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.

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 ?
Prabhakar, "A singular integral equation with a generalized Mittag-Leffler func-tion in the Kernel," Yokohama Mathematical Journal, 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.
Direct singular integral equation methods in scattering from strip-loaded dielectric cylinders," Journal of Electromagnetic Wave and Applications, Vol.
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.
w] can be found as a solution of the singular integral equation
In this paper we consider a polynomial collocation method for the numerical solution of a singular integral equation over the interval.
RATHSFELD, A wavelet algorithm for the solution of a singular integral equation over a smooth two-dimensional manifold, J.
Jalilian, Existence of solutions for a class of nonlinear Volterra singular integral equations, Comput.
Mathematical modeling of controlling the coating process of the structural elements based on singular integral equations.
Satisfying boundary conditions (5) by the functions (13)-(14) on the faces of cracks with end zones, we get a system of singular integral equations with respect to the unknown functions [g.