1st Edition

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

By Evgenii A. Volkov Copyright 1994
    238 Pages
    by CRC Press

    240 Pages
    by CRC Press

    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.

    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

    Biography

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