Synopses & Reviews
A lucid and comprehensive introduction to general PDE systems for graduates and researchers.
Review
"The book provides a lucid and comprehensive introduction to the differential geometric study of partial differential equations; it will be a valuable resource for graduate students and researchers in related fields." Mathematical Reviews
Synopsis
This book describes a general approach to systems of partial differential equations based on ideas developed by Lie, Cartan and Vessiot. It provides a lucid and comprehensive introduction and will be a valuable resource for graduate students and researchers in partial differential equations, Lie groups and related fields.
Table of Contents
Preface; 1. Introduction and summary; 2. PDE systems, pfaffian systems and vector field systems; 3. Cartan's local existence theorem; 4. Involutivity and the prolongation theorem; 5. Drach's classification, second order PDEs in one dependent variable and Monge characteristics; 6. Integration of vector field systems n satisfying dim n' = dim n + 1; 7. Higher order contact transformations; 8. Local Lie groups; 9. Structural classification of 3-dimensional Lie algebras over the complex numbers; 10. Lie equations and Lie vector field systems; 11. Second order PDEs in one dependent and two independent variables; 12. Hyperbolic PDEs with Monge systems admitting 2 or 3 first integrals; 13. Classification of hyperbolic Goursat equations; 14. Cartan's theory of Lie pseudogroups; 15. The equivalence problem; 16. Parabolic PDEs for which the Monge system admits at least two first integrals; 17. The equivalence problem for general 3-dimensional pfaffian systems in five variables; 18. Involutive second order PDE systems in one dependent and three independent variables, solved by the method of Monge; Bibliography; Index.