Teacher

prof. Sebastiano SONEGO

Credits

6 CFU

Language

Italian

Objectives

Familiarise the student with the fundamental ideas and the main solving techniques for problems involving partial differential equations.

Acquired skills

- Vector algebra and calculus.
- Linear systems of ordinary differential equations.
- Equations of mathematical physics.
- Problems involving partial differential equations.
- Special functions.
- Basic concepts in functional analysis.

Lectures and exercises (topics and specific content)

Vector algebra: scalar and vector product; Kronecker and Levi-Civita symbols; vector identities (3 hours).
Vector calculus in three dimensions: scalar and vector fields; directional derivative and gradient; vector differential operators; vector calculus identities; derivation of the wave equation from Maxwell's equations; Gauss' and Stokes' theorems; coordinate changes; orthogonal curvilinear coordinates (8 hours).
Linear systems with a finite number of degrees of freedom: OHarmonic oscillator; coupled oscillators; normal modes and normal frequences; systems of linear differential equations, eigenvalues, eigenvectors, and general solution; system of N coupled oscillators (9 hours).
Linear partial differential equations: continuum limit; wave and diffusion equations; linear second order partial differential equations with two independent variables, classification and canonical forms (4 hours).
Wave equation in one space dimension: separation of variables; normal modes; initial value problem and its general solution; reflection; resolution of problems with inhomogeneous boundary conditions; wWell-posed problems (7 hours).
Basic functional analysis: systems of orthonormal functions; Hilbert spaces and linear operators; Hermitian and self-adjoint operators; eigenvalue problem; Sturm-Liouville problem (6 hours).
Wave equation in three space dimensions: plane waves; spherical waves; separation of variables; Legendre polynomials and associated functions; spherical harmonics; Bessel equation and Bessel functions; definition and properties of the gamma function (8 hours).
Distributions and Green's functions: basic distribution theory; Green's functions for linear differential equations; causality; Dirac's delta in three dimensions (5 hours).
Laplace equation: integral representation for the electrostatic potential; Green's function for the Laplace operator; Dirichlet's  problem; finite differences; Neumann's problem (6 hours).
Diffusion equation: problems in one space dimension; maximum-minimum theorem; initial value problem and its general solution; extreme values theorem; problems in three space dimensions (4 hours).

References

- G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, San Diego, 2005
- S. J. Farlow, Partial Differential Equations for Scientists and Engineers (Dover, New York, 1982)
- P. C. Matthews, Vector Calculus (Springer, London, 1998)
- E. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differential Equations with Applications (Dover, New York, 1986)

Type of exam

Written and oral