Summary: We give details of the numerical analysis of an integral equation of the form \[ y(t)=f(t)+\int^t_0\frac{s^{\mu-1}}{t^\mu}y(s)ds\;\mu>0. \tag{1} \] The distinctive feature of the equation is the presence of a singularity at \(t=0\) for all values of \(\mu >0\) and at \(s=0\) for all values of \(t>0\) for \(0<\mu<1\). This means that conventional analytical and numerical theory does not apply. In fact, for \(0<\mu <1\), the equation has an infinite family of solutions.
As background, we give details of the analytical results on existence and uniqueness of solutions to (1) and we give new results on the use of numerical schemes that will yield approximations to any specific solution. We conclude with some numerical results that show that our methods enable us to find approximations of any chosen order to any of the infinite class of exact solutions to (1).