1. Basic tools of Computational Modeling. - Cartesian, cylindrical and spherical orthogonal coordinates - Change of coordinates: "Jacobian" - Elements of matrix calculus Elements of analytic geometry Derivatives: "substantial", Eulerians and Lagrangians, physical meaning of partial and total derivative with respect to time, derivative with respect to time of a vector, - Introduction to differential operators – Important functions (Dirac; Heaviside) - 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 - 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, 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 Consolidation Law - Notes on wave interaction electromagnetic with matter (Georadar).
4. Notes on Navier Stokes equations. Equations in dimensionless form. 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
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. General principles of analysis and numerical calculation. Evaluation of truncation and rounding errors and their propagation.
9. Methods of numerical solution of partial differential equations in time and space. Newmark approach - Notes on the Finite Element Method (FEM) - Notes on the Finite Difference Method (FDM) - Delaunay and Voronoy Meshing.
10. Solution of linear and non-linear systems (general concepts and definitions). Conjugate Gradient and Cholesky Method - Solution of nonlinear systems (Newton Rapshon).
11. Examples of modeling and numerical computation. Problems carried out by the finite element method (F.E.M.): pollution of a lake in stationary conditions