# Applied Mathematics

**Term**: 1^{st} Semester – Academic Year 2018-2019

**Instructor**: Adrien Lefieux (adrien.lefieux@emory.edu), Alessandro Veneziani (alessandro.veneziani@iusspavia.it)

**Institution**: Emory University and IUSS

**CFU**: 6

**SSD**: MAT05/MAT08

**Duration**: 4 weeks (50hrs lectures + 20hrs tutoring)

**Schedule**: TBA

**Office hours**: by appointment

**OBJECTIVES**

The goal of the course is to provide the mathematical background for an appropriate understanding of models and techniques relevant for the program.

**DESCRIPTION**

Theoretical lectures will be complemented by tutorials(aiming at the practical application of the concepts and methods developed during the lectures).

__Course contents:__

- Recall of multivariable calculus. Multivariable differentiation and integration. Curves, curvilinear integrals, multiple integrals (Riemann).
- Optimization: free and constrained (equalities and inequalities). Lagrange multipliers. Elements of Calculus of Variations.
- Functional spaces, metric spaces. Approximation of functions. Fourier series and transform.
- Recall of Ordinary Differential Equations and Systems. Cauchy problem. Autonomous Systems, Stability.
- Partial Differential Equations. Classification. Initial/Boundary Value Problems. Method of the Characteristics for Conservation Laws and Hyperbolic Problems. Method of Separation of Variables for the Wave Equation, the Heat Equation, the Laplace Equation. Sturm Liouville Problem. Weak formulation, the Lax-Milgram Lemma.
- Introduction to the numerical approximation of partial differential equations. Finite Differences and Finite Elements. Some practical examples with FreeFem++.

**REQUIREMENTS**

Undergraduate Calculus, Multivariable Calculus, Linear Algebra, Undergraduate Numerical Analysis or Scientific Computing

** **

**REFERENCES**

- Handouts and notes made available during the course

- Courant, Differential and Integral Calculus, Blackie & Son
- Ciarlet, Introduction to Numerical Linear Algebra and Optimization, Cambridge
- Salsa, F.M.G. Vegni, A. Zaretti, P. Zunino, A Primer on PDEs, Springer
- Formaggia, F. Saleri, A. Veneziani, Solving Numerical PDEs: Problems, Applications, Exercises, Springer
- Italian Students: C.D. Pagani, S. Salsa, AnalisiMatematica 2, Vol. 2, Masson (replacing 1 and 2)

**ASSESSMENT**

Assignments will be handed over and graded during the course. The final examination will consist of a written test. Grading: 40% assignments, 60% final exam.

**COURSE ORGANIZATION:**

Adrien Lefieux: Chapters 1-2-3 (weeks 1-2)

Alessandro Veneziani: Chapter 4-5-6 (weeks 3-4)