PlanetMath: integral equation

(more info)

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integral equation

(Definition)

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

$\displaystyle \alpha(t)\,y(t)+\!\int_a^bk(t,\,x)\,y(x)\,dx = f(t),$

(1)

where $\alpha,\,k,\,f$ are given functions. The function $t \mapsto y(t)$ is to be solved.

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 and .

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 . 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.