PDE - Partial Differential Equations

Ref: 1SL1500
Télécharger la fiche de cours

Description

Partial differential equations (or PDEs for short) are equations whose solutions are functions. They appear naturally in modeling in physics, mechanics, biology, economics, finance, and more generally in all engineering fields.
In this course, you will learn the basics of PDEs. We will start with recalling the situation of the ordinary differential equations (ODE) for which we will examine the well-posedness of the questions. Then, you will understand the different classes of PDEs, including elliptic, parabolic and hyperbolic. You will see how one can prove the existence and uniqueness of solutions of some elliptic equations.

You will see how to numerically approximate the solutions of elliptic and parabolic partial differential equations using two standard techniques: the Finite Element Method and the Finite Difference Method. Both of these techniques lead to a huge linear system, so we will see the basics of numerical linear algebra to tackle this problem. You will also learn a software to approximate the solutions of a PDE.

In this course, you will also see the theory of distributions which generalizes the concept of functions. You will learn how to successfully use distributions and apply them. You will also learn about Sobolev spaces which are useful in the context of PDEs.

Période(s) du cours

ST2 and SG3

Prérequis

  • Convergence, Integration and Probabability
  • Modeling (co-requirement)
  • Information Systems and Programming

Syllabus

Chapter I - Ordinary Differential Equations
Chapter II - Classification of PDEs and Modeling
Chapter III - Distributions
Chapter IV - The Variational Formulation
Chapter V - The Finite Element Method
Chapter VI - The Finite Difference Method
Chapter VII - Numerical Linear Algebra
Chapter VIII - Parabolic PDEs

Chapters I and V are conducted over two sessions each.
Chapter II will be covered by students working on their own.

Composition du cours

The course is available in:

French

Face-to-face (sections 3 to 6) or blended learning (section 1)

English

blended leaning (section 2)

Blended learning means that the lectures take place in the form of video capsules and the TD in person.

Students who need additional help are enrolled in MR (support sessions). They benefit from additional sessions led by students enrolled in the 2nd year "Teaching Assistant" elective class under the responsibility of the teaching staff. The support sessions are compatible with face-to-face French and distance or mixed English.

A special section might be available for a few students with an exceptional mathematical skills who wish to study partial differential equations in greater depth. Admission is subject to approval.

The section and lab group of CIP determines the section and lab group for PDE (not applicable for the special section or in case of compelling reasons).

Ressources

This course is composed of nine lectures and nine lab sessions.

Courses are given in one section. Each section is subdivisez in lab groups. A support session is assigned to certain students.
It is possible that a special "free maths" section will be created. If this is the case, students will be informed by e-mail and the purpose of this section will be explained.

In addition, students who have to work on a modeling problem leading to a PDE in the context of a project, a ST, an associative activity or a personal interest, can ask to benefit from a companion project. This is the completion of additional work. This work is optional and subject to acceptation. It is under the responsibility of Lionel Gabet.

Résultats de l'apprentissage couverts par le cours


Support de cours, bibliographie

Erick Herbin & Pauline Lafitte
CIP and PDE Lecture Notes

Haïm Brézis
Functional Analysis, Sobolev Spaces and Partial Differential Equations.
Springer, 2011.

Grégoire Allaire
Numerical Analysis and Optimization:
An Introduction to Mathematical Modelling and Numerical Simulation
Oxford University Press, USA, 2007
Responsable(s) :
John Cagnol
Langues du cours :
FRENCH
ENGLISH
Type de cours :
Cours hors séquence
Nombre d'heure :
50 h
Nombre d'heure sur site :
30 h
Année académique :
2024-2025
Niveau avancé :
No