Nonlinear Partial Differential Equations

List of ContributorsForewordPrefaceGeneralized Similarity Analysis of Partial Differential Equations Introduction Types of Similarity Analyses Free Parameter Analysis Separation of Variables Method The Group Theory Approach ReferencesVector Eigenfunction Expansions for the Growth of Taylor Vortices in the Flow between Rotating Cylinders 1. Introduction 2. The Governing Equations 3. The Linear Problem 4. The Growth of Taylor Vortices 5. Eigenfunction Expansions 6. Discussion ReferencesNew Methods for the Solution of Partial Differential Equations 1. Introduction 2. Partial Differential Equations in Dynamic Programming 3. Quasilinearization 4. Novel Difference Techniques 5. Novel Difference Techniques 6. Infinite Systems of Ordinary Differential Equations 7. Laplace Transform Techniques 8. Quadrature Techniques 9. Perturbation Techniques ReferencesAd hoc Exact Techniques for Nonlinear Partial Differential Equations 1. Introduction 2. Separation of Variables 3. Further Specific Forms 4. Assumed Relations between Dependent Variables 5. Equations Equivalent to Linear Forms 6. Equation Splitting 7. Equation Splitting and the Navier-Stokes Equations ReferencesThe Lubrication Approximation Applied to Non-Newtonian Flow Problems: a Perturbation Approach 1. Introduction 2. The Lubrication Approximation 3. Equations of State for Non-Newtonian Fluids 4. Perturbation and Iterative Solution Scheme 5. Extension to Include Unsteadiness, Compressibility, and Heat Effects 6. Discussion ReferencesThe Computation of Compressible Boundary-Layer Flow Text ReferencesIntegral Equations for Nonlinear Problems in Partial Differential Equations Introduction 1. Boundary Value Problems for Elliptic Equations 2. Upper and Lower Function for Volterra Equations with Monotonic Integrands 3. A Nonlinear Initial Value Problem ReferencesElectrical Problems Modeled by Nonlinear Partial Differential Equations Text ReferencesDifference Methods and Soft Solutions 1. Soft Solutions 2. Weak Solutions 3. Exact Difference Methods 4. Second Order Equations ReferencesNumerical Solution of the Nonlinear Equations for Two-Phase Flow through Porous Media Introduction The Differential Equations Solution by Finite Difference Equations Evaluation of the Nonlinear Coefficients S' Limiting Form of Equations at Zero Capillary Pressure Use of "Upstream" Values of Coefficients KN and KW Existence of Discontinuity Possible Improvements ReferencesAn Extrapolated Crank-Nicolson Difference Scheme for Quasilinear Parabolic Equations Text ReferencesHeat Transfer to the Endwall of a Shocktube. A Variational Analysis Introduction A Least-Error Problem for Transport Experiments The Shocktube Experiment The Thermal Conduction Model Energy Equation Transformations Variational Formulation A Computational Procedure Some Numerical Results Discussion ReferencesA Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction I. Introduction II. The Synergetic Approach III. The Nonlinear One-Dimensional Lattice IV. Solitons, the Korteweg-de Vries Equation, and Some Computational Results V. Synergetics-Future Directions ReferencesUniformization of Asymptotic Expansions I. Introduction II. The Uniformization Method III. Results and Open Problems ReferencesHigh Order Accurate Difference Methods in Hydrodynamics 1. Introduction 2. Trends in Lagrange Calculations 3. Eulerian Calculations in Three Independent Variables 4. Two Step Lax-Wendroff Schemes 5. Instabilities of the Nonlinear Type 6. Navier-Stokes Equations 7. Conclusions ReferencesNonlinear Problems in the Dynamics of Thin Shells Text ReferencesIndex