1st Edition

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





ISBN 9781138557796
Published January 25, 2019 by CRC Press
238 Pages

USD $59.95

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Book 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

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Author(s)

Biography

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