# Nonlinear Partial Differential Equations

## 1st Edition

### A Symposium on Methods of Solution

**Editors:**W. F. Ames

**eBook ISBN:**9781483221502

**Imprint:**Academic Press

**Published Date:**1st January 1967

**Page Count:**332

## Description

Nonlinear Partial Differential Equations: 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 Symposia: 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

List of Contributors

Foreword

Preface

Generalized Similarity Analysis of Partial Differential Equations

Introduction

Types of Similarity Analyses

Free Parameter Analysis

Separation of Variables Method

The Group Theory Approach

References

Vector 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

References

New 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

References

Ad 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

References

The 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

References

The Computation of Compressible Boundary-Layer Flow

Text

References

Integral 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

References

Electrical Problems Modeled by Nonlinear Partial Differential Equations

Text

References

Difference Methods and Soft Solutions

1. Soft Solutions

2. Weak Solutions

3. Exact Difference Methods

4. Second Order Equations

References

Numerical 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

References

An Extrapolated Crank-Nicolson Difference Scheme for Quasilinear Parabolic Equations

Text

References

Heat 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

References

A 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

References

Uniformization of Asymptotic Expansions

I. Introduction

II. The Uniformization Method

III. Results and Open Problems

References

High 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

References

Nonlinear Problems in the Dynamics of Thin Shells

Text

References

Index

## Details

- No. of pages:
- 332

- Language:
- English

- Copyright:
- © Academic Press 1967

- Published:
- 1st January 1967

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483221502