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 I_∊Φ 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 IΦ is expressed by the equation I²Φ ≡ I²I(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.