Nonlinear Partial Differential Equations in Engineering

PrefaceChapter 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 ReferencesChapter 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 ReferencesChapter 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 ReferencesChapter 4. Further Analytic Methods 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 Hodograph 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 ReferencesChapter 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 ReferencesChapter 6. Further Approximate Methods 6.0 Introduction 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 ReferencesChapter 7. Numerical Methods 7.0 Introduction 7.1 Terminology and Computational Molecules A. Parabolic Equations 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 B. Elliptic Equations 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 C. Hyperbolic Equations 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 Interfaces 7.34 Shocks 7.35 Additional Methods D. Mixed Systems 7.36 The Role of Mixed Systems 7.37 Hydrodynamic Flow and Radiation Diffusion 7.38 Nonlinear Vibrations of a Moving Threadline ReferencesChapter 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 ReferencesAPPENDIX. Elements of Group Theory A.1 Basic Definitions A.2 Groups of TransformationsAuthor IndexSubject Index