Abel's integral equation

Example

Let $b > 0$ and let $f$ be a given function on $[0, b]$. Abel's integral equation is the equation $$\int_0^x \frac{\varphi(t)}{\sqrt{x - t}} \, dt = f(x), \quad 0 \le x \le b,$$ for an unknown function $\varphi$ on $[0, b]$. It is a Volterra equation of the first kind, of convolution type, with the weakly singular kernel $k(x - t) = (x - t)^{-1/2}$. More generally, Abel's equation with exponent $\alpha \in (0, 1)$ has the kernel $(x - t)^{-\alpha}$.

Check on an instance: take $\varphi = 1$. Then $\int_0^x (x - t)^{-1/2} \, dt = 2 \sqrt{x}$, so $\varphi = 1$ solves the equation with $f(x) = 2\sqrt{x}$. Take $\varphi(t) = t$. The substitution $t = xu$ gives $\int_0^x t (x - t)^{-1/2} \, dt = x^{3/2} \int_0^1 u (1 - u)^{-1/2} \, du = \tfrac43 x^{3/2}$, using the Beta function value $B(2, \tfrac12) = \tfrac43$.