# Nonlinear Partial Differential Equations in Engineering

## 1st Edition

### Mathematics in Science and Engineering: A Series of Monographs and Textbooks, Vol. 18

**Authors:**W. F. Ames

**Editors:**Richard Bellman

**eBook ISBN:**9781483222929

**Imprint:**Academic Press

**Published Date:**1st January 1965

**Page Count:**526

## Description

*Nonlinear Partial Differential Equations in Engineering* discusses methods of solution for nonlinear partial differential equations, particularly by using a unified treatment of analytic and numerical procedures. The book also explains analytic methods, approximation methods (such as asymptotic processes, perturbation procedures, weighted residual methods), and specific numerical procedures associated with these equations. The text presents exact methods of solution including the quasi-linear theory, the Poisson-Euler-Darboux equation, a general solution for anisentropic flow, and other solutions obtained from ad hoc assumptions. The book explores analytic methods such as an ad hoc solution from magneto-gas dynamics. Noh and Protter have found the Lagrange formulation to be a convenient vehicle for obtaining "soft" solutions of the equations of gas dynamics. The book notes that developing solutions in two and three dimensions can be achieved by employing Lagrangian coordinates. The book explores approximate methods that use analytical procedures to obtain solutions in the form of functions approximating solutions of nonlinear problems. Approximate methods include integral equations, boundary theory, maximum operation, and equations of elliptic types. The book can serve and benefit mathematicians, students of, and professors of calculus, statistics, or advanced mathematics.

## Table of 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 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

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 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

References

Chapter 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

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

## Details

- No. of pages:
- 526

- Language:
- English

- Copyright:
- © Academic Press 1965

- Published:
- 1st January 1965

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483222929

## About the Author

### W. F. Ames

## About the Editor

### Richard Bellman

### Affiliations and Expertise

Departments of Mathematics, Electrical Engineering, and Medicine University of Southern California Los Angeles, California