Integral Equations

equations containing unknown functions under the integral sign. Numerous problems of physics and mathematical physics lead to various types of integral equations. Suppose, for example, we are required to obtain with the help of an optical instrument an image of a linear object A occupying the interval 0 ≤ x ≤ l on the x-axis, where the illumination of the object is characterized by a density function u(x). The image B is a certain interval on another axis x₁. By a suitable choice of origin and unit of length the latter interval can be made to coincide with the interval 0 ≤ x₁ ≤ l. If an infinitesimal part (x,x + Δ x) of the object A causes an illumination of the image B with density K(xi,x) u(x) dx, where the function K(x₁,x) is determined by the properties of the optical instrument, then the total illumination of the image will have density

Depending on whether we wish to obtain a given illumination v(x,₁) of the image, or an “accurate” photographic image [v(x) = ku(x), where the constant k is not fixed in advance], or, finally, a specified difference of the illumination of A and B [ u(x)— v(x) = f(x)], we arrive at various integral equations for the function u(x):

In general, an integral equation of the form

is called a linear integral equation of the first kind, whereas an equation of the form

is called a linear integral equation of the second kind, or a Fredholm equation [when f(x) ≡ 0 it is called a homogeneous Fredholm equation]. A Fredholm equation usually involves a parameter λ:

In all equations, the function

K(x,j) (a≤ x ≤ b, a≤ y ≤ b)

the kernel of the integral equation, is given as is the function f(x) (a ≤ x ≤ b); the unknown function is u(x) (a ≤ x ≤ b).

The functions K(x, y), f(x), and u(x) and the parameter λ can take on real or complex values. In the particular case when the kernel K(x, y) vanishes for y > x, we obtain the Volterra equation:

An integral equation is called singular if at least one of the limits of integration is infinite or if the kernel K(x, y) becomes infinite at one or several points of the square a ≤ x ≤ b, a ≤ y ≤ b or on a certain curve. Integral equations can involve functions of several variables. A relevant example is furnished by the equation

Nonlinear integral equations are also considered, for example, equations of the form

Linear integral equations of the second kind are solved by the following methods. (1) The solution u(x) is obtained in the form of a power series in λ (convergent in a certain circle |λ| < K) with coefficients that depend on x (the Volterra-Neumann method). (2) The solution u(x), for values of λ for which it exists, is expressed in terms of certain entire functions of λ (the Fredholm method). (3) In the case when the kernel is symmetric, that is, K(x, y) ≡ K(y,x), the solution u(x) is expressed in the form of a series of orthogonal functions u_k(x), which are nonzero solutions of the corresponding homogeneous equation

(the latter has nontrivial solutions only for certain values λ = λ_kk = 1, 2, …, of the parameter λ) (the Hilbert-Schmidt method). (4) In certain exceptional cases, a solution is comparatively easily obtained with the help of the Laplace transform. (5) In the case when

(the case of the so-called degenerate kernel), the determination of u(x) reduces to the solution of a system of algebraic equations. Approximate solutions may be obtained, either by applying some formula for numerical integration to ∫_a^bK(x,t) u(t) dt or by substituting for the given kernel K(x,y). some degenerate kernel that differs slightly from K(x,y). Boundary value problems for ordinary and partial differential equations often reduce to integral equations; such a reduction is of theoretical and practical value.