Nonlinear Partial Differential Equations in Engineering

✍ Scribed by William F. Ames (editor)

• Publisher: Academic Press
• Year: 1965
• Tongue: English
• Leaves: 527
• Series: Mathematics in Science and Engineering; 18-I
• Category: Library

✦ Synopsis

Nonlinear Partial Differential A Symposium on Methods of Solution is a collection of papers presented at the seminar on methods of solution for nonlinear partial differential equations, held at the University of Delaware, Newark, Delaware on December 27-29, 1965. The sessions are divided into four Analytic Methods, Approximate Methods, Numerical Methods, and Applications. Separating 19 lectures into chapters, this book starts with a presentation of the methods of similarity analysis, particularly considering the merits, advantages and disadvantages of the methods. The subsequent chapters describe the fundamental ideas behind the methods for the solution of partial differential equation derived from the theory of dynamic programming and from finite systems of ordinary differential equations. These topics are followed by reviews of the principles to the lubrication approximation and compressible boundary-layer flow computation. The discussion then shifts to several applications of nonlinear partial differential equations, including in electrical problems, two-phase flow, hydrodynamics, and heat transfer. The remaining chapters cover other solution methods for partial differential equations, such as the synergetic approach. This book will prove useful to applied mathematicians, physicists, and engineers.

✦ Table of Contents

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

6.5 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