Is there a method for solving the following system of generalized Abel's integral equation:?

$(x^2 -1)\int_0^x \frac{u(t)}{(x-t)^{\frac{1}{2}}}\; dt + x\int_0^x \frac{v(t)}{(x-t)^{\frac{1}{3}}}\; dt =g_1 (x),\\ x^3 \int_0^x \frac{u(t)}{(x-t)^{\frac{1}{4}}}\; dt + (1-x)\int_0^x \frac{v(t)}{(x-t)^{\frac{1}{5}}}\; dt =g_2 (x),$

where

$\begin{cases} g_1 (x)&=\frac{16}{15}x^{\frac{9}{2}}-\frac{16}{15}x^{\frac{5}{2}}+\frac{27}{40}x^{\frac{11}{3}}+\frac{243}{440}x^{\frac{14}{3}}, \\
g_2 (x)&= \frac{128}{231}x^{\frac{23}{4}}+\frac{125}{252}x^{\frac{14}{5}}-\frac{125}{1197}x^{\frac{19}{5}}-\frac{625}{1596}x^{\frac{24}{5}}\end{cases}$

with $0\leq x\leq 1$?