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)

An exact expression is derived for the general finite-part integral 






over an inclined ellipticaldomain Ω. r denotes the distance of a point in Ω to the singular point is a general function of the Cartesian co-ordinates x0,y0. The boundary of the region Ω represents the equation Z(x0, y0)=O. These integrals appear during the numerical solution of plane crack problems in three-dimensional elasticity where they are the dominant part of a hypersingular integral equation. The availability of exact expressions for the integrals with arbitrary integers i and j will increase the accuracy of the numerical results and, simultaneously, lead to quicker numerical results.



The considered finite-part integral can be expressed in closed form as function of complete elliptical integrals or Gauss hypergeometric functions, respectively.



Formuias for special cases and some i, j values and their numerical verification are given in Appendices II and III.

Derivation of a new analytical solution for a general two-dimensional finite-part integral applicable in fracture mechanics

Non-linear singular integral equations are investigated in connection with some basic applications in two-dimensional fluid mechanics. A general existence and uniqueness analysis is proposed for non-linear singular integral equations defined on a Banach space. Therefore, the non-linear equations are defined over a finite set of contours and the existence of solutions is investigated for two different kinds of equations, the first and the second kind. Moreover, the existence of solutions is further studied for non-linear singular integral equations over a finite number of arbitrarily ordered arcs. An application to fluid mechanics theory is finally given for the determination of the form of the profiles of a turbomachine in two-dimensional flow of an incompressible fluid.

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Existence and uniqueness for non-linear singular integral equations used in fluid mechanics

Non-linear singular integral approximations in Banach spaces

Quadrature rules for weakly singular, strongly singular, and hypersingular integrals in boundary integral equation methods

A new technique is proposed for the numerical solution of the Cauchy-type singular integral equations, by using the well known Gegenbauer polynomials. A large class of problems of mathematical physics, and especially several plane and antiplane elasticity problems, not possessing closed-form solutions, can be reduced to the solution of certain systems of such a type of singular integral equations. Also by using a certain method the new formula which is used for the numerical solution of the Cauchy-type integral equations can be extended for the general type of the finite-part singular integrals, too.

E. G. Ladopoulos

Papers

On a new integration rule with the Gegenbauer polynomials for singular integral equations used in the theory of elasticity