Integral Equations

ds}¹/²

1

Classification of integral equations

1.1 Historical introduction

The name integral equation for any equation involving the unknown function ϕ(x) under the integral sign was introduced by du Bois-Reymond in 1888. However, the early history of integral equations goes back a considerable time before that to Laplace who, in 1782, used the integral transform

to solve linear difference equations and differential equations.

In connection with the use of trigonometric series for the solution of heat conduction problems, Fourier in 1822 found the reciprocal formulae

and

where the Fourier sine transform (3) and the cosine transform (5) provide the solutions ϕ(x) of the integral equations (2) and (4) respectively in terms of the known function f(x).

In 1826 Abel solved the integral equation named after him having the form

where f(x) is a continuous function satisfying f(a) = 0, and 0 < α < 1.

Abel’s integral equation corresponds to the famous tautochrone problem first solved by Huygens, namely the determination of the shape of the curve with a given end point along which a particle slides under gravity in an interval of time which is independent of the starting position on the curve. Huygens showed that this curve is a cycloid.

An integral equation of the type

in which the unknown function ϕ(s) occurs outside as well as before the integral sign and the variable x appears as one of the limits of the integral, was obtained by Poisson in 1826 in a memoir on the theory of magnetism. He solved the integral equation by expanding ϕ(s) in powers of the parameter λ but without establishing the convergence of this series. Proof of the convergence of such a series was produced later by Liouville in 1837.

Dirichlet’s problem, which is the determination of a function ψ having prescribed values over a certain boundary surface S and satisfying Laplace’s equation ∇²ψ = 0 within the region enclosed by S, was shown by Neumann in 1870 to be equivalent to the solution of an integral equation. He solved the integral equation by an expansion in powers of a certain parameter λ. This is similar to the procedure used earlier by Poisson and Liouville, and corresponds to a method of successive approximations.

In 1896 Volterra gave the first general treatment of the solution of the class of linear integral equations bearing his name and characterized by the variable x appearing as the upper limit of the integral.

A more general class of linear integral equations having the form

which includes Volterra’s class of integral equations as the special case given by K(x, s) = 0 for s > x, was first discussed by Fredholm in 1900 and subsequently, in a classic article by him, in 1903. He employed a similar approach to that introduced by Volterra in 1884. In this method the Fredholm equation (8) is regarded as the limiting form as n → ∞ of a set of n linear algebraic equations

where δn = (b − a)/n and xr = a + rδn. The solution of these equations can be readily obtained and Fredholm verified by direct substitution in the integral equation (8) that his limiting formula for n → ∞ gave the true solution.

1.2 Linear integral equations

Integral equations which are linear involve the integral operator

having the kernel K(x, s). It satisfies the linearity condition

where

and λ1 and λ2 are constants.

Linear integral equations are named after Volterra and Fredholm as follows:

The Fredholm equation of the first kind has the form

Examples are given by Laplace’s integral (1) and the Fourier integrals (2) and (4).

The Fredholm equation of the second kind has the form

while its corresponding homogeneous equation is

The Volterra equation of the first kind has the form

An example of such an equation is Abel’s equation (6) which, however, has a special feature arising from the presence of a singular kernel

with a singularity at s = x.

The Volterra equation of the second kind has the form

We see that the Volterra equations can be obtained from the corresponding Fredholm equations by setting K(x, s) = 0 for

It can be readily seen also that the Volterra equation (16) of the first kind can be transformed into one of the second kind by differentiation, for we have

so that provided K(x, x) does not vanish in a ≤ x ≤ b we obtain

1.3 Special types of kernel

1.3.1 Symmetric kernels

Integral equations with symmetric kernels satisfying

possess certain simplifying features. For this reason it is valuable to know that the integral equation

with the unsymmetrical kernel P(x, s)μ(s), where however P(s, x) = P(x, s), can be transformed into the integral equation (14) with symmetric kernel

by multiplying (and setting

and

Real symmetric kernels are members of a more general class known as Hermitian kernels which are not necessarily real and are characterized by the relation

the bar denoting complex conjugate. We shall investigate the properties of integral equations with Hermitian kernels in chapter 10.

1.3.2 Kernels producing convolution integrals

A class of integral equation which is of particular interest has a kernel of the form

depending only on the difference between the two coordinates x and s. The type of integral which arises is

called a convolution or folding. This nomenclature comes from the situation which occurs in Volterra equations where the integral (25) becomes

and the range of integration can be regarded as if it were folded at s = x/2 so that the point s corresponds to the point x − s as shown in Fig. 1.

Fig. 1. Convolution or folding

Integral equations of the convolution type can be solved by using various kinds of integral transform such as the Laplace and Fourier transforms and will be discussed in detail in chapter 3.

1.3.3 Separable kernels

Another interesting type of integral equation has a kernel possessing the separable form

The Fredholm integral equation of the second kind (14) now becomes

and can be readily solved exactly. Thus we have, on multiplying (and integrating:

which gives

so that

We see that the solution (30) can be expressed in the form

where

and is called the solving kernel or resolvent kernel.

The homogeneous equation (15) corresponding to (14) becomes in the present case

The solution ϕ(x) of equation (33) must satisfy

The values of λ for which the homogeneous equation has solutions are called characteristic values. There exists just one value of λ for which (33) possesses a solution. This characteristic value λ1 is given by

the associated characteristic solution of (33) being ϕ1(x) = cu(x) where c is an arbitrary constant.

The simple separable form (27) is a special case of the class of degenerate kernels

giving rise to integral equations which can be solved in closed analytical form as we shall show in chapter 9.

Example 1. As a simple illustration of an integral equation with separable kernel (27) we consider

Here f(x) = 1, K(x, s) = λxs and so u(x) = x and v(s) = s. It follows that the resolvent kernel