An integral equation involves an unknown function under the integral sign. Most common of them is a linear integral equation

where α, k, f are given functions. The function ty(t) is to be solved.

Any linear integral equation is equivalent to a linear differential equation; e.g. the equation y(t)+∫ ₀^t(2t− 2x− 3) y(x) dx = 1 + t− 4 sin t to the equation y′′(t) − 3y′(t) + 2y(t) = 4 sin t with the initial conditions y(0) = 1 and y′(0) = 0.

The equation (1) is of

• 1st kind if α(t) ≡ 0,
• 2nd kind if α(t) is a nonzero constant,
• 3rd kind else.

If both limits of integration in (1) are constant, (1) is a Fredholm equation, if one limit is variable, one has a Volterra equation. In the case that f(t) ≡ 0, the linear integral equation is homogeneous.

Example. Solve the Volterra equation y(t)+∫ ₀^t(t−x) y(x) dx = 1 by using Laplace transform.

Using the convolution, the equation may be written y(t) + t ∗ y(t) = 1. Applying to this the Laplace transform, one obtains Y (s) + Y (s) = , whence Y (s) = . This corresponds the function y(t) = cos t, which is the solution.

Solutions on some integral equations in EqWorld.