Symmetry analysis based on Lie group theory is the most important method for solving nonlinear problems aside from numerical computation. The method can be used to find the symmetries of almost any system of differential equations and the knowledge of these symmetries can be used to reduce the complexity of physical problems governed by the equations. This is a broad, self-contained, introduction to the basics of symmetry analysis for first and second year graduate students in science, engineering and applied mathematics. Mathematica-based software for finding the Lie point symmetries and Lie-Bäcklund symmetries of differential equations is included on a CD along with more than forty sample notebooks illustrating applications ranging from simple, low order, ordinary differential equations to complex systems of partial differential equations. MathReader 4.0 is included to let the user read the sample notebooks and follow the procedure used to find symmetries.
• Introduces symmetry analysis within an engineering framework • Includes ready-to-use software • Includes many exercises and examples
Preface; 1. Introduction to symmetry; 2. Dimensional analysis; 3. Systems of ODE's, first order PDE's, state-space analysis; 4. Classical dynamics; 5. Introduction to one-parameter Lie groups; 6. First order ordinary differential equations; 7. Differential functions and notation; 8. Ordinary differential equations; 9. Partial differential equations; 10. Laminar boundary layers; 11. Incompressible flow; 12. Compressible flow; 13. Similarity rules for turbulent shear flows; 14. Lie-Bäcklund transformations; 15. Invariance condition for integrals, variational symmetries; 16. Bäcklund transformations and non-local groups; Appendix 1. Review of calculus and the theory of contact; Appendix 2. Invariance of the contact conditions under Lie point transformation groups; Appendix 3. Infinite-order structure of Lie-Bäcklund transformations; Appendix 4. Symmetry analysis software.
'… an easy to read informal introduction to symmetry methods for ordinary and partial differential equations.' Monatshefte für Mathematik
'Overall, this was an enjoyable book to read, which I could recommend to someone to tackle as background reading. … if one's interest is in the application of symmetry analysis to problems arising in fluids, then this book discusses 'how' in detail.' Peter Clarkson, University of Kent