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 $0\leq x\leq 1$?



A system of generalized Abel's integral equation
The system in question is solved this article: *A new operational method to solve Abel’s and generalized Abel’s integral equations*.[see here](https://www.researchgate.net/publication/319910584_A_new_operational_method_to_solve_Abel's_and_generalized_Abel's_integral_equations)


