Lie's group theory of differential equations unifies the many ad hoc methods known for solving differential equations and provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations and is not restricted to linear equations. Applications of Lie's Theory of Ordinary and Partial Differential Equations provides a concise, simple introduction to the application of Lie's theory to the solution of differential equations. The author emphasizes clarity and immediacy of understanding rather than encyclopedic completeness, rigor, and generality. This enables readers to quickly grasp the essentials and start applying the methods to find solutions. The book includes worked examples and problems from a wide range of scientific and engineering fields.
Conventions Used in This Book One-Parameter Groups Groups of transformations Infinitesimal transformations Group invariants Invariant curves and families of curves Transformation of derivatives: the extended group Transformation of derivatives (continued) Invariant differential equations of the first order First-Order Ordinary Differential Equations Lie's integrating factor The converse of Lie's theorem Invariant integral curves Singular solutions Change of variables Tabulation of differential equations Notes to chapter two Second-Order Ordinary Differential Equations Invariant differential equations of the second order Lie's reduction theorem Stretching groups Streching groups (continued) Stretching groups (continued) Other groups Equations invariant to two groups Two-parameter groups Noether's theorem Noether's theorem (continued) Similarity Solutions of Partial Differential Equations One-parameter families of stretching groups Similarity solutions The associated group The asymptotic behavior of similarity solutions Proof of the ordering theorem Functions invariant to an entire family of stretching groups A second example Further use of the associated group More wave propagation problems Wave propagation problems (continued) Shocks Traveling-Wave Solutions One-parameter families of translation groups The diffusion equation with source Determination of the propagation velocity a Determination of the propagation volocity: role of the initial condition The approach to traveling waves The approach to traveling waves (continued) A final example Concluding remarks Notes of chapter five Approximate Methods Introduction Superfluid diffusion equation with a slowly varying face temperature Ordinary diffusion equation with a nonconstant diffusion coefficient Check on the accuracy of the approximate formula Epilogue Appendix 1: Linear, First-Order Partial Differential Equations Appendix II: Riemann's Method of Characteristics Appendix III: The Calculus of Variations and the Euler-Lagrange Equation Appendix IV: Computation of Invariants and First Differential Invariants from the Transformation Equations Solutions to the Problems References Symbols and Their Definitions