list 9 sequences
assignment Level : Introductive
chat_bubble_outline Language : French
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About the content

Une fonction discontinue peut-elle être solution d'une équation différentielle? Comment définir rigoureusement la masse de Dirac (une "fonction" d'intégrale un, nulle partout sauf en un point) et ses dérivées? Peut-on définir une notion de "dérivée d'ordre fractionnaire"? Cette initiation aux distributions répond à ces questions - et à bien d'autres.

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Syllabus

Semaine 1 :
EDP d'ordre 1 et méthode des caractéristiques
L'équation de Hopf
Fonctions test
 
Semaine 2 :
Définition des distributions 
Dérivation des distributions 

Semaine 3 :
Autres opérations sur les distributions
Formule des sauts en dimension 1
Intégration sur les courbes et les surfaces 

Semaine 4 :
Formule des sauts en dimension >1 et applications 
Distributions à support compact
Convolution des distributions par des fonctions régulières 

Semaine 5 :
Régularisation des distributions 
Convolution des distributions 
Distributions tempérées 

Semaine 6 :
Transformation de Fourier sur la classe de Schwartz
Transformation de Fourier pour les distributions tempérées
 
Semaine 7 :
Introduction à l'étude des EDP
Exemples de solutions élémentaires
Détermination de quelques solutions élémentaires 

Semaine 8 :
Fonctions harmoniques 
Equation de Poisson

Semaine 9 :
Problème de Cauchy au sens des distributions
Exemples d'EDP d'évolution d'ordre 1
Un exemple d'EDP d'évolution d'ordre 2 
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Instructors

Prof. François Golse
Professeur à l'Ecole polytechnique
Département de mathématiques

Prof. Yvan Martel
Professeur à l'Ecole polytechnique
Département de mathématiques

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École Polytechnique

École polytechnique (also known by the nickname " X ") is a French public institution of higher education and research, located in Palaiseau near Paris. It is one of the French Grandes écoles, and is renowned for its four-year undergraduate Ingénieur Polytechnicien degree in science and engineering. Students are usually admitted after two years of selective university-level preparation in mathematics and physics or after a Bachelor of Sciences (Licence in French educative system).

Polytechnique was established in 1794 by the mathematician Gaspard Monge during the French Revolution, and became a military academy under Napoleon I in 1804. Today, the institution still runs under the supervision of the French ministry of Defence. Initially located in the Latin Quarter of central Paris, the establishment was moved in 1976 to Palaiseau on the Saclay Plateau, southwest of Paris.

Polytechnique is a founding member of ParisTech, a grouping of leading Paris-area engineering colleges established in 2007. In 2014 it became a founding member as well as the head of multidisciplinary technology and innovation institute, the confederal "University of Paris in Saclay". Among its alumni are three Nobel prize winners, one Fields Medalist, three Presidents of France and many CEOs of French and international companies.

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Coursera

Coursera is a digital company offering massive open online course founded by computer teachers Andrew Ng and Daphne Koller Stanford University, located in Mountain View, California. 

Coursera works with top universities and organizations to make some of their courses available online, and offers courses in many subjects, including: physics, engineering, humanities, medicine, biology, social sciences, mathematics, business, computer science, digital marketing, data science, and other subjects.

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