Nonlinear Partial Differential Equations in Engineering
Published on 1. January 1965
510 pages
978-0-08-095524-7 (ISBN)
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Nonlinear Partial Differential Equations in Engineering
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William F. Ames
Nonlinear Partial Differential Equations in Engineering: v. 1
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01/1965
Saunders College Publishing/Harcourt Brace
€45.10
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Content
- Front Cover
- Nonliner Partial Differential Equations in Engineering, Volume 18
- Copyright Page
- Contents
- Preface
- CHAPTER 1. The Origin of Nonlinear Differential Equations
- 1.0 Introduction
- 1.1 What is Nonlinearity?
- 1.2 Equations from Diffusion Theory
- 1.3 Equations from Fluid Mechanics
- 1.4 Equations from Solid Mechanics
- 1.5 Miscellaneous Examples
- 1.6 Selected References
- References
- CHAPTER 2. Transformation and General Solutions
- 2.0 Introduction
- 2.1 Transformations on Dependent Variables
- 2.2 Transformations on Independent Variables
- 2.3 Mixed Transformations
- 2.4 The Unknown Function Approach
- 2.5 General Solutions
- 2.6 General Solutions of First-Order Equations
- 2.7 General Solutions of Second-Order Equations
- 2.8 Table of General Solutions
- References
- CHAPTER 3. Exact Methods of Solution
- 3.0 Introduction
- 3.1 The Quasi-Linear System
- 3.2 An Example of the Quasi-Linear Theory
- 3.3 The Poisson-Euler-Darboux Equation
- 3.4 Remarks on the PED Equation
- 3.5 One-Dimensional Anisentropic Flows
- 3.6 An Alternate Approach to Anisentropic Flow
- 3.7 General Solution for Anisentropic Flow
- 3.8 Vibration of a Nonlinear String
- 3.9 Other Examples of the Quasi-Linear Theory
- 3.10 Direct Separation of Variables
- 3.11 Other Solutions Obtained by Ad Hoc Assumptions
- References
- CHAPTER 4. Further Analytic Method
- 4.0 Introduction
- 4.1 An Ad Hoc Solution from Magneto-Gas Dynamics
- 4.2 The Utility of Lagrangian Coordinates
- 4.3 Similarity Variables
- 4.4 Similarity via One-Parameter Groups
- 4.5 Extensions of the Similarity Procedure
- 4.6 Similarity via Separation of Variables
- 4.7 Similarity and Conservation Laws
- 4.8 General Comments on Transformation Groups
- 4.9 Similarity Applied to Moving Boundary Problems
- 4.10 Similarity Considerations in Three Dimensions
- 4.11 General Discussion of Similarity
- 4.12 Integral Equation Methods
- 4.13 The Hdograph
- 4.14 Simple Examples of Hodograph Application
- 4.15 The Hodograph in More Complicated Problems
- 4.16 Utilization of the General Solutions of Chapter 2
- 4.17 Similar Solutions in Heat and Mass Transfer
- 4.18 Similarity Integrals in Compressible Gases
- 4.19 Some Disjoint Remarks
- References
- CHAPTER 5. Approximate Methods
- 5.0 Introduction
- 5.1 Perturbation Concepts
- 5.2 Regular Perturbations in Vibration Theory
- 5.3 Perturbation and Plasma Oscillations
- 5.4 Perturbation in Elasticity
- 5.5 Other Applications
- 5.6 Perturbation about Exact Solutions
- 5.7 The Singular Perturbation Problem
- 5.8 Singular Perturbations in Viscous Flow
- 5.9 The "Inner-Outer'' Expansion (a Motivation)
- 5.10 The Inner and Outer Expansions
- 5.11 Examples
- 5.12 Higher Approximations for Flow past a Sphere
- 5.13 Asymptotic Approximations
- 5.14 Asymptotic Solutions in Diffusion with Reaction
- 5.15 Weighted Residual Methods: General Discussion
- 5.16 Examples of the Use of Weighted Residual Methods
- 5.17 Comments on the Methods of Weighted Residuals
- 5.18 Mathematical Problems of Approximate Methods
- References
- CHAPTER 6. Further Approximate Methods
- 6.0 Introducrion
- 6.1 Integral Methods in Fluid Mechanics
- 6.2 Nonlinear Boundary Conditions
- 6.3 Integral Equations and Boundary Layer Theory
- 6.4 Iterative Solutions for ?2u = bu2
- 6.5 The Maximum Operation
- 6.6 Equations of Elliptic Type and the Maximum Operation
- 6.7 Other Applications of the Maximum Operation
- 6.8 Series Expansions
- 6.9 Goertler's Series
- 6.10 Series Solutions in Elasticity
- 6.11 "Traveling Wave" Solutions by Series
- References
- CHAPTER 7. Numerical Methods
- 7.0 Introduction
- 7.1 Terminology and Computational Molecules
- 7.2 Explicit Methods for Parabolic Systems
- 7.3 Some Nonlinear Examples
- 7.4 Alternate Explicit Methods
- 7.5 The Quasi-Linear Parabolic Equation
- 7.6 Singularities
- 7.7 A Treatment of Singularities (Example)
- 7.8 Implicit Procedures
- 7.9 A Second-Order Method for Lu = f(x, t, u)
- 7.10 Predictor Corrector Methods
- 7.11 Traveling Wave Solutions
- 7.12 Finite Differences Applied to the Boundary Layer Equations
- 7.13 Other Nonlinear Parabolic Examples
- 7.14 Finite Difference Formula for Elliptic Equations in Two Dimensions
- 7.15 Linear Elliptic Equations
- 7.16 Methods of Solution of Au = v
- 7.17 Point Iterative Methods
- 7.18 Block Iterative Methods
- 7.19 Examples of Nonlinear Elliptic Equations
- 7.20 Singularities
- 7.21 Method of Characteristics
- 7.22 The Supersonic Nozzle
- 7.23 Properties of Hyperbolic Systems
- 7.24 One-Dimensional Isentropic Flow
- 7.25 Method of Characteristics: Numerical Computation
- 7.26 Finite Difference Methods: General Discussion
- 7.27 Explicit Methods
- 7.28 Explicit Methods in Nonlinear Second-Order Systems
- 7.29 Implicit Methods for Second-Order Equations
- 7.30 "Hybrid" Methods for a Nonlinear First-Order System
- 7.31 Finite Difference Schemes in One-Dimensional Flow
- 7.32 Conservation Equations
- 7.33 Intetfaces
- 7.34 Shocks
- 7.35 Additional Methods
- 7.36 The Role of Mixed Systems
- 7.37 Hydrodynamic Flow and Radiation Diffusion
- 7.38 Nonlinear Vibrations of a Moving Threadline
- References
- CHAPTER 8. Some Theoretical Considerations
- 8.0 Introduction
- 8.1 Well-Posed Problems
- 8.2 Existence and Uniqueness in Viscous Incompressible Flow
- 8.3 Existence and Uniqueness in Boundary Layer Theory
- 8.4 Existence and Uniqueness in Quasi-Linear Parabolic Equations
- 8.5 Uniqueness Questions for Quasi-Linear Elliptic Equations
- References
- APPENDIX. Elements of Group Theory
- A.1 Basic Definitions
- A.2 Groups of Transformations
- Author Index
- Subject Index
