Riemann–Liouville fractional integral equation

Example

Let $b > 0$ and $0 < \alpha < 1$. The Riemann–Liouville integral of order $\alpha$ of a continuous function $\varphi$ on $[0, b]$ is $$(I^\alpha \varphi)(x) = \frac{1}{\Gamma(\alpha)} \int_0^x (x - t)^{\alpha - 1} \varphi(t) \, dt,$$ where $\Gamma$ is the Gamma function. The Riemann–Liouville fractional integral equation of order $\alpha$ is $I^\alpha \varphi = f$ for a given $f$ on $[0, b]$. It is a Volterra equation of the first kind of convolution type, with the weakly singular kernel $k(u) = u^{\alpha - 1} / \Gamma(\alpha)$.

Check on power functions: let $\beta > -1$. The substitution $t = xu$ gives $$I^\alpha (t^\beta)(x) = \frac{x^{\alpha + \beta}}{\Gamma(\alpha)} \int_0^1 (1 - u)^{\alpha - 1} u^\beta \, du = \frac{x^{\alpha + \beta} B(\alpha, \beta + 1)}{\Gamma(\alpha)} = \frac{\Gamma(\beta + 1)}{\Gamma(\alpha + \beta + 1)} \, x^{\alpha + \beta},$$ using the Beta function. So for $\mu > \alpha - 1$, the equation $I^\alpha \varphi = x^\mu$ is solved by $$\varphi(t) = \frac{\Gamma(\mu + 1)}{\Gamma(\mu + 1 - \alpha)} \, t^{\mu - \alpha}.$$ For instance, $\alpha = \tfrac12$ and $\mu = 1$ give $I^{1/2} \varphi = x$ with $\varphi(t) = \frac{\Gamma(2)}{\Gamma(3/2)} t^{1/2} = \frac{2}{\sqrt{\pi}} \sqrt{t}$.