# integral equation

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

 $\displaystyle\alpha(t)\,y(t)+\!\int_{a}^{b}k(t,\,x)\,y(x)\,dx=f(t),$ (1)

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

Any linear integral equation is equivalent (http://planetmath.org/Equivalent3) 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  $y^{\prime\prime}(t)-3y^{\prime}(t)+2y(t)=4\sin{t}$  with the initial conditions  $y(0)=1$  and  $y^{\prime}(0)=0$.

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 (http://planetmath.org/UpperLimit) 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 .

Example.  Solve the Volterra equation  $\displaystyle y(t)\!+\!\int_{0}^{t}(t\!-\!x)\,y(x)\,dx=1$  by using Laplace transform  .

Using the convolution (http://planetmath.org/LaplaceTransformOfConvolution), 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.

http://eqworld.ipmnet.ru/en/solutions/ie.htmSolutions on some integral equations in EqWorld.

Title integral equation IntegralEquation 2013-03-22 18:03:57 2013-03-22 18:03:57 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 45D05 msc 45A05 Equation linear integral equation Volterra equation