# Applied Mathematics

**Major**: HYRIS+ROSE

**Term**: 1^{st} 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 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 the program of the course, as well as exercises and Matlab scripts discussed in class

**COURSE WEBSITE****:**https://sites.google.com/view/tamellini-applied-mathematics