# Analogues for the Solution of Boundary-Value Problems

### International Tracts in Computer Science and Technology and Their Application

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Analogues for the Solution of Boundary-Value Problems considers the simulation of integral methods of solving boundary-value problems. This book is organized into 11 chapters. After the introduction provided in Chapter I, the formulation of some important engineering problems that reduce to the solution of partial differential equations is reviewed in Chapter II. Chapter III covers the mathematical methods for the solution of problems, such as the thermal problem of electrode graphitization and underground coal gasification. The theory of the physical processes of electrical simulation and principles involved in the construction of analogues is elaborated in Chapter IV, while the measurements in electrical analogues is deliberated in Chapter V. Chapters VI to VIII describe the construction of network analyzers and star-integrating networks. The methods of physical simulation for the solution of certain boundary-value problems are analyzed in Chapter IX. Chapters X and XI are devoted to future improvements and developments in analogues for the solution of boundary-value problems. This publication is intended for college students and specialists engaged in solving boundary-value problems.

## Table of Contents

Preface

Preface to the English Edition

Chapter I. Introduction

Chapter II. Formulation of Problems

§ 1 Heating of Electrodes in Graphitization Furnaces

§ 2 Heating of Metals in Continuous Furnaces

§ 3 Efficient Exploitation of Oil-Fields

§ 4 The Problem of Underground Gasification of Coal

§ 5 The Plane Contact Problem of the Theory of Elasticity

§ 6 The Computation of a Magnetic Field

§ 7 Flow of an Incompressible Fluid Past a Body

§ 8 Computation of an Electron Lens

§ 9 Investigating the Operation of Deep-Pumping Installations

Chapter III. Mathematical Principles for the Solution of Boundary-Value Problems

§ 1 Types of Equations

§ 2 Boundary and Initial Conditions

§ 3 Some Exact Solutions of Boundary-Value Problems

§ 4 The Finite-Difference Method for the Solution of Boundary-Value Problems

§ 5 The Method of Straight Lines

§ 6 The Basic Formulae of the Theory of Potential Functions and the Basic Properties of Harmonic Functions

§ 7 The Integral Form of Solution

§ 8 Integral Equations

§ 9 The Monte Carlo Method

§ 10 Connexion Between the Methods of Solution of Boundary-Value Problems and the Domain of their Application

§ 11 Combination of Finite-Difference and Integral Methods in the Solution of Boundary-Value Problems

§ 12 The Characteristic Equation. Characteristic Values and Characteristic Functions

§ 13 Variational Principles for the Solution of Boundary-Value Problems

Chapter IV. Instrumental Methods of Solving Boundary-Value Problems

§ 1 Fundamental Laws of Electricity Employed in Solving Boundary-Value Problems

§ 2 The Principles of Constructing Electrical Circuits for the Solution of Boundary-Value Problems

§ 3 Approximation to the Laplacian by Electrical Networks

§ 4 Poisson's Equation

§ 5 Fourier's Equation

§ 6 The Equation of Heat Conduction

§ 7 The Wave Equation

§ 8 Pockels' Equation

§ 9 The Telegraph Equation

§ 10 The General Case

§ 11 The Biharmonic Equation

§ 12 Assigning the Boundary and Initial Conditions

§ 13 The Use of Two Potential Dividers in the Solution of Boundary-Value Problems

§ 14 Integrating Stars

§ 15 Instrumental Solution of Integral Equations

§ 16 The Characteristic Equation

§ 17 The Schrodinger Equation

§ 18 Instrumental Approximation to Complex Expressions

§ 19 Determination of Green's Function by Instrumental Means

§ 20 The Limiting Precision of Measurements on Grids

Chapter V. Measurements

§ 1 Methods of Measurement

§ 2 The Null Indicator

§ 3 Determination of the Scale Factor

§ 4 Measurements in "Absolute" Units

Chapter VI. Universal Grid Analogues

§ 1 The EI-11 Electrical Analogue

§ 2 The EI-12 Electro-Integrator

§ 3 The EI-22 Electrical Analogue

§ 4 The EI-21 Electrical Analogue

§ 5 The EI-31 Electrical Analogue

Chapter VII. Special-Purpose Grid Analogues

§ 1 The EM-1 and EM-2 Analogues

§ 2 The EM-5, EM-7 and EM-8 Analogues

§ 3 The EP-41 Electrical Analogue for Determining a Magnetic Field in a Halfspace

§ 4 Solution of Plane Problems of the Theory of Elasticity. The EM-6-BU Analogue

Chapter VIII. Star Integrating Networks

§ 1 The Need for a Further Type of Instrument to Cover the Solution of Boundary-Value Problems for Unbounded Domains

§ 2 Design Principles of Circuits of the "Integrating Star" Instruments

§ 3 The IZ-3 Instrument

§ 4 Errors Arising in the Use of IZ Instruments

§ 5 Problems Solved

Chapter IX. Methods of Physical Simulation for the Solution of Certain Boundary-Value Problems

§ 1 Analogues for the Electrohydrodynamical Analogy EGDA

§ 2 Magnetic Analogue

§ 3 Capacitor Analogue

§ 4 Optical Analogues

§ 5 Hydraulic Integrator

Chapter X. Future Improvements to Analogues for the Solution of Boundary-Value Problems

§ 1 Extension of the Class of Problems which can be Solved

§ 2 Acceleration of the Process of Solution

§ 3 Modernization of the Design of Analogues and their Components

Chapter XI. Further Developments-Supplementary Chapter to the English Edition

§ 1 Introduction

§ 2 Development of Integral Analogue Methods of Solving Boundary-Value Problems

§ 3 Iterative Method of Defining Boundary Conditions in the Solution of Some Boundary-Value Problems

§ 4 Electronic Apparatus for Defining Variable Boundary Conditions in the Solution of Non-Stationary Boundary-Value Problems

§ 5 On the Theory of Substitution in Electrical Network Analogues for the Solution of Boundary-Value Problems

Bibliography

Index

Other Titles in the Series

## Product details

- No. of pages: 474
- Language: English
- Copyright: © Pergamon 2013
- Published: January 1, 1965
- Imprint: Pergamon
- eBook ISBN: 9781483181370

## About the Authors

### B. A. Volynskii

### V. Ye. Bukhman

## About the Editors

### N. Metropolish

### E. E.

### S. Ulam

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