Author | Shivamoggi, Bhimsen K. author |
---|---|
Title | Perturbation Methods for Differential Equations [electronic resource] / by Bhimsen K. Shivamoggi |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2003 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0047-5 |
Descript | XIV, 354 p. online resource |
1 Asymptotic Series and Expansions -- 1.1 Introduction -- 1.2 Taylor Series Expansions -- 1.3 Gauge Functions -- 1.4 Asymptotic Series and Expansions -- 1.5 Asymptotic Solutions of Differential Equations -- 1.6 Exercises -- 2 Regular Perturbation Methods -- 2.1 Introduction -- 2.2 Algebraic Equations -- 2.3 Ordinary Differential Equations -- 2.4 Partial Differential Equations -- 2.5 Applications to Fluid Dynamics: Decay of a Line Vortex -- 2.6 Exercises -- 2.7 Appendix. Review of Partial Differential Equations -- 3 The Method of Strained Coordinates/Parameters -- 3.1 Introduction -- 3.2 Poincarรฉ-Lindstedt-Lighthill Method of Perturbed Eigenvalues -- 3.3 Eigenfunction Expansion Method -- 3.4 Lighthillโs Method of Shifting Singularities -- 3.5 Prituloโs Method of Renormalization -- 3.6 Wave Propagation in an Inhomogeneous Medium -- 3.7 Applications to Solid Mechanics: Nonlinear Buckling of Elastic Columns -- 3.8 Applications to Fluid Dynamics -- 3.9 Applications to Plasma Physics -- 3.10 Limitations of the Method of Strained Parameters -- 3.11 Exercises -- 3.12 Appendix 1. Fredholmโs Alternative Theorem -- 3.13 Appendix 2. Floquet Theory -- 3.14 Appendix 3. Bifurcation Theory -- 4 Method of Averaging -- 4.1 Introduction -- 4.2 Krylov-Bogoliubov Method of Averaging -- 4.3 Krylov-Bogoliubov-Mitropolski Generalized Method of Averaging -- 4.4 Whithamโs Averaged Lagrangian Method -- 4.5 Hamiltonian Perturbation Method -- 4.6 Applications to Fluid Dynamics: Nonlinear Evolution of Modulated Gravity Wave Packet on the Surface of a Fluid -- 4.7 Exercises -- 4.8 Appendix 1. Review of Calculus of Variations -- 4.9 Appendix 2. Hamilton-Jacobi Theory -- 5 The Method of Matched Asymptotic Expansions -- 5.1 Introduction -- 5.2 Physical Motivation -- 5.3 The Inner and Outer Expansions -- 5.4 Hyperbolic Equations -- 5.5 Elliptic Equations -- 5.6 Parabolic Equations -- 5.7 Interior Layers -- 5.8 Lattaโs Method of Composite Expansions -- 5.9 Turning Point Problems -- 5.10 Applications to Fluid Dynamics: Boundary-Layer Flow Past a Flat Plate -- 5.11 Exercises -- 5.12 Appendix 1. Initial-Value Problem for Partial Differential Equations -- 5.13 Appendix 2. Review of Nonlinear Hyperbolic Equations -- 6 Method of Multiple Scales -- 6.1 Introduction -- 6.2 Differential Equations with Constant Coefficients -- 6.3 Strubleโs Method -- 6.4 Differential Equations with Slowly Varying Coefficients -- 6.5 Generalized Multiple-Scale Method -- 6.6 Applications to Solid Mechanics: Dynamic Buckling of a Thin Elastic Plate -- 6.7 Applications to Fluid Dynamics -- 6.8 Applications to Plasma Physics -- 6.9 Exercises -- 7 Miscellaneous Perturbation Methods -- 7.1 A Quantum-Field-Theoretic Perturbative Procedure -- 7.2 A Perturbation Method for Linear Stochastic Differential Equations -- 7.3 Exercises