integral equationIntegralEquation2008-05-17 16:59:472008-09-21 14:12:07Definitionequationlinear integral equationVolterra equation% this is the default PlanetMath preamble. as your knowledge
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An {\em integral equation} involves an unknown function under the \PMlinkescapetext{integral sign}.\, Most common of them is a {\em linear integral equation}
\begin{align}
\alpha(t)\,y(t)+\!\int_a^bk(t,\,x)\,y(x)\,dx = f(t),
\end{align}
where $\alpha,\,k,\,f$ are given functions.\, The function\, $t \mapsto y(t)$\, is to be solved.
Any linear integral equation is \PMlinkname{equivalent}{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\, $y''(t)-3y'(t)+2y(t) = 4\sin{t}$\, with the initial conditions \,$y(0) = 1$\, and\, $y'(0) = 0$.\\
The equation (1) is of
\begin{itemize}
\item {\em 1st kind} if\, $\alpha(t) \equiv 0$,
\item {\em 2nd kind} if $\alpha(t)$ is a nonzero constant,
\item {\em 3rd kind} else.
\end{itemize}
If both \PMlinkname{limits}{UpperLimit} of integration in (1) are constant, (1) is a {\em Fredholm equation}, if one limit is variable, one has a {\em Volterra equation}.\, In the case that\, $f(t) \equiv 0$,\, the linear integral equation is \PMlinkescapetext{{\em homogeneous}}.\\
\textbf{Example.}\, Solve the Volterra equation\, $\displaystyle y(t)\!+\!\int_0^t(t\!-\!x)\,y(x)\,dx = 1$\, by using Laplace transform.
Using the \PMlinkname{convolution}{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.\\
\PMlinkexternal{Solutions on some integral equations}{http://eqworld.ipmnet.ru/en/solutions/ie.htm} in EqWorld.