operators.
• Introduction to the theory of unbounded operators. Linear differential operators. Applications.
• Introduction to infinite-dimensional differential calculus and variational methods.

Pre-requisites:

Basic calculus and analysis in several variables, linear algebra.

Reading list:

• John K. Hunter, Bruno Nachtergaele, Applied Analysis. World Scientific.
• H. Brezis, Funtional Analysis, Sobolev Spaces, and partial differential equations. Springer.

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: I0183

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing basic properties and main techniques to solve basic partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematical Modelling was verified at the time of the planning of the master program.

LEARNING OUTCOMES.
At the end of the course, the student should:

1. know basic properties (existence, uniqueness, etc.) and main techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic partial differential equations (semilinear first order PDEs, heat, Laplace, wave equations);
2. understand and be able to explain thesis and proofs in the field of basic partial differential equations;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

First order partial differential equations. Definition of characteristic vectors and characteristic surfaces. Characteristics for (semi)linear partial differential equations of first order in two independent variables. Existence and uniqueness to initial value problems for first order semilinear partial differential equations in two independent variables Duhamel’s principle for non homogeneous first order partial differential equations.

Second order partial differential equations. Classification of second order semilinear partial differential equations in two independent variables. Canonical form for second order semilinear partial differential equations in two independent variables. Classification for second order semilinear partial differential equations in many independent variables.

Heat equation. Derivation of heat equation and well–posed problems in one space dimension. Solution of Cauchy–Dirichlet problem for one dimensional heat equation by means of Fourier method of separation of variables. Energy method and uniqueness. Maximum principle. Fundamental solution. Solution of global Cauchy problem. Non homogeneous problem: Duhamel’s principle.

Laplace equation. Laplace and Poisson equation: well-posed problems; uniqueness by means of energy method. Mean value property and maximum principles. Laplace equation in a disk by means of separation of variables. Poisson’s formula. Harnack’s inequality and Liouville’s Theorem. Fundamental solution of Laplace operator. Solution of Poisson’s equation in the whole space. Green’s functions and Green’s representation formula.

Wave equation. Transversal vibrations of a string. Well–posed problems in one space dimension. D’Alembert formula. Characteristic parallelogram. Domain of dependence and range of influence. Fundamental solution for one dimensional wave equation. Duhamel’s principle for non homogeneous one dimensional wave equations. Special solutions of multi–d wave equation: planar and spherical waves. Well–posedness for initial, boundary value problems: uniqueness by means of energy estimates. Separation of variables. Domain of dependence and range of influence in several space variables. Fundamental solution for multi–dimensional wave equation. Solution of 3–d wave equation: Kirchhoff’s formula and strong Huygens’ principle. Wave equation in two dimensions: method of descent. Fundamental solution in 2–d. Duhamel’s principle for non homogeneous wave equation in 3–d: delayed potentials.

Pre-requisites:

Students must know the basic notions of mathematical analysis, including Fourier series and ordinary differential equations, and the basic notions of continuum mechanics.

Reading list:

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS, 2010.

- S. Salsa. Partial Differential Equations in Actions: from Modelling to Theory. Springer–Verlag Italia, 2008.

- S. Salsa, G. Verzini. Equazioni a derivate parziali: complementi ed esercizi. Springer–Verlag Italia. 2005.

- W.A. Strauss. Partial Differential Equations: An Introduction. John Wiley & Sons Inc., 2008.

- E.C. Zachmanoglou, D.W. Thoe. lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc., 1986.

Additional info:

http://people.disim.univaq.it/~corrado/

ECTS Credits: 6   |   Semester: 1   |   Year: 1   |   Campus: University of L'Aquila   |   Language: English   |   Code: I0062

Unit Coordinator: Alessandro Borri

Aims:

The course provides basic and advanced methodologies for the modeling, analysis and design of control systems.

Content:

Frequency domain models of Linear Systems: Laplace Transform, Transfer Functions, Block diagrams.
Time domain models of Linear Systems: State space representation.
BIBO stability.
Control specifications for transient and steady-state responses. Polynomial and sinusoidal disturbances rejection.
Digital control: Z-Transform, discretization of continuous-time linear systems, finite-time response, deadbeat response.
Analysis and control design using the eigenvalues assignment: controllability, observability, the separation principle.
Controller design using MATLAB.

Pre-requisites:

Mathematical analysis.

Reading list:

[1] K. J. Astrom, R. M. Murray, "Feedback Systems: An Introduction for Scientists and Engineers". Princeton Univ. Pr., 2nd Edition, 2021
[2] R. C. Dorf, R. H. Bishop. "Modern Control Systems". Prentice Hall, 12th Edition, 2008 
[3] G. F. Franklin, J. D. Powell, A. Emami-Naeini. "Feedback control of dynamic systems". Prentice Hall, 4th Edition, 2002
[4] G. F. Franklin, J. D. Powell, M. L. Workman. "Digital Control of Dynamic Systems". Addison-Wesley, 3rd Edition, 1998
[5] K. Ogata. "Discrete-Time Control Systems". Prentice Hall, 2nd Edition, 1995
[6] A. Isidori. "Sistemi di controllo". Siderea, 1996
[7] Lectures slides

ECTS Credits: 9   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Corrado Lattanzio

Aims:

LEARNING OBJECTIVES.
The course aims at providing advanced mathematical notions used in the field of (applied) mathematical analysis and their applications to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Those objectives contribute to the learning goals of the entire course of studies, as the inner coherence of the master degree in Mathematics was verified at the time of the planning of the master program.
LEARNING OUTCOMES.
At the end of the course, the student should:
1. know the advanced mathematical notions used in the field of (applied) mathematical analysis, as measure theory, Sobolev Spaces, distributions, and their applications to the theory of linear and non-linear partial differential equations;
2. understand and be able to explain thesis and proofs in advanced mathematical analysis;
3. have strengthened the logic and computational skills;
4. be able to read and understand other mathematical texts on related topics.

Content:

Distributions. Locally integrable functions. The space of test function D(Ω). Distributions associated to locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions. Fourier transform and tempered distributions. Convolution between distributions and regularization of distributions. Denseness of D(Ω) in D′(Ω).

Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(Ω) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo–Nirenberg–Sobolev inequality and embedding theorem for p < n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich–Kondrachov theorem, Poincaré inequalities. Embedding theorem for p = n. Characterization of the dual space H-1.

Second order parabolic equations. Definition of parabolic operator. Weak solutions for linear parabolic equations. Existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions.

First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem.

Measures. System of sets, Positive Measures, Outer Measures, Construction of Measures, Signed Measures, Borel and Radon Measures
Integration. Measurable Functions, Simple Functions, Convergence Almost Everywhere, Integral of Measurable Functions, Convergences of Integrals, Fubini-Tonelli Theorems.
Differentiation. The Radon-Nikodym Theorem, Differentiation on Euclidean space, Differentiation of the real line.
Radon measures and continuous functions. Spaces of continuous functions, Riesz Theorem.

Pre-requisites:

Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.

Reading list:

- L. Ambrosio, G. Da Prato, A. Mennucci. Introduction to measure theory and integration. Edizioni della Normale.

- L. Ambrosio, N. Fusco, D. Pallara. Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs.

- V.I. Bogachev. Measure theory, Volume I, Springer.

- H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer.

- P. Cannarsa, T. D’Aprile. Introduction to Measure Theory and Functional Analysis. Springer.

- C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, Springer.

- L.C. Evans. Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, AMS.

- L. Evans, R. Gariepy. Measure Theory and Fine Properties of Functions, CRC Press. Revised Edition.

- G.B. Folland. Real analysis: Modern techniques and their applications. New York Wiley

- G. Gilardi. Analisi 3. McGraw–Hill.

- L. Grafakos, Classical Fourier Analysis. Springer.

- V.S. Vladimirov. Equations of Mathematical Physics. Marcel Dekker, Inc

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Carmela Scalone

Aims:

Learning objectives.

The aim of the course is to provide knowledge and skills in the field of numerical modelling in the epidemiological field, creating an effective bridge between the understanding of the continuous model, its qualitative properties and their counterparts in the discrete.

Learning outcomes.

At the end of the course, the student should

- have a deep knowledge and understanding of the most relevant computational techniques for the treatment of models in the epidemiological field, together with aspects related to their implementation in accurate and efficient mathematical software;
- demonstrate the ability to evaluate the most appropriate discretization in relation to the problem to be solved and ability in theoretical analysis and mathematical software design;
- demonstrate ability to read and understand other texts on related topics.

Content:

- Compartmental models in Epidemiology.
- Stochastic models.
- Epidemiological models based on pds.
- An introduction to network models for the spread of epidemics. 

For each theme, the presentation will be twofold: presenting the continuous model, its qualitative properties and, jointly, presenting the most appropriate numerical approach to the approximation of solutions and the discrete conservation of the properties of the model. In addition, advanced techniques of numerical linear algebra and optimization for large linear systems arising from semi-discretization of PDEs in epidemiology will be presented.

Pre-requisites:

Basic knowledge of Numerical Analysis, differential equations, linear algebra.

Reading list:

- F. Brauer et al., Mathematical Epidemiology, Springer-Verlag (2008).
- J. D. Lambert, Numerical methods for ordinary differential systems: the initial value problem, John Wiley (1991).
- A. Quarteroni, Numerical Models for Differential Problems, Springer (2017).
- E. Isaacson, H.Keller, Analysis of numerical methods, Dover Publications (1994).
- D.J. Higham and P. E. Kloeden, An Introduction to the Numerical Simulation of Stochastic Differential Equations, SIAM, 2021.
--Walter Gander, Martin J. Gander, Felix Kwok. Scientific Computing
An introduction using Maple and MATLAB. Springer.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Marco Di Francesco, Antonio Esposito

Aims:

This course contributes to one of the learning objectives of the degree programmes in Mathematical Modelling and Mathematical Engineering, namely the formation of the student on advanced mathematical modelling in an interdisciplinary context, in particular in the biology/medicine area, more specifically in the field of mathematical modelling applied to the diffusion of epidemics, which is the main goal of the curriculum "Modelling and simulation of infectious diseases" of the degree program in Mathematical Modelling. Moreover, the course contributes to forming the student in the context of mathematical modelling applied to population dynamics, in agreement with the objectives of the curriculum "Mathematicla Models in Social Sciences" of the degree programme in Mathematical Modelling.

At the end of the course, the student 

1) will be equipped with a sound knowledge on population dynamics modelling, particularly compartmental models which can also be applied in epidemiology.

2) will be able to formulate "ad-hoc" deterministic models, such as ordinary differential equations, partial differential equations, interacting particle systems, that describe the dynamics of an epidemics in specific situations.

3) will possess analytical techniques allowing to resolve the models studied and to determine the qualitative behaviour of the solutions to those models.

4) complement the models with auxiliary terms in order to plan specific "containment strategies" for epidemiological models.

Content:

1) Introduction to population dynamics modelling via ODEs.

2) Introduction to epidemic modelling. The Kermack-McKendrick models and its variants.

3) Modelling of vector-borne diseases. Models with delay.

4) Computing the basic reproduction number.

5) Complex epidemics modelling, multi-group modelling.
Control strategies in ODE models.

6) Age structured population models and applications in epidemiology. Class-Age structured models in epidemiology.

7) Spatial heterogeneity in population dynamics. Models with diffusion.

8) Reaction-diffusion systems. Travelling waves.
Lagrangian movements. Particle models. Nonlocal models. Discrete vs Continuum modeling using integro-differential equations.

10) SIR (and variants) models with local and nonlocal diffusion.

11) Graph modeling for population dynamics

Pre-requisites:

Dynamical systems, analytical methods for to resolutoin of ordinary differential equations.
Basics of linear partial differential equations.

Reading list:

James D. Murray; Mathematical biology I: An introduction; Springer.

James D. Murray; Mathematical biology II: Spatial Models and biomedical applications; Springer.

Fred Brauer, Pauline van den Driessche, Jianhong Wu; Mathematical Epidemiology; Lecture notes in mathematics; Springer.

Maia Martcheva - An Introduction to Mathematical Epidemiology; Texts in Applied Mathematics 61, Springer.

Lecture notes by the lecturer.

ECTS Credits: 6   |   Semester: 1   |   Year: 2   |   Campus: University of L'Aquila   |   Language: English

Unit Coordinator: Giordano Pola

Aims:

The aim of this course is to pro