Mathematical Modelling and Programming

The course is offered in the first term of the PhD and Master’s Programmes in Science and Management of Climate Change. It is one of the core courses aimed at providing students with the mathematical and computational tools necessary to understand the dynamics of socio-economic and environmental systems and apply dynamic models to describe and forecast their functioning and evolution. This course is fundamental for understanding environmental and socio-economic modeling, which will be further developed in the applications during subsequent courses.

1. Knowledge and understanding:
* Understand the fundamentals of multivariable calculus, linear algebra, integration, and optimization, with a specific focus on their application in dynamic models;
* Acquire the foundational knowledge needed to comprehend and solve ordinary differential equations (ODEs) and dynamic systems in both continuous and discrete time;
* Be able to define and describe complex phenomena related to environmental and socio-economic systems, interpreting them through system dynamics;
* Acquire the theoretical and methodological tools necessary to develop a systemic view of environmental and socio-economic phenomena.
2. Applying knowledge and understanding:
* Be able to select and use appropriate mathematical tools to describe and analyze complex phenomena;
* Solve practical problems related to dynamic systems using mathematical models;
* Be capable of addressing specific applied problems related to climate change using cross-disciplinary mathematical tools.
3. Judgment skills:
* Be able to identify the strengths and limitations of the mathematical models used in environmental and socio-economic modeling.

The student is expected to have a thorough understanding of the following topics: set theory, elementary algebra, equations and inequalities (first and second degree, including inequalities with absolute values), analytic geometry, functions and their properties (domain, codomain, injectivity, surjectivity, composition, and inverse), trigonometry, exponential and logarithmic functions, and their respective properties.

Additionally, the student should possess a basic knowledge of the fundamentals of calculus, including limits, continuity, derivatives, integrals, monotonicity, convexity, as well as essential notions of vectors and matrices.

It is INDISPENSABLE that the student arrives at the course with these competencies already well established.

1. Introduction:
* Course presentation, review of prior knowledge.
2. Elements of calculus:
* Review: derivatives, integrals, Taylor series expansions;
* Definition of multivariable functions, examples. Partial derivatives, gradient, Jacobian matrix, and Hessian matrix.
3. Linear algebra applied to dynamic systems:
* Review: vectors, dot and cross products, linear dependence and independence, matrices, matrix addition and multiplication, transpose matrix, determinant, rank;
* Solving systems of linear equations;
* Eigenvalues, eigenvectors, and their role in dynamic systems.
4. Dynamical systems models:
* Definition of dynamic system, complex system, continuous and discrete time systems, stability, equilibria. Examples;
* Ordinary Differential Equations (ODE) 1: types of ODEs, Cauchy problem, existence and uniqueness of solutions;
* Ordinary Differential Equations (ODE) 2: solution methods: separable variables, Lagrange's general integral. Finite difference equations. Introduction to second-order differential equations;
* Linear dynamic systems;
* Introduction to nonlinear dynamics, chaos theory, and catastrophe theory.
5. Numerical methods:
* Application of Euler and Runge-Kutta methods.
6. Optimization and optimal control:
* Unconstrained and constrained optimization, FOC, SOC, in one and multiple variables. Lagrange multipliers;
* Introduction to control theory for dynamic systems.

Attendance at the lectures is mandatory.
Relevant teaching materials will be provided during the lectures and made available on Moodle.
Some reference books will be recommended throughout the course.

The assessment will include a final written exam, possibly followed by an oral examination. During the course, some intermediate assessment tests may be administered.

The lectures will take place in the classroom, with explanations given on the board and supported by multimedia materials to enhance intuition and understanding of the concepts.

English

written and oral