Ernst Hairer Livres

Cet ouvrage traite des théorèmes fondamentaux de l'analyse (convergence, continuité, calcul différentiel et intégral à une et plusieurs variables) avec grand soin pédagogique, une centaine de dessins, d'exemples et de contre-exemples. Pour une meilleure compréhension du sujet, il commence avec des calculs anciens de problèmes géometriques et mécaniques, qui ont conduit aux séries infinies, dérivées, intégrales, équations différentielles. Ainsi ce volume nous fait découvrir "l'analyse au fil de l'histoire". De nombreuses motivations et remarques historiques enrichissent le texte.

The book offers a distinctive approach to KAM theory through a numerical perspective, setting it apart from other texts in the field. It delves into the intricacies of this mathematical theory, providing insights and methodologies that are not commonly found in existing literature. This focus on numerical analysis makes it a valuable resource for those looking to deepen their understanding of KAM theory.

Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches.