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. Applications of the General Similarity Solution of the Heat Equation. - 2. 9. -Axially-Symmetric Wave Equation. - 2. 10. Similarity Solutions of the One-Dimensional Fokker-Planck Equation. - 2. 11. The Green's Function for an Instantaneous Line Particle Source Diffusing in a Gravitational Field and Under the Influence of a Linear Shear Wind An Example of a P. D. E. in Three Variables Invariant Under a Two-Parameter Group. - 2. 12. Infinite Parameter Groups Derivation of the Poisson Kernel. - 2. 13. Far Field of Transonic Flow. - 2. 14. Nonlinear and Other Examples. - 2. 15. Construction of Partial Differential Equations Invariant Under a Given Multi-parameter Group. - Appendix. Solution of Quasilinear First-Order Partial Differential Equations. - Bibliography. Part 1. - Bibliography. Part 2. Language: English