PlanetPhysics/Integral Equation

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

${\begin{matrix}\alpha (t)\,y(t)+\!\int _{a}^{b}k(t,\,x)\,y(x)\,dx=f(t),\end{matrix}}$

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\, $y(t)\!+\!\int _{0}^{t}(2t-2x-3)\,y(x)\,dx=1+t-4\sin {t}$\, 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\, $y(t)\!+\!\int _{0}^{t}(t\!-\!x)\,y(x)\,dx=1$\, 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\, $Y(s)+{\frac {1}{s^{2}}}Y(s)={\frac {1}{s}}$,\, whence\, $Y(s)={\frac {s}{s^{2}+1}}$.\, This corresponds the function \,$y(t)=\cos {t}$,\, which is the solution.\\

Solutions on some integral equations in EqWorld.