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integral equation


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Any equation for an unknown function f(x),

axb

, involving integrals of the function. An equation of the form

f(x) = ∫xaK(x,y) f(y) dy + g(x)

is a Volterra equation of the second kind.

The analogous equation with constant limits

f(x) = ∫baK(x,y) f(y) dy + g(x)

is a Fredholm equation of the second kind. If the required function only appears under the integral sign it is a Volterra or Fredholm equation of the first kind; these are more difficult to treat both theoretically and numerically. The Volterra equation can be regarded as a particular case of the Fredholm equation where

K(x,y) = 0 for y > x

Fredholm equations of the second kind occur commonly in boundary-value problems in mathematical physics. Numerical techniques proceed by replacing the integral with a rule for numerical integration, leading to a set of linear algebraic equations determining approximations to f(x) at a set of points in

axb

.

Subjects: Computing.


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