Mathématiques et applications
Partial Differential Equations and Numerical Analysis

This course presents introductory lectures on Partial Differential Equations (PDE) and their numerical analysis. The objectives are to study linear PDE's limited to one space dimension, the time being also a possible variable. The basic theory of existence, uniqueness and regularity of solutions is presented for :

1) the Cauchy problem for linear transport (or advection) equation with constant velocity as a model of first-order hyperbolic equations

2) the boundary-value problem for diffusion (or advection-diffusion) equation as a model of second-order elliptic equations

3) the initial and boundary-value problem for the heat evolution equation as a model of second-order parabolic equations

4) the Cauchy problem for the wave propagation equation as a model of second-order hyperbolic equations.

Moreover, the basic discretization methods by finite differences (or possibly finite volumes) to solve these PDE's are studied with their properties of consistency, stability, convergence and error estimates.

An important part is devoted to practical training, exercises and coding of the numerical schemes with Python to get a deeper understanding of the topics.
Cours de logique et calculabilité, option du M1 Mathématiques et applications.