# Nonlinear Problems of Engineering

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Nonlinear Problems of Engineering reviews certain nonlinear problems of engineering. This book provides a discussion of nonlinear problems that occur in four areas, namely, mathematical methods, fluid mechanics, mechanics of solids, and transport phenomena. Organized into 15 chapters, this book begins with an overview of some of the fundamental ideas of two mathematical theories, namely, invariant imbedding and dynamic programming. This text then explores nonlinear integral equations, which have long occupied a prominent place in mathematical analysis. Other chapters consider the phenomena associated with essentially divergent small-divisor series, such as may occur in the formal solution of differential equations that represent the oscillations of conservative dynamical systems. This book discusses as well the mechanics of idealized textiles consisting of inextensible filaments. The final chapter deals with the use of the Peaceman–Rachford alternating direction implicit method for solving the finite difference analogs of boundary value problems. This book is a valuable resource for engineers and mathematicians.

## Table of Contents

Contributors

Foreword

Preface

New Methods in Nonlinear Mechanics

I. Introduction

II. The Comparative Approach

III. Classical Approach

IV. Discussion

V. The New Approach of Invariant Imbedding

VI. Discussion

VII. Variational Processes

VIII. Geodesics

IX. Dynamic Programming and the Calculus of Variations

X. Approximation in Policy Space

XI. Discussion

References

Probability as a Science

On Random Solutions of Nonlinear Integral Equations

Text

References

A Singular Case of Iteration of Analytic Functions: A Contribution to the Small-Divisor Problem

I. The "Center Problem" in the Iteration of Analytic Functions

II. Contexts Leading to Small-Divisor Series

III. Generalities Regarding Iteration of Analytic Functions

IV. Construction of Singular Iterations Zn+1 = f(Zn)

V. Properties of Singular Iterations

References

Networks of Inextensible Cords

I. Introduction

II. Pure Homogeneous Deformation

III. Plane Strain of a Net—Displacement Boundary Conditions

IV. Plane Strain of a Net—Force Boundary Conditions

V. Further Developments

Bibliography

On Integral Operators Generating Stream Functions of Compressible Fluids

Introduction

I. Partial Differential Equations Arising in the Theory of Compressible Fluids

II. The Determination of the Stream Function of a Vortex in the Subsonic Region

III. A Method for Generating Solutions of Equation (1.3.5)

References

Bibliography

Some Problems Concerning Linear Differential Equations Made Nonlinear by Unknown or Moving Boundaries

A. Melting or Solidification

B. Consolidation of an Earth Dam

C. Seepage through a Porous Dam

D. Lubrication in a Journal Bearing, with Oil Cavitation

References

General References

Nonlinear Problems of Combined Conductive-Radiative Heat Transfer

Nomenclature

Introduction

The Single Plate-Type Fin

Radiant Interaction Between Fin and Tube Surfaces

Radiant Interaction Between Fin Surfaces

Radiant Interaction Between Fin and Tube and Between Neighboring Fins

Concluding Remarks

References

Statistical Problems Connected with the Solution of a Nonlinear Partial Differential Equation

I. A Particular Nonlinear Partial Differential Equation and Its Solution by Means of a Geometrical Construction

II. Formulation of a Statistical Problem

III. The Solution of the Auxiliary Differential Equation

Notes

Melting Decelerating Bodies

Introduction

Symbols

Analysis

Results

Concluding Remarks

References

Thermal Radiation Effects on Hypersonic Flow Fields

I. Introduction

II. Fundamentals of Radiant Energy Transfer

III. Fundamental Equations of Radiation Gas Dynamics

IV. Solutions for Radiation Transfer Equation

V. New Parameters of Radiation Gas Dynamics

VI. Wave Motions in Radiation Gas Dynamics

VII. Shock Waves in Radiation Gas Dynamics

VIII. Jet Mixing in Radiation Gas Dynamics

IX. Plane Couette Flow in Radiation Magnetogas Dynamics

References

Mathematical Theory of Compressible Fluids

Introduction

List of Symbols

1. Two-Dimensional Compressible Flow in the Physical Plane

2. Two-Dimensional Compressible Flow in the Hodograph Plane

3. Two-Dimensional Compressible Flow About a Closed Body in a Uniform Subsonic Flow

4. Two-Dimensional Flow through the Sonic Line

References

Problems in Nonlinear Nonmechanics

Introduction

Information and Communication

Symbols and Symbolic Structure

Handicaps and Communication

Other Observations

Concluding Remarks

References

Nonlinear Problems in Ballistics

Text

References

Alternating Direction Methods for Solving Partial Difference Equations

1. Introduction

2. The Difference Equation

3. The Linear System

4. The Peaceman-Rachford Method

5. The Cummutative Case

6. Laminar Flow Problem

7. Numerical Results and Conclusions

References

Subject Index

## Product details

- No. of pages: 266
- Language: English
- Copyright: © Academic Press 1964
- Published: January 1, 1964
- Imprint: Academic Press
- eBook ISBN: 9781483225814

## About the Editor

### William F. Ames

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