In their simplest form, integral equations are equations in
one variable (say t) that involve an integral over a domain of another variable
(s) of the product of a kernel function K(s,t) and
another (unknown) function (f(s)). The purpose of the numerical solution is to
determine the unknown function f. If the limit(s) on the integration domain are
fixed then it is said to be a Fredholm Equation. If
the limit(s) on the integration domain are not fixed then it is said to be a Volterra Equation.
Integral equations are solved
by replacing the integral by a numerical integration or
quadrature formula. The integral equation is then reduced to a linear
equation with the values of f at the quadrature
points being unknown at the outset. The solution of the linear equation(s)
gives the approximate values of f at the quadrature
points.
Fredholm Equations
Fredholm equations occur in two forms: Fredholm equations of the first kind and Fredholm equations of the second kind. There is a close
analogy between Fredholm equations and linear systems of equations; the functions can be viewed
as vectors, the integration over the kernel function as a matrix. The solution
generally involves replacing the integral equation by a linear system and then
solving it. Clearly the accuracy of this method depends partially on the
accuracy of the numerical integration method; the more quadrature points the
better.
Fredholm Equations of the First Kind have the form

where
only the function f is unknown at the outset. Usually an approximation to f(t) for a £
t £ b is required.
On applying an integral
equation method, such an equation is replaced by a linear system of equations

where
K is a matrix and f and g are vectors.
Unfortunately, such equations
have poor numerical properties if the kernel function is smooth at s = t. The
matrix is often severely illconditioned; solution is hopeless.
In the special cases where K(s,t) is nonsmooth at s = t, for
example a singular function (e.g. K(s,t) = [1/(
st)]), then the condition of the matrix is better and solution is possible.
Fredholm Equations of the Second
Kind have the form

where
only the function f is unknown at the outset. Usually an approximation to f(t) for a £
t £ b is required.
On applying an integral
equation method, such an equation is replaced by a linear system of equations

where
K is a matrix and f and g are vectors.
Solution of Fredholm Integral Equations by Collocation :: Excel Spreadsheet solution of example problem
Fredholm integral equations formulate a range
of physical problems. Elliptic partial diffential
equations like the Laplace or Helmholtz equations can be reformulated as Fredholm integral equations and are solved as part of the boundary element method.
Volterra Equations
A Volterra
equation has the form

usually
we need to find f(t) in a domain [a,b].
Most methods for solving volterra equations divide the range into discrete time
windows with time points t_{0} = a, t_{1},...,
t_{n} = b. Starting with f(t_{0}) =
g(t_{0}), by using some integration method, f(t_{1}), f(t_{2}),...
can be found in turn.