# PlanetPhysics/Integral Equation

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

where are given functions.\, The function\, \, is to be solved.

Any linear integral equation is equivalent to a linear differential equation; e.g. the equation\, \, to the equation\, \, with the initial conditions \,\, and\, .\\

The equation (1) is of

*1st kind*if\, ,*2nd kind*if 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\, ,\, the linear integral equation is *homogeneous* .\\

**Example.** \, Solve the Volterra equation\, \, by using Laplace transform.

Using the convolution, the equation may be written\, .\, Applying to this the Laplace transform, one obtains\, ,\, whence\, .\, This corresponds the function \,,\, which is the solution.\\

Solutions on some integral equations in EqWorld.