I read a solution of an integral inequality. The solution uses condition $$f(1)=f(0)=f'(0)=0$$ to derive that $$f(x)=\int_0^1k(x,y)f'''(y)dy$$, $$k(x,y)=\begin{cases}-\frac{x^2(1-y)}{2} & x\leq y\leq 1\\\frac{1}{2}(1-x^2)y^2+(x^2-x)y & 0\leq y<x\end{cases} $$

It seems that it designs a kernel to deal with the problem.

However, I cannot recognize what technique it applied to construct this kernel.

Any help is appreciated.