Elearn - Université de Pau et des Pays de l'Adour

The course is composed of fours chapters:
- Harmonic functions and Laplace equation;
- Second order Elliptic equations: existence, properties of weak solutions;
- Variational and nonvariational methods for elliptic PDE;
- Linear heat equations: existence of weak solutions and regularity.

More precisely, in chapter 1 we study the harmonic functions in order to obtain qualitative properties (as Maximum principle; regularity or uniqueness) of solutions of Laplace equation. These properties are the goals of chapter 2 considering a more general class of elliptic PDEs and the weak solutions of these equations and discuss properties (regularity, maximum principles and uniqueness). For the last chapters, we are going to give methods to solve elliptic and parabolic PDEs. In the case of elliptic PDE, for the variational method we study the minimization method (without or with constraints) and concerning the nonvariational method, we consider the method of sub- and supersolution and Fixed point methods. Finally, we study the Heat equation treating the questions of existence, energy estimates by discretization methods and we discuss the regularity of weak solutions. 

This course is intended for 2nd year students of MSID Master.

Objectives
One major aim of reliability theory is to predict the ability of an industrial system to perform its required functions. This ability is measured through different indicators such as the system reliability, availability, mean residual life, ... The evolution of a system over time is not fixed in advance and stochastic models are used to model its randomness. Once fitted from experimental data, these stochastic models are used to quantify the indicators of interest and make some prediction over the future evolution of the system. They are also of interest to propose well-adapted preventive maintenance strategies, to enlarge the system lifetime and prevent unexpected failures.

The objective of this course is to present basic notions of reliability theory, together with stochastic models and methods, useful in this context.

Outline

1. Introduction to reliability theory
   
2. Multi-unit system modeling
   
3. Markovian systems
4. Renewal theory and regenerative systems
5. Gamma wear process
   

Prerequisites 
Continuous-time Markov processes with finite state space (included in UE Poisson and Markov processes of MSID M1)