Revival: Block Method for Solving the Laplace Equation and for Constructing Conformal Mappings (1994): 1st Edition (Paperback) book cover

Revival: Block Method for Solving the Laplace Equation and for Constructing Conformal Mappings (1994)

1st Edition

By Evgenii A. Volkov

CRC Press

238 pages

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Paperback: 9781138557796
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Description

This book presents a new, efficient numerical-analytical method for solving the Laplace equation on an arbitrary polygon. This method, called the approximate block method, overcomes indicated difficulties and has qualitatively more rapid convergence than well-known difference and variational-difference methods. The block method also solves the complicated problem of approximate conformal mapping of multiply-connected polygons onto canonical domains with no preliminary information required. The high-precision results of calculations carried out on the computer are presented in an abundance of tables substantiating the exponential convergence of the block method and its strong stability concerning the rounding-off of errors.

Table of Contents

Approximate Block Method for Solving the Laplace Equation on Polygons

Setting up a Mixed Boundary Value Problem for the Laplace Equation on a Polygon

A Finite Covering of a Polygon by Blocks of Three Types

Representation of the Solution of a Boundary Value Problem on Blocks

An Algebraic Problem

The Main Result - Theorem on the Convergence of the Block Method

Proofs of Theorem and Lemmas

The Stability and the Labor Content of Computations Required by the Block Method

Approximation of a Conjugate Harmonic Function on Blocks

Neumann's Problem

The Case of Arbitrary Analytic Mixed Boundary Conditions

Approximate Block Method of Conformal Mapping of Polygons onto Canonical Domains

Approximate Conformal Mapping of a Simply-Connected Polygon onto a Disk

Basic Harmonic Functions

Approximate Conformal Mapping of a Multiply-Connected Polygon onto a Plane with Cuts along Parallel Line Segments

Approximate Conformal Mapping of a Multiply-Connected Polygon onto a Ring with Cuts along the Arcs of Concentric Circles

Development and Application of the Approximate Block Method for Conformal Mapping of Simply-Connected and Doubly-Connected Domains

Approximate Conformal Mapping of Some Polygons onto a Strip

Scheme of Constructing a Conformal Mapping of a Doubly-connected Domain onto a Ring

Mapping a Square Frame onto a Ring

Mapping a Square with a Circular Hole Using Circular Lune Block

Representation of a Harmonic Function on a Ring

Using a Block-Ring for Mapping Domain (18.1) onto a Ring

A Block-Bridge

Limit Cases

Mapping a Disk with an Elliptic Hole or with a Retro-Section onto a Ring

Mapping a Disk with a Regular Polygonal Hole

Mapping the Exterior of a Parabola with a Hole onto a Ring

Approximate Conformal Mapping of Domains with a Periodic Structure by the Block Method

Mapping a Domain of the Type of Half-Plane with a Periodic Structure onto a Half-plane

Mapping a Domain of the Type of Strip with a Periodic Structure onto a Strip

Mapping the Exterior of a Lattice of Ellipses onto the Exterior of a Lattice of Plates

References

Index

About the Author

Evgenii A. Volkov is a professor at the Steklov Mathematical Institute in Moscow, Russia.

About the Series

CRC Press Revivals

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations