Full detailed program:
1. Basic tools of Computational Modeling. Types of quantities used - Cartesian, cylindrical and spherical orthogonal coordinates - Change of coordinates: "Jacobian" - Elements of matrix calculus (eigenvalues, eigenvectors, hints on tensors, rigid rototranslations, matrices as operators, inversion of matrices and LU Theorem) - Elements of analytic geometry (conic as quadratic forms) - Derivatives: "substantial", Eulerians and Lagrangians, physical meaning of partial and total derivative with respect to time, derivative with respect to time of a vector, Poisson formulas, derivative of an integral, notes on fractional derivatives (anomalous diffusion) - Introduction to differential operators (gradient, divergence, rotor and their physical meaning, Green-Gauss and Stokes' theorems, Symbolic calculus) - Notable functions (Gamma; Dirac; Heaviside; derivatives of discontinuous functions) - Inertial, non-inertial reference systems.
2. Differential equations. General definitions - Ordinary differential equations - Partial differential equations - Application notes on the stability of differential equations and on the Mathematical Theory of Stability of Dynamic Systems - Notes on the onset of deterministic chaos also in climatology (Lorentz equations, Strange attractors) - Notes on the Principles of Minimum and Elements of Variation Calculus - Example of vibrating bar.
3. General form of the balance equation of scalar, vector and tensor quantities and equations derived from it by means of experimental laws. Diffusion of a substance in a moving and non-moving medium (Fick's laws, advection), with transport of pollutants, sediments - heat exchange by conduction and convection - wave equation - discrete model and continuous model (notes on the propagation of seismic perturbations) - infiltration of water and air in unsaturated soil (Richards, Fokker-Plank) - Notes on the Biot Consolidation Law - Notes on the "debris flow" equations, using the Shallow Water model - Notes on wave interaction electromagnetic with matter (Georadar).
4. Navier Stokes equations. Equations in dimensionless form (Buckingham's theorem and emergence of "Dimensional Numbers" Reynolds, Froude, Rayleigh, Grashov) - Dynamic Reynolds equation - Conservation of mass, of momentum; energy; notes on turbulence and models for its treatment.
5. Notes on the methods of analytical solution of the main ordinary and partial differential equations relating to Environmental Computational Modeling. Initial and boundary conditions - Meaning and utility of the series development of functions: introductory concepts on Hilbert Spaces as formal basis for the use of series expansion of orthogonal functions (Fourier expansion) - Method of Variable Separation, with examples : (diffusion; forced oscillations of discrete systems seismic and electromagnetic waves, vibration of a one-dimensional bar, Consolidation and conductive heat exchange equation).
6. General problem of Sturm_Liouville and series expansions of Fourier orthogonal functions. Some examples - Expansion of the solutions in eigenfunctions with the relative eigenvalues.
7. Notes on the Transformation Method. Fourier and Laplace.
8. Interpolation of numerical data. Lagrange, Hermite, Spline polynomial.
9. Elements of statistics and probability calculus. Chebyshev's inequality - Notes on the theory of errors and the problem of experimental measurement - Notes on the Probability Calculus and the main statistical distributions - Test of a statistical hypothesis with examples.
10. Processing of observation data with linear and non-linear regression. Least squares methods - Fourier analysis with applications - Numerical data analysis using the FFT (Fast Fourier Transform) method - Time Domain and Frequency Domain analysis - Outline of stochastic investigation with the Monte-Carlo approach.
11. General principles of analysis and numerical calculation. Evaluation of truncation and rounding errors and their propagation.
12. Methods of numerical solution of partial differential equations in time and space. Newmark and Bathe's approach - Notes on Smoothed Particles Hydrodynamics (SPH) - Finite Element Method: FEM - Notes on the Finite Element Method (PFEM) - Notes on the Finite Difference Method (FDM) - Meshatura Delaunay and Voronoy.
13. Solution of linear and non-linear systems (general concepts and definitions). Conjugate Gradient and Cholesky Method - Solution of nonlinear systems (Newton Rapshon and United Point Method).
14. Examples of modeling and numerical computation. Problems carried out by the finite element method (F.E.M.): pollution of a marsh in stationary conditions
15. Examples of routines in MATLAB and FORTRAN95 language.
