On polygonal approximation in solving Abel's equation (Q1912434)

On the interval \([a, b]\) of the real line we consider Abel's integral equation \[ \int^x_a {\varphi(t) dt\over (x- t)^\alpha}= g(x),\quad 0< \alpha< 1,\tag{1} \] where \(g(x)\) is a given real-valued function of Hölder class: \[ |g(x)- g(t)|\leq A|x- t|^\mu,\quad \mu> 1- \alpha;\quad x, t\in [a,b]. \] The solution \(\varphi(t)\) is sought in the class of functions that have integrable singularities. It is known that the solution of equation (1) can be written in the form \[ \varphi(x)= {\sin \alpha\pi\over \pi} \Biggl[ {g(x)\over (x- a)^{1- \alpha}}+ (1- \alpha){\mathcal J}(x)\Biggr],\text{ where } {\mathcal J}(x)= \int^x_a {g(x)- g(t)\over (x- t)^{2- \alpha}} dt.\tag{2} \] In the present paper the function \(g(x)\) in formula (2) is replaced by a polygonal line, and the requirement of differentiability is weakened to a Hölder condition. We obtain a formula which can be easily implemented on a computer.