Abel’s Integral Equation

Abstract

The following Volterra integral equation of the first kind is due to Abel (1823):

$$g(x) = \int\limits_a^x {\frac{{f(y)}} {{\sqrt {x - y} }}dy\;for\;x \geqslant a}$$

(6.1.1)

. Since the denominator $\sqrt {x - y}$ has a zero at y=x, the integral in (1) is to be understood in the improper sense (cf. §6.1.3) and Abel’s integral equation is an example of a weakly singular equation.