Lecturer: Ugo Pietro Gianazza   Marco Veneroni  

Course name: Basic Tools For Partial Differential Equations
Course code: 502939
Degree course: Ingegneria Civile
Disciplinary field of science: MAT/05
The course relates to:
University credits: ECTS 6
Course website: http://www.imati.cnr.it/~gianazza/comp_anmat.html

Specific course objectives

The course naturally complements the Calculus courses of the First Level Degree, and aims at giving the students further analytical notions and tools, which turn out to be particularly useful for the applications: finding the local maxima and minima of a function subject to equality constraints, ordinary differential equations and systems, first elements of Calculus of Variations, Fourier series. The main focus will be on the understanding of definitions and main results, although some of the most interesting proofs will be presented in details. A lot of time will be devoted to examples and exercises.

Course programme

Differential Calculus and Applications
Multivariate Calculus: continuity, differentiability, gradient, directional derivatives, tangent plane, Hessian matrix; minimum and maximum points. Implicit functions and the implicit function theorem. Local maxima and minima of a function subject to equality constraints; Lagrange's multiplier method.

Ordinary differential equations and systems
Introduction to the theory of ordinary differential equations. Cauchy and Boundary Value problems. Nonlinear ordinary differential equations of first order; existence and uniqueness theorems, local and global statements. Systems. Linear differential equations of order n. Some particular instances of differential equations of first and second order. Eigenvalues and eigenfunctions for Boundary Value Problems.

Calculus of Variations
Functionals; minima and maxima of functionals. The Euler-Lagrange equation. Conditions for a solution of the Euler-Lagrange equation to be a minimum or a maximum. Examples and applications. Isoperimetric problems.

Fourier Analysis
Periodic Functions. Trigonometric polynomilas and trigonometric series. Fourier series; exponential form of the Fourier series. Main properties and examples. Convergence theorems for Fourier series: pointwise, uniform, mean square convergence. Applications to ordinary differential equations.

Course entry requirements

The contents studied in the Calculus, Geometry and Linear Algebra courses.

Course structure and teaching

Lectures (hours/year in lecture theatre): 22
Practical class (hours/year in lecture theatre): 43
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

R.A. Adams. Calcolo differenziale 2. Casa Editrice Ambrosiana. (for the first part of the course) .

N. Fusco, P. Marcellini e C. Sbordone. Analisi matematica 2. Liguori. (for the second, third, fourth parts of the course9.

C.D. Pagani, S. Salsa. Analisi Matematica 2. Masson. (for the second, third, fourth parts of the course).

Testing and exams

The final test consists of a written and an oral part, which have to taken in the same session.