Similarity Methods for Differential Equations

1. Ordinary Differential Equations.- 1.0. Ordinary Differential Equations.- 1.1. Example: Global Similarity Transformation, Invariance and Reduction to Quadrature.- 1.2. Simple Examples of Groups of Transformations; Abstract Definition.- 1.3. One-Parameter Group in the Plane.- 1.4. Proof That a One-Parameter Group Essentially Contains Only One Infinitesimal Transformation and Is Determined by It.- 1.5. Transformations; Symbol of the Infinitesimal Transformation U.- 1.6. Invariant Functions and Curves.- 1.7. Important Classes of Transformations.- 1.8. Applications to Differential Equations; Invariant Families of Curves.- 1.9. First-Order Differential Equations Which Admit a Group; Integrating Factor; Commutator.- 1.10. Geometric Interpretation of the Integrating Factor.- 1.11. Determination of First-Order Equations Which Admit a Given Group.- 1.12. One-Parameter Group in Three Variables; More Variables.- 1.13. Extended Transformation in the Plane.- 1.14. A Second Criterion That a First-Order Differential Equation Admits a Group.- 1.15. Construction of All Differential Equations of First-Order Which Admit a Given Group.- 1.16. Criterion That a Second-Order Differential Equation Admits a Group.- 1.17. Construction of All Differential Equations of Second-Order Which Admit a Given Group.- 1.18. Examples of Application of the Method.- 2. Partial Differential Equations.- 2.0. Partial Differential Equations.- 2.1. Formulation of Invariance for the Special Case of One dependent and Two Independent Variables.- 2.2. Formulation of Invariance in General.- 2.3. Fundamental Solution of the Heat Equation; Dimensional Analysis.- 2.4. Fundamental Solutions of Heat Equation Global Affinity.- 2.5. The Relationship Between the Use of Dimensional Analysis and Stretching Groups to Reduce theNumber of Variables of a Partial Differential Equation.- 2.6. Use of Group Invariance to Obtain New Solutions from Given Solutions.- 2.7. The General Similarity Solution of the Heat Equation.- 2.8.