Abel Integral Equation

An Abel integral equation of the first kind is a Volterra integral equation of the first kind with a power-law integral kernel,

(1)

where , is given, and is unknown. The integral is times the left-sided fractional integral of order , where is the gamma function.

For an integrable solution on a finite interval , the inversion formula recovers almost everywhere as

(2)

The two fractional integrations have orders and , so their composition is ordinary integration. Differentiating and applying the reflection relation for the gamma function gives the coefficient in the inverse (Gorenflo and Vessella 1991).

In the tautochrone problem, let be the arc length from the bottom to height along a monotone branch of a frictionless track, and let be the constant gravitational acceleration. Conservation of energy gives the descent time from rest at height as

(3)

This is an Abel integral equation with . A constant descent time gives proportional to (Podlubny et al. 2017).