# Singular Integral Equation

## singular integral equation

[′siŋ·gyə·lər ¦int·ə·grəl i′kwā·zhən]
(mathematics)
An integral equation where the integral appearing either has infinite limits of integration or the kernel function has points where it is infinite.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Singular Integral Equation

an integral equation with a kernel that becomes infinite in the domain of integration so that the corresponding improper integral containing the unknown function diverges and is replaced by its Cauchy principal value. An example of a singular integral equation with the Hilbert-type kernel is

whose solution is the function

where a ≠ 0 and a2+ b2 ≠ 0; the first integral here is also understood as a Cauchy principal value.

The best studied class of singular integral equations consists of equations with a Cauchy-type kernel of the form

Here, a(t), b(t), and f(t) are given continuous functions of the point t of the path of integration L in the complex plane; L may consist of a finite number of smooth non-self-intersecting open or closed curves with continuous curvature. The singular integral

is understood as the limit, as ∊ → 0, of the integral of IΦ along the path L obtained from L after eliminating an arc of length 2 that is symmetric with respect to t. The kernel K(t, z) is assumed to belong to one of the classes considered in the theory of non-singular integral equations. Many problems of, for example, the theory of analytic functions, elasticity theory, and hydrodynamics, reduce to singular integral equations of the form (*).

The singular integral equation (*) is studied on the basis of the properties of the singular integral IΦ, which depend on the assumptions made regarding Φ. Singular integral equations have been investigated in detail in the space of continuing functions Φ and in the space of square-integrable functions. The fundamental property of the singular integral is expressed by the equation I2ΦI2I(IΦ) = Φ, which is valid for a broad class of functions.

Many results in the theory of singular integral equations can be carried over virtually without change to systems of singular integral equations. Such systems can be written in the form (*) if a and b are understood as matrix functions, and f and Φ as vectors (single-column matrices). The theory can also be generalized to the case of a system of singular integral equations with discontinuous coefficients and a piecewise smooth path of integration. Certain classes of singular integral equations are also studied in multidimensional domains.

Singular integral equations were first used in the early 20th century in works by H. Poincaré (on the theory of tides) and D. Hubert (on boundary value problems). A number of important properties of singular integral equations were established by the German mathematician F. Noether. Studies by T. Carleman and I. I. Privalov were of great importance in the development of the theory of singular integral equations. The most complete results have been obtained by such Soviet scientists as N. I. Muskhelishvili, I. N. Vekua, and V. D. Kupradze.

### REFERENCES

Muskhelishvili, N. I. Singuliarnye integral’nye uravneniia: Granichnye zadachi teorii funktsii i nekotorye ikh prilozheniia k malematicheskoi fizike, 3rd ed. Moscow, 1968.
Vekua, N. P. Sistemy singuliarnykh integral’nykh uravnenii i nekotorye granichnye zadachi, 2nd ed. Moscow, 1970. [23–1230–]
References in periodicals archive ?
It is well known [6, 12,26,32] (see also [11, 15,20]) that the derivatives of a solution to a weakly singular integral equation such as (1.1), in general, have certain boundary singularities.
Prabhakar, "A singular integral equation with a generalized Mittag Leffler function in the kernel," Yokohama Mathematical Journal, vol.
By assuming perfect adhesion between film and half plane together with membrane behaviour of the film, the compatibility condition between the coating and substrate leads to a singular integral equation with Cauchy kernel.
Prabhakar, "A singular integral equation with a generalized Mittag-Leffler func-tion in the Kernel," Yokohama Mathematical Journal, vol.
Results of [11,12] are based on the volume singular integral equation (VSIE) method [8, 9] for the system of Maxwell's equations.
Ioakimidis, "Two methods for the numerical solution of Bueckner's singular integral equation for plane elasticity crack problems," Computer Methods in Applied Mechanics and Engineering, vol.
It is a straightforward matter to verify that the solution to Equation (41) with the logarithmic kernel automatically satisfies singular integral Equation (43) with the Cauchy kernel [10].
Then, we use the Toeplitz matrix method and product Nystrom method, as the best two methods, to obtain the solution of the singular integral equation numerically.
L., "Direct singular integral equation methods in scattering from strip-loaded dielectric cylinders," Journal of Electromagnetic Wave and Applications, Vol.
and a solution is of the form w(z) = [G.sup.0,0.sub.m,n-m]g(z) where g [member of] [L.sup.p](D), p > 2, is a solution of the singular integral equation
He covers Volterra integral equations, Fredholm integral equations, nonlinear integral equations, the singular integral equation, integro-differential equations, symmetric kernals and orthogonal systems of functions, and a range of applications.
By virtue of the singular integral equation theory, we can obtain the solution of (21); namely,

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