## Mathématiques appliquées 2

### 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)