Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

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Methods of Numerical Integration

Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

Boundary value problems

WHAT is the theory of functions about? This question may be heard now and again from a mathematical student; and if, by way of a pattial reply, it be said that the elements of the theory of functions forms the basis on which the whole of that part of pure mathematics which deals with continuously varying quantity rests, the answer would not be too wide nor would it always imply too much. Theory of Functions of a Complex Variable. By Dr. A. R. Forsyth. (Cambridge University Press, 1893.)

Theory of Functions of a Complex Variable

Generalized Analytic Functions. By I. N. Vekua. English Translation Editor I. N. Sneddon. Pergamon Press. 1962. Pp. 668. 105s.

The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions

We present a unified boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region $G$ onto the five classical canonical slit domains. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann kernel. Using the proposed method, the approximate mapping functions onto the five canonical slit domains can be computed in a unified way by solving linear systems with a common coefficient matrix. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The performance of the method is illustrated by several examples for regions with smooth boundaries and with piecewise smooth boundaries.

Numerical Conformal Mapping via a Boundary Integral Equation with the Generalized Neumann Kernel

A boundary integral method is presented for constructing approximations to the mapping functions of bounded multiply connected regions to the standard canonical slits domains given by Nehari [11]. The method is based on expressing the mapping function in terms of the solution of a Riemann-Hilbert problem which can be solved by a uniquely solvable boundary integral equation with the generalized Neumann kernel. Three numerical examples are presented to show the effectiveness of the present method.

Mohamed M. S. Nasser

Papers

The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions

Numerical Conformal Mapping via a Boundary Integral Equation with the Generalized Neumann Kernel

A Boundary Integral Equation for Conformal Mapping of Bounded Multiply Connected Regions

Numerical conformal mapping of multiply connected regions onto the second, third and fourth categories of Koebeʼs canonical slit domains

Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions