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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  $y(0) = 1$  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.



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Cross-references: Laplace transform, differential equation, function

This is version 1 of integral equation, born on 2009-04-17.
Object id is 642, canonical name is IntegralEquation.
Accessed 253 times total.

Classification:
Physics Classification: 02.30.Rz (Integral equations)
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