# Integral equations, numerical methods

Methods for finding approximate solutions of integral equations.

It is required to find the solution of a one-dimensional Fredholm equation of the second kind, (1)

where is continuous on , is a numerical parameter and is continuous on .

Suppose that is not an eigen value of . Then equation (1) has a unique solution , which is continuous on . Under these conditions one can give the following methods for obtaining an approximate solution.

## First method.

Let and be finite numbers. The integral in (1) is replaced by an integral sum over a grid , , while the variable takes the values . One then obtains a system of linear algebraic equations with respect to the , (2)

where , and are the coefficients of the quadrature formula by means of which the integral in (1) is replaced by the integral sum. For sufficiently large , the system (2) has a unique solution . As an approximate solution to (1) one can take the function since as and , the sequence of functions converges uniformly on to the required solution of equation (1). See .

When replacing the integral by a quadrature formula, one has to bear in mind that the greater the precision of the quadrature formula used, the greater the smoothness of the kernel and solution (and hence also ) needs to be.

In the case when the range of integration is infinite, it is replaced by a finite interval by using a priori information concerning the behaviour of the required solution for large values of . The equation so obtained is then approximately solved by the above method. Alternatively, by a change of the integration variable the range of integration is reduced to a finite range. As another alternative, quadrature formulas for an infinite range can be applied.

## Second method.

In equation (1) the kernel is replaced by a degenerate kernel approximating it: in which the functions are linearly independent. The equation (3)

obtained in this way has a solution of the form (4)

in which the constants have to be determined. On substituting the function into equation (3) and comparing the coefficients at the functions one obtains a system of linear algebraic equations for the : where Having determined the from the above system and by substituting them in (4) one obtains the function , which is taken as the approximate solution of (1), since for a sufficiently good approximation of the kernel by a degenerate kernel the solution of equation (3) differs by an arbitrarily small amount from the required solution on any interval , as well as on in the case when is a finite interval (see , ).

## Third method.

As approximate solutions one takes functions obtained by iteration based on the formula  ; the sequence converges uniformly to the required solution as , provided that , where . Convergence of to the exact solution also holds for kernels with an integrable singularity (see ). Estimates of the errors of these methods are given in . In  the question is considered of the minimum number of arithmetical operations required in order to obtain an approximate value of the integral with a prescribed precision. The solution of this problem is equivalent to finding the size of the minimal error of an approximate solution of the problem under a prescribed number of arithmetical operations.

For the solution of Fredholm integral equations of the first kind special methods need to be applied, since these problems are ill-posed. If in equation (1) is one of the eigen values of the kernel , then the problem of finding a solution of (1) is ill-posed and requires special methods (see Ill-posed problems).

Non-linear equations of the second kind are usually solved approximately by an iterative method (see ).

The Galerkin method for obtaining approximate solutions of linear and non-linear equations is also used.

Similar methods can also be applied for obtaining approximate solutions of multi-dimensional Fredholm integral equations of the second kind. However, their numerical implementation is more complicated. See  for cubature formulas for the approximate computation of multiple integrals and their error estimates. A Monte-Carlo method of approximate numerical computation of multiple integrals is discussed in .