Author | Bluman, G. W. author |
---|---|
Title | Similarity Methods for Differential Equations [electronic resource] / by G. W. Bluman, J. D. Cole |
Imprint | New York, NY : Springer New York, 1974 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-6394-4 |
Descript | IX, 333 p. online resource |
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 the Number 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 Greenโs Function for an Instantaneous Line Particle Source Diffusing in a Gravitational Field and Under the Influence of a Linear Shear Wind โ An Example of a P.D.E. in Three Variables Invariant Under a Two-Parameter Group -- 2.12. Infinite Parameter Groups โ 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