Ects : 5
Volume horaire : 39

Description du contenu de l'enseignement :

This course introduces the functional framework of partial differential equations, including distribution theory and Sobolev spaces. It then studies the Laplace, heat, and wave equations, together with analytical and numerical methods for their resolution.

• Distribution theory and Sobolev spaces (~5 weeks)
  • The space of test functions and of distributions: examples (locally integrable functions, Dirac mass, principal value distributions, etc.) ; jump formula (e.g. Rankine–Hugoniot condition in problem sessions) ; support of a distribution ; convergence of sequences ; operations on distributions ; applications (fundamental solution of the Laplacian, representation of solutions to -?u=f in the whole space when f has compact support).
  • Tempered distributions: Schwartz space ; Fourier transform on tempered distributions (Plancherel theorem, inversion formula, etc.) ; important examples.
  • Sobolev spaces (Hilbert framework): spaces H^s(?) (definition, Fourier characterization when ?=R^d, simple case of compact embedding, Rellich theorem) ; the space H^1_0(?) and Poincaré inequality.
• Laplace, heat, and wave equations (~7 weeks)
  • Laplace equation: solutions in a bounded domain, Lax–Milgram theorem ; solutions (definitions, existence and uniqueness) of -?u=f in H^1_0 ; H^2 regularity (optional) ; eigenfunction basis of the Laplacian.
  • Heat equation: Fourier method in the whole space (possible reminder) ; Galerkin method ; explicit resolution using the eigenfunction basis of the Laplacian.
  • Numerical simulation of the Laplace and heat equations: time and space discretization ; implicit and explicit Euler schemes ; notions of stability ; consistency ; and convergence.
  • Wave equation: modeling ; particular solutions via Fourier methods and separation of variables ; light cone ; classical solutions ; weak solutions ; superposition principle.

Compétence à acquérir :

Introduction to the fundamental partial differential equations — Laplace, heat, and wave — within the functional framework of distribution theory and Sobolev spaces.