Mathématiques appliquées 2

Dernière mise à jour le 27 février 2024

Présentation

This course aims at introducing exact and approximate solutions of partial dfferential equations to the students. In particular, parabolic (diffusion) equations, hyperbolic (wave and transport) equations and elliptic (Poisson and Laplace) equations are introduced, sand are solved in finite or in infinite domains. Functional transforms (Fourier and Laplace) are introduced and are used to solve these equations. A link with domains of physics in which these equations are used is made. A connection with numerics is also given at the end of the course, to relate the theoretical results with numerical results.

This course is taught in English.

Pré-requis nécessaires

mathematical analysis : ODEs, real functions of several variables, geometry of curves and surfaces, vector analysis ;
physics : classical mechanics, electromagnetism.

Objectifs

This is an essential course for all fluid dynamics courses (GFD, fluid-solid interactions) and more generally for all of physics (electromagnetism for instance). Partial differential equations govern the evolution of physical phenomena in many fields (subatomic physics, microscopic physics, physics of solds and of fluids, electromagnetism...)

Compétences visées

ability to validate numerical results with theoretical results
problem solving in fluids
global approach (holistic approach) to problem solving
building numerical algorithms for professional purposes

Descriptif

Introduction/ description of PDEs, basic theory

Part I / Parabolic equations

solution in a finite domain
solution in a semi infinite or an infinite domain

Part II / Hyperbolic equations

A) second order wave equations

solution in a finite domain
solution in a semi infinite or an infinite domain

B) first order transport equations

linear equations
nonlinear equations / shocks
systems of equations

Part III / Elliptic equations

Laplace equation
Poisson equation

Complement : first integrals, nonlinear diffusion equations, Korteweg-DeVries equation

Bibliographie

Partial differential equations for scientists and engineers (available on the web)