How can solve the following special integral equation:
$$\int_0^x \frac{x^2 t^3 + t^4 +1}{(x-t)^{\frac{1}{4}}}u(t)\; dt=\frac{128}{231}x^{\frac{11}{4}}+\frac{32768}{100947}x^{\frac{31}{4}}+\frac{262144}{908523}x^{\frac{27}{4}}$$ with $0\leq x\leq 1$?
Which method is suitable for solving it?
I suppose you don't want one to 'show' the solution $u$, but if you would like to know what bibliography has to offer, consider this article on AMS. It mentions that Volterra considers
$$\phi(x) - \phi(a) = \int_a^x\,\frac{G(x,t)}{{(x-t)}^\lambda}\,u(t)\,dt,$$
where $\lambda<1$, $G(x,t)$ and $\partial G/\partial x$ are continuous in the triangular region $a \leq t \leq x \leq b$ and $G(x,x)\neq 0$. It says
Hence, with $a=0$, $b=1$, $\lambda = 1/4$, $G(x,t) = x^2t^3+t^4+1$ and
$$\phi(x) = \frac{128}{231}x^{\frac{11}{4}}+\frac{32768}{100947}x^{\frac{31}{4}}+\frac{262144}{908523}x^{\frac{27}{4}},$$
your equation has a unique continuous solution $u$.
The reference cited is