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

(1)

Any linear integral equation is equivalent to a linear differential equation; e.g. the equation $\displaystyle y(t)\!+\!\int_0^t(2t-2x-3)\,y(x)\,dx = 1+t-4\sin{t}$ 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 $\alpha(t) \equiv 0$ ,
• 2nd kind if $\alpha(t)$ 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 $f(t) \equiv 0$ , the linear integral equation is homogeneous.

Example. Solve the Volterra equation $\displaystyle y(t)\!+\!\int_0^t(t\!-\!x)\,y(x)\,dx = 1$ 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 $\displaystyle Y(s)+\frac{1}{s^2}Y(s) = \frac{1}{s}$ , whence $\displaystyle Y(s) = \frac{s}{s^2+1}$ . This corresponds the function $y(t) = \cos{t}$ , which is the solution.

Solutions on some integral equations in EqWorld.