How to find the Green's function $G(t,x)$ for the BVP consisting of the equation :
$$u'''(t)=0 , \quad t\in (0,1)$$
and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < p<q<1$ are constants .
they say that $$G(t,s)=-t(p-s)\chi_{[0,p]}(s)+\frac{(t-s)^2}{2}W(s)\chi_{[0,t]}(s)+\frac{t(2p-t)}{2}W(s)\chi_{[q,1]}(s)+\frac{t(2p-t)}{2}\chi_{[0,q]}(s)$$
Where $W(s)=(\int_q^1w(v)dv)^{-1}\int_s^1 w(v)dv , s\in [q,1]$
but how to prove it ? Please,
Thank you