How can solve the following system of nonlinear generalized Abel's integral equation:
$\begin{cases} u(x)-2v(x)+\int_0^x \frac{u^2 (t)+v^2 (t)}{(x-t)^{\frac{1}{5}}}\; dt=g_1 (x), \\
v(x)-u(x)-\int_0^x \frac{u(t)v(t)}{(x-t)^{\frac{1}{3}}}\; dt=g_2 (x) \end{cases}$
where
$\begin{cases} g_1 (x)&=x^2 -2x^3 +\frac{390625}{1573656}x^{\frac{34}{5}}+\frac{3125}{9576}x^{\frac{24}{5}},\\ g_2 (x)&=x^3 -x^2 -\frac{2187}{5236}x^{\frac{17}{3}} \end{cases}$
with $0\leq x\leq 1$?
The system in question is solved this article: A new operational method to solve Abel’s and generalized Abel’s integral equations. see here