Numerical inversion of Abel's equation with unbounded right-hand side (Q1912458)

On the closed interval \([a,b]\) of the real line we consider Abel's integral equation \[ \int^x_a {\varphi(t)dt\over (x- t)^\alpha}= {g(x)\over (x- a)^{\gamma_a}(b- x)^{\gamma_b}},\quad 0< \alpha< 1, \max(\gamma_a, \gamma_b)< \alpha, \min (\gamma_a, \gamma_b)> 0,\tag{1} \] where \(g(x)\) is a given real-valued function satisfying a Lipschitz condition: \(|g(x)- g(t)|\leq A|x- t|^\mu\), \(x, t\in [a, b]\), and \[ \mu> 1- \alpha.\tag{2} \] We obtain an inversion formula for equation (1) that does not involve differentiation. We show that under a certain strengthening of the requirement (2) the resulting formula assumes a form convenient for numerical implementation.