# Applied Mathematics

Term: 1st Semester – Academic Year 2018-2019

Instructor: Lorenzo Tamellini (tamellini@imati.cnr.it)

Institution: Emory University and IUSS

CFU: 6

SSD: MAT05/MAT08

Duration: 4 weeks

Schedule

Office hours: by appointment

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 (35 hours of lectures + 18h hours of exercises/Matlab sessions).

1. Ordinary Differential Equations (ODE) (7h lectures + 3h exercises/Matlab = 10h). Scalar ODEs and system of ODEs. Analytic solutions of linear system of ODEs (exponential matrix) Equilibria and bifurcations. Numerical methods for ODEs (Euler, Crank-Nicholson, Runge-Kutta, …) and absolute stability issues.
2. Optimization of N-variate functions (6h lectures + 3h exercises/Matlab = 9h). Free and constrained optimization of N-variate functions. Lagrange multipliers and KKT conditions. Optimization algorithms (gradient, Newton, square penalty, log-barrier…).
3. Function approximation, transforms, numerical quadrature (8h lectures + 4h exercises/Matlab = 12h). Space of square-summable functions, orthonormal bases and Parseval’s identity, Fourier and Legendre expansions, least squares. Fourier transform, FFT. Numerical schemes for quadrature of univariate and multivariate functions (midpoint, trapezoidal rule, Gauss quadrature, Monte Carlo and probabilistic methods)
4. Partial Differential Equations (PDE) (10h lectures + 3h exercises = 13h).
1. Elliptic and parabolic PDEs: separation of variables, fundamental solution, Dirac’s delta, maximum principle.
2. Hyperbolic PDEs: method of lines, weak solution, RH condition, entropy, systems of hyperbolic PDEs, wave PDE, D’Alambert principle.
5. Numerical Methods for PDEs (4h lectures + 3h exercises/Matlab = 7h). Well-posedness of elliptic PDEs in Sobolev spaces by Lax-Milgram Lemma. Finite Elements method.

REFERENCES:

Hand-outs and notes made available during the course. For backup and further readings:

1. Ordinary Differential Equations (ODE): Teschl, Ordinary Differential Equations and Dynamical Systems. American Mathematical Society;
2. Optimization of N-variate functions: Nocedal, S. Wright. Numerical Optimization. Springer;
3. Function approximation, transforms, numerical quadrature: Quarteroni, R. Sacco, F. Saleri. Numerical Mathematics. Springer; D. Kammler, A First Course in Fourier Analysis, Cambridge University Press;
4. Partial Differential Equations (PDE): Salsa, Partial Differential Equations in Action, Springer; L. Evans, Partial Differential Equations. American Mathmatical Society
5. Numerical Methods for PDEs: Quarteroni, Numerical Models for Differential Problems, Springer.

Italian-speaking students can also use these books:

1. Chapters 1,2,3: Analisi Matematica 2, M. Bramanti, C. Pagani, Salsa, Zanichelli ed.;
2. Chapter 4: Equazioni a Derivate Parziali – Metodi, modelli e applicazioni,  Salsa,  Springer;
3. Chapter 5: Modellistica Numerica per Problemi Differenziali, Quarteroni,  Springer.

ASSESSMENT:

The final grade will be composed as follows:

• 30% homework assignments (three in total) graded during the course;
• 50% oral discussion over
• exercises and Matlab scripts discussed in class;
• one chapter of choice of the student.
• 20% oral discussion of a small additional topic (not covered during class but related to the class topics), to be preliminarily agreed upon with the teacher.