34605 - Mathematical Methods for Energetics (Graduate Course)

Learning outcomes

At the end of the course, the student has acquired competences on the mathematical tools of widest use in modeling energy systems, and more generally in treating problems in Physics. More specifically, the pupil is able to make us of: analytic functions; Fourier series; Fourier and Laplace integral transforms; PDE's. He/She will be acquainted with probability calculus: laws, random variables, main distributions and densities; including basic notions of statistics.

Course contents

Review of Linear Algebra

      Fields. Matrices over a field. Vector spaces over a field. Linear transformation between vector spaces. Matrix representation of linear transformations. Coordinate representation of vector spaces. Eigenvalues and eigenvectors. Normed spaces, inner product spaces. Orthogonal vectors, orthogonal subspaces.

 

Foundations of Real and Functional Analysis

  Vector spaces of infinite dimension. Normed spaces and completeness: Banach spaces. Hilbert spaces. Cauchy-Schwarz inequality. Spaces of continuous functions. L-p spaces. Hölder's inequality. Embeddings of L-p spaces. Sobolev spaces.

 

Solution of Differential Problems: ODEs

    First order ODEs and the main solution methods. Exact ODEs, separable ODEs. Substitution methods.

   Higher-order linear ODEs, decomposition of the solution set. Linear homogeneous ODEs. Linear nonhomogeneous ODEs: method of variation of parameters, method of undetermined coefficients.  Series solutions. Solutions by integral transforms. Applications to problems in science and engineering.

Solution of Differential Problems: PDEs

    Preliminary concepts: Sturm-Liouville problems for linear ODEs; Fourier series. Classifications of PDEs: elliptic, parabolic, hyperbolic. The heat equation.   The wave equation. Laplace's equation. Solution by separation of variables. Solution by integral transforms. Formulation in various coordinate systems. Other solution methods, D'Alembert method. Applications to problems in science and engineering.

 

 

 

Readings/Bibliography

G.C. Barozzi, Matematica per l'Ingegneria dell'Informazione

Adams, Sobolev Spaces

Spiga, Problemi matematici della fisica e dell'ingegneria

 

Teaching methods

Chalk over blackboard

Assessment methods

Written examination followed by verbal examination

Teaching tools

Projector to show applications of mathematical methods

Office hours

See the website of Giorgio Bornia