Full detailed program:
1.0 General principles of mathematical modeling.
1.1 Complex numbers and vector quantities.
1.2 Cartesian, cylindrical and spherical orthogonal coordinates. Change of coordinates: "Jacobian". Angles of Euler.
1.3 Matrix Calculation; search for main components (eigenvalues, eigenvectors); notes on the Tensors; rigid roto-translations and associated matrix, as an operator
1.4 Elements of analytical geometry.
1.5 Taylor series expansion. Notes on differential geometry; concept of curvature of a line and of a surface (applications to slopes and to capillary phenomena).
1.6 Introduction to differential Operators: gradient with symbolic calculation; divergence, rotor and their physical meaning; Theorems of Green-Gauss and Stokes.
1.7 Gamma, Dirac and Heaviside Functions; derivatives of discontinuous functions.
1.8 "Material" derivative; physical meaning of partial and total derivative with respect to time; derivative of a vector with respect to the time: Poisson formulas; derivative of an integral; notes on fractional derivatives.
1.9 General definitions of differential equations; Ordinary Differential Equations; Partial Differential Equations.
1.10 Overview of the Minimum Principles and Elements of Variational Calculus; Example of a vibrating bar.
1.11 Meaning and usefulness of the expansion in series of functions: introductory concepts on Hilbert Spaces as formal bases for the use of orthogonal functions expansion (Fourier).
2.0 Models related to static equilibrium (processes independent of time).
3.0 Dynamic models (processes that evolve over time).
3.1 General form of the balance equation of scalar, vectorial and tensor quantities and equations derived from it, by means of experimental laws (reasons for which a mathematical model may or may not provide useful information):
3.1.1 Diffusion of a substance in a non-moving medium (Fick's laws);
3.1.2 Diffusion of a substance in a moving medium: advective equation;
3.1.3 Dynamic equation (Cauchy-Beltrami) for slope stability analysis;
3.1.4 Equations of heat exchange (Fourier Law): conduction and convection;
3.1.5 Wave equation; discrete model and continuous model;
3.1.6 Navier Stokes equation;
3.1.7 Equations of seismic perturbations propagation.
3.2 Application of the Navier Stokes equation to justify Darcy's law.
3.3 Reynolds dynamic equation.
3.4 Equation of water and air infiltration in unsaturated soil (Richards, Fokker-Plank)
3.5 Overview of the Law of Consolidation of Biot in 2D and 3D.
3.6 Overview of the "debris flow" equations and of the Shallow Water model.
3.7 Notes on models pertaining to Geophysics investigation techniques using geoelectric, geomagnetic and georadar methods; interaction of electromagnetic waves with matter.
4.0 Dimensional analysis: Reynolds, Froude, Rayleigh, Grashov numbers.
5.0 Overview of Earth's electromagnetism.
6.0 Analytical aspects of a mathematical model expressed by differential equations.
7.0 General approach on the correct setting of a mathematical model by means of differential equations; Notes on the stability of differential equations and on the mathematical theory of the Stability of Dynamical systems.
8.0 Notes on the methods of analytical solution of the main Ordinary and Partial differential equations related to Geology and Environmental Modeling:
8.1 Partial Differential Equations:
8.1.1 Method of the variable separation. Solution of the following equations:
8.1.1.1 Solution of the diffusion equation;
8.1.1.2 Solution of the equation of forced oscillations of discrete systems;
8.1.1.3 Solution of the wave equation (seismic, electromagnetic);
8.1.1.4 Solution of the Consolidation equation;
8.1.1.5 Solution of the vibration equation of a one-dimensional bar;
8.1.1.6 Solution of the vibration equation of a membrane; Expansion of solutions into eigenfunctions with their eigenvalues;
8.1.1.7 Solution of the telegraphic equation;
8.1.1.8 Sturm_Liouville problem and expansion in orthogonal Fourier functions (trigonometric, Bessel, Hankel, Legendre, Chebyshev-Hermite, Chebyshev-Laguerre);
8.1.1.9 Solution of Schrodinger Equation for the hydrogen atom;
8.1.1.10 Fokker Plank Equation.
9.0 Method of the Transforms: Fourier and Laplace.
10.0 Interpolation of numerical data: Lagrange, Hermite polynomial, spline, serendipity.
11.0 Elements of statistics and probability calculation in Geology and Environmental Engineering:
11.1 Inequality of Chebyshev;
11.2 Overview of the theory of errors and the problem of experimental measurement;
11.3 Basics of Probability Calculation (Binomial or Bernoulli Distribution, Poisson Distribution, Normal or Gauss Distribution or Error Distribution, Standard Normal Distribution, Lognormal Probability Distribution, Weibull probability distribution;
11.4 Multivariate Analysis;
11.5 Verification of a statistical hypothesis;
11.5.1 Example 1: statistical analysis of the belonging of rock samples to a specific site, based on the measurement of porosity;
11.5.2 Example 2: Membership of a statistical sample at a given Distribution; analysis of the randomness of the concentration values of a pollution emitted into the sea by a river.
12.0 Processing of observation data:
12.1 Least square methods;
12.2 Fourier analysis with applications;
12.3 Analysis of numerical data using the FFT (Fast Fourier Transform) method;
13.0 General principles of numerical calculus:
13.1 Evaluation of truncation and rounding errors and their propagation;
13.2 Overview of the numerical analysis of seismic data:
13.2.1 Application of the "Fast Fourier Transform";
13.2.2 Numerical filters;
13.3 Solution of linear and non-linear systems (concepts and general definitions):
13.3.1 Method of conjugate or Cholesky gradient;
13.3.2 Solution of non-linear systems (Newton Rapshon method);
13.4 Numerical integration of differential equations:
13.4.1 Overview of the Mesh-less Method;
13.4.2 Finite difference method;
13.4.3 Finite element method (F.E.M.);
13.4.4 Numerical solution of time-dependent problems;
13.4.5 Comparison of methods.
13.5 Numerical modeling of pollution of a wetland under stationary conditions by means of the Finite Element Method;
13.6 Examples of routines in FORTRAN95 language.
