Differential Equations

This book provides an introduction to the theory and application of the solution of differential equations using symmetries, a technique of great value in mathematics and the physical sciences.In many branches of physics, mathematics, and engineering, solving a problem means a set of ordinary or partial differential equations. Nearly all methods of constructing closed form solutions rely on symmetries. The theory and application of such methods have therefore attracted increasing attention in the last two decades. In this text the emphasis is on how to find and use the symmetries in different cases. Many examples are discussed, and the book includes more than 100 exercises.This book will form an introduction accessible to beginning graduate students in physics, applied mathematics, and engineering. Advanced graduate students and researchers in these disciplines will find the book an invaluable reference.

Preface; 1. Introduction; Part I. Ordinary Differential Equations: 2. Point transformations and their generators; 3. Lie point symmetries of ordinary differential equations: the basic definitions and properties; 4. How to find the Lie point symmetries of an ordinary differential equation; 5. How to use Lie point symmetries: differential equations with one symmetry; 6. Some basic properties of Lie algebras; 7. How to use Lie point symmetries: second order differential equations admitting a G2; 8. Second order differential equations admitting a G3IX; 9. Higher order differential equations admitting more than one Lie point symmetry; 10 Systems of second order differential equations; 11. Symmetries more general than Lie point symmetries; 12. Dynamical symmetries: the basic definitions and properties; 13. How to find and use dynamical symmetries for systems possessing a Lagrangian; 14. Systems of first order differential equations with a fundamental system of solutions; Part II. Partial Differential Equations: 15. Lie point transformations and symmetries; 16. How to determine the point symmetries of partial differential equations; 17. How to use Lie point symmetries of partial differential equations I: generating solutions by symmetry; 18. How to use Lie point symmetries of partial differential equations II: similarity variables and reduction of the number of variables; 19. How to use Lie point symmetries of partial differential equations III: multiple reduction of variables and differential invariants; 20. Symmetries and the separability of partial differential classification; 21. Contact transformations and contact symmetries of partial differential equations, and how to use them; 22. Differential equations and symmetries in the language of forms; 23. Lie-Backlund transformations; 24. Lie-Backlund symmetries and how to find them; 25. How to use Lie-Backlund symmetries; Appendices; Index.