differential equations

Cet ouvrage présente les modèles d'interaction onde-matière faisant intervenir une description quantique de la matière (optique quantique), ainsi que des modèles classiques qui en sont dérivés. L'objectif est de décrypter pour des lecteurs mathématiciens ces modèles habituellement décrits dans des livres de physique et de donner les résultats mathématiques et les méthodes numériques existants. Ces résultats, reflets de sujets de recherche actuels faisant intervenir des outils mathématiques variés, sont détaillés pour être accessibles à des étudiants ayant un niveau DEA. Les parties numériques de ce livre peuvent également intéresser des physiciens désirant effectuer des simulations.

The book provides a pedagogic and comprehensive introduction to homogenization theory with a special focus on problems set for non-periodic media. The presentation encompasses both deterministic and probabilistic settings. It also mixes the most abstract aspects with some more practical aspects regarding the numerical approaches necessary to simulate such multiscale problems. Based on lecture courses of the authors, the book is suitable for graduate students of mathematics and engineering. Cet ouvrage est une introduction pédagogique à la théorie de l’homogénéisation, et aux approches numériques associées, pour la compréhension et la simulation des problèmes à plusieurs échelles. La présentation est axée sur les différentes hypothèses possibles pour mettre en œuvre la théorie, selon que le milieu ambiant a une géométrie périodique ou non. Le public visé est celui des cycles M et D des universités ainsi que celui des écoles d’ingénieurs et formations équivalentes.

This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg–de Vries equation and the nonlinear Schrödinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schrödinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schrödinger equation, taking the reader to the forefront of recent research.The second edition of Introduction to Nonlinear Dispersive Equations builds upon the success of the first edition by the addition of updated material on the main topics, an expanded bibliography, and new exercises. Assuming only basic knowledge of complex analysis and integration theory, this book will enable graduate students and researchers to enter this actively developing field.

This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise. Within each section the author creates a narrative that answers the five questions:

What is the scientific problem we are trying to understand? How do we model that with PDE?What techniques can we use to analyze the PDE?How do those techniques apply to this equation?What information or insight did we obtain by developing and analyzing the PDE?

The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods.