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- INTRODUCTION TO DISTRIBUTION THEORY

# INTRODUCTION TO DISTRIBUTION THEORY

Start

11/11/2019

Duration

9 semaines

Rhythm

3 to 5 hours

Prerequisites

Course on demand

Discipline

Mathematics

Course language

English

PRESENTATION:

Can a discontinuous function solve a differential equation? How can we strictly define Dirac mass (a “function” of integral one that is equal to zero everywhere except at one point) and its derivatives? Can we define the notion of a “fractional order derivative”? This introduction to distributions addresses these questions and many more.

ABOUT THE COURSE

In numerous scientific disciplines (physics, mechanics, numerical analysis etc.), there is a need to develop a “generic” differential calculus for irregular functions, for example discontinuous ones. The first lesson presents such a situation. The purpose of this course is to offer an introduction to distribution theory, which provides an elegant answer to this problem.

Firstly, we present the notion of distribution, and in doing so, genericize the notion of function, and we look at the initial properties of these mathematical objects. We then demonstrate how some well-known operations on functions, in particular derivation, can be naturally applied to the field of distributions. The course concludes with an introduction to some classic examples of linear partial differential equations.

COURSE SYLLABUS

Week 1 :

First-order PDEs and method of characteristics

Hopf equation

Test functions

Week 2 :

Definition of distributions

Derivation of distributions

Week 3 :

Other operations on distributions

Compact support distributions

Convolution of distributions by regular functions

Week 4 :

Formula of multi-dimensional jumps and applications

Compact support distributions

Convolution of distributions by regular functions

Week 5 :

Regularization of distributions

Convolution des distributions

Tempered distributions

Week 6 :

Fourier transform on the Schwartz class

Fourier transform for tempered distributions

Week 7 :

Introduction to the study of PDEs

Examples of elementary solutions

Determination of some elementary solutions

Week 8 :

Harmonic functions

Examples of elementary solutions

Poisson’s equation

Week 9 :

Cauchy problem in the context of distributions

Examples of first-order evolutionary partial differential equations

Example of a second-order evolutionary partial differential equation

REQUIREMENTS

Multivariable differential calculus

Basic notions of topology

Knowledge of the Lebesgue integral

FAQ

Is it possible to get a certificate upon completion of the course?

Yes. All students who follow the entire course and complete the final multiple-choice quiz will receive a certificate. However, this certificate does not entitle you to ECTS credits.

What do I need for this course?

You just need to watch the videos of the course to be able to benefit from it. Additional information can be downloaded if necessary.

Teacher(s)