Category:Examples of Integral Equations

This category contains examples of Abel's Integral Equation.

Abel's integral equation is an integral equation whose purpose is to solve Abel's mechanical problem, which finds how long it will take a bead to slide down a wire.

The purpose of Abel's integral equation is to find the shape of the curve into which the wire is bent in order to yield that result:

Let $\map T y$ be a function which specifies the total time of descent for a given starting height.

$\ds \map T {y_0} = \int_{y \mathop = y_0}^{y \mathop = 0} \rd t = \frac 1 {\sqrt {2 g} } \int_0^{y_0} \frac 1 {\sqrt {y_0 - y} } \frac {\d s} {\d y} \rd y$

where:

$y$ is the height of the bead at time $t$

$y_0$ is the height from which the bead is released

$g$ is Acceleration Due to Gravity

$\map s y$ is the distance along the curve as a function of height.