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%%% Primary Title: integral equation
%%% Primary Category Code: 02.30.Rz
%%% Filename: IntegralEquation.tex
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\begin{document}
An {\em integral equation} involves an unknown \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} under the 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 equivalent to a linear \htmladdnormallink{differential equation}{http://planetphysics.us/encyclopedia/DifferentialEquations.html}; 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 limits 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 {\em homogeneous}.\\
\textbf{Example.}\, Solve the Volterra equation\, $\displaystyle y(t)\!+\!\int_0^t(t\!-\!x)\,y(x)\,dx = 1$\, by using \htmladdnormallink{Laplace transform}{http://planetphysics.us/encyclopedia/2DLT.html}.
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.\\
\htmladdnormallink{Solutions on some integral equations}{http://eqworld.ipmnet.ru/en/solutions/ie.htm} in EqWorld.
\end{document}
