Partial differential equations
ECTS Credits: 4
Duration: 42 hours
Person(s) in charge:
Antoine HENROT, Professor, email@example.com
Keywords: Partial differential equations, variational formulation, boundary-value problems, spectral theory
Prerequisites: First year Math I and Math II courses of the core curriculum
Objective: To be able to analyze a well-posed problem modeled by partial differential equations
Program and contents:
Introduction to elliptic, parabolic and hyperbolic partial differential equations. Examples: Laplace and Poisson equations, heat equation and wave equation. The different kinds of boundary conditions (Dirichlet, Neumann, Robin). Initial conditions for evolution problems.
Some methods for an explicit resolution: separation of variables, Fourier or Laplace transform.
Tools for the mathematical study of p.d.e.: distributions and the Sobolev spaces, Poincaré inequalities, Sobolev and compact embedding, trace.
Variational formulation of elliptic problems. The Lax-Milgram theorem. Application to different kinds of boundary conditions. Regularity results, relation between weak and classical solutions. Examples of the Laplace operator, the Stokes system, the plate equation.
Maximum principle. Spectral theory for elliptic operators, Galerkin method.
Evolution equation: the heat equation and the wave equation. Existence and regularity results, asymptotic behavior.
2 written tests
Description and operational vocabulary
To be able to recognize the different kind of partial differential equations, of boundary conditions
Understand if the problem is well posed (existence, uniqueness of a solution, stability with respect to data)
To be able to write a variational formulation in order to apply the fundamental Lax-Milgram Theorem
To be able to study this variational formulation to prove the well-posedness of the problem
State a correct answer to the problem, analyze a model to decide whether it is a good model
To be able to analyze the (numerical) solution of an equation to decide if it is pertinent
Gestion des contenus