 # Mathematical tools for the engineer II

## Presentation

Programme (detailed contents):

Introduction to ODEs

Theoretical Part:

I Some examples of model problems involving ODEs: single pendulum, chains of oscillators, chemistry.

II Second order linear ODEs: characteristic equations, separation of variables, resonance. Examples.

III Systems of ODEs: linear systems with constant coefficients, phase portrait. Nonlinear systems: existence of a solution, linearization, stability of steady points, periodic solutions.

Numerical Part:

I Euler Methods for ODEs

II Convergence, consistency and stability, order

III Numerical Schemes and Qualitative properties: stiff problems and convservative problems.

Introduction to Probability and Statistics

Space of probability, random variables, limit theorems, statistical tests.

Organisation: Standard for the theoretical part (lecture notes and exercises classes). For the numerical part, the students follow videos available on the platform Moodle. A brief review of the course is done at the beginning of each labwork class. Numerical simulations with Python.

Main difficulties for students: Calculus and Programmation.

## Objectives

At the end of this module, the student will have understood and be able to solve explicitely simple ordinary differential equations (ODE): second order linear equations with constant coefficients. In more complex situations, he must be able to provide qualitative properties of solutions (existence, uniqueness, existence time, large time behaviour: phase portrait, stability of critical points). He also must be able to propose adapted numerical  algorithms to solve systems of ODEs.

Concerning probability and statistics, the student must be able to explain the notion of randomness, confidence interval, hypothesis tests. He must be able to identify the main features of random modeling (randomness, risks) and carry out an elementary statistical test for real life problems. In both cases, the numerical simulations are carried out with Python.

## Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...

## Bibliography

Calcul Différentiel et équations différentielles, Sylvie Benzoni, Dunod

Analyse Numérique, Francis Filbet, Dunod.

Probabilités et statistiques, analyse des données, Gilbert Saporta, Editions Technip.