# Applied Mathematics

Major: HYRIS+ROSE

Term: 1st Semester

Duration: From 28.11.2022 to 23.12.2022

INSTRUCTOR: Prof. M. Martinelli

INSTITUTION: Consiglio Nazionale delle Ricerche – Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”  (CNR-IMATI)

OBJECTIVES: To provide advanced mathematical tools that will be used throughout the rest of the program.

DESCRIPTION:
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

REFERENCES:
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 approximationFourier 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 the program of the course, as well as exercises and Matlab scripts discussed in class