Applied Mathematics a.y. 2020/21

Programma

The course is divided in five chapters as follows. Each chapter includes also exercises and hands-on Matlab sessions
• Introduction to Matlab and basic programming  Basic Matlab commands, if/for/while instructions. Functions and @-functions. Design of basic algorithms.
• Optimization of N-variate functions Free and constrained optimization of N-variate functions. Lagrange multipliers and KKT conditions, duality theory. Optimization algorithms (Gradient, Newton, Log-barrier).
• Ordinary Differential Equations (ODE) Scalar ODEs and system of ODEs. Analytic solutions of linear systems of ODEs (exponential matrix). Equilibria of linear and non-linear systems (linearization, Liapunov function) and bifurcations.
• Function approximation and Fourier transform  Space of square-summable functions, orthonormal bases and Parseval’s identity, Fourier and Legendre expansions, least squares. Fourier transform, DFT/FFT, Dirac’s delta.
• Partial Differential Equations (PDE) Elliptic and parabolic PDEs: separation of variables, maximum and mean principle. Fundamental solution of heat equation, Dirac’s delta. Finite differences 1d. Hyperbolic PDEs: method of lines for 1st order hyperbolic PDEs, inflow and outflow; D’Alambert formula for wave equation on the line and semiline, separation of variables

REQUIREMENTS:
• Basics of linear algebra
• Basics of multivariate calculus and ODE solution
• Elementary programming skills not necessary but highly welcome

Svolgimento

 

Week	Date	Lecture hours	Tot h
1	19.11.2020	09.30 – 12.00	2.5
	19.11.2020	14.00 – 16.00	2
	20.11.2020	09.30 – 11.00	1.5
	20.11.2020	14.00 – 17.00	3
2	25.11.2020	09.30 – 12.00	2.5
	25.11.2020	14.00 – 16.00	2
	26.11.2020	09.30 – 12.00	2.5
	26.112020	14.00 – 16.00	2
	27.11.2020	09.30 – 11.00	1.5
	27.11.2020	14.00 – 17.00	3
3	02.12.2020	09.30 – 12.00	2.5
	02.12.2020	14.00 – 16.00	2
	03.12.2020	09.30 – 12.00	2.5
	03.12.2020	14.00 – 16.00	2
	04.12.2020	09.30 – 11.00	1.5
	04.12.2020	14.00 – 17.00	3
4	10.12.2020	09.30 – 12.00	2.5
	10.12.2020	14.00 – 16.00	2
	11.12.2020	09.30 – 11.00	1.5
	11.12.2020	14.00 – 17.00	3
5	15.12.2020	09.30 – 12.00	2.5
	15.12.2020	14.00 – 16.00	2
	17.12.2020	FINAL EXAM	-
	18.12.2020	FINAL EXAM	-

Bibliografia

Class notes made available during the course. For backup and further readings:
• Optimization of N-variate functions: J. Nocedal, S. Wright. Numerical Optimization. Springer;
• Ordinary Differential Equations (ODE):
• G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society;
• Blanchard, Devaney, Hall. Differential Equations, Cengage Learning.
• Function approximation, Fourier transforms:
• A. Quarteroni, R. Sacco, F. Saleri. Numerical Mathematics. Springer;
• D. Kammler, A First Course in Fourier Analysis, Cambridge University Press;
• Partial Differential Equations (PDE):
• S. Salsa, Partial Differential Equations in Action, Springer;
• L. Evans, Partial Differential Equations. American Mathematical Society

Italian-speaking students can also use these books:
• ODE, optimization,  Function approximation and Fourier transform: Analisi Matematica 2, M. Bramanti, C. Pagani, S. Salsa, Zanichelli ed.;
• PDE: Equazioni a Derivate Parziali – Metodi, modelli e applicazioni, S. Salsa, Springer;
ASSESSMENT:
The final grade will be composed as follows:
• 30% homework assignments (four in total) graded during the course;
• 70% oral discussion over
• exercises and Matlab scripts discussed in class;
• one chapter of choice of the student

 

Esame

The final grade will be composed as follows:
• 30% homework assignments (four in total) graded during the course;
• 70% oral discussion over
• exercises and Matlab scripts discussed in class;
• one chapter of choice of the student