integral equation
An integral equation^{} involves an unknown function under the . Most common of them is a linear integral equation
$\alpha (t)y(t)+{\displaystyle {\int}_{a}^{b}}k(t,x)y(x)\mathit{d}x=f(t),$  (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 $y(t)+{\displaystyle {\int}_{0}^{t}}(2t2x3)y(x)\mathit{d}x=1+t4\mathrm{sin}t$ to the equation ${y}^{\prime \prime}(t)3{y}^{\prime}(t)+2y(t)=4\mathrm{sin}t$ with the initial conditions^{} $y(0)=1$ and ${y}^{\prime}(0)=0$.
The equation (1) is of

•
1st kind if $\alpha (t)\equiv 0$,

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2nd kind if $\alpha (t)$ is a nonzero constant,

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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 $y(t)+{\displaystyle {\int}_{0}^{t}}(tx)y(x)\mathit{d}x=1$ 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 $Y(s)+{\displaystyle \frac{1}{{s}^{2}}}Y(s)={\displaystyle \frac{1}{s}}$, whence $Y(s)={\displaystyle \frac{s}{{s}^{2}+1}}$. This corresponds the function $y(t)=\mathrm{cos}t$, which is the solution.
http://eqworld.ipmnet.ru/en/solutions/ie.htmSolutions on some integral equations in EqWorld.
Title  integral equation 

Canonical name  IntegralEquation 
Date of creation  20130322 18:03:57 
Last modified on  20130322 18:03:57 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  8 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 45D05 
Classification  msc 45A05 
Related topic  Equation 
Defines  linear integral equation 
Defines  Volterra equation 