Green function of a fourth order ode

Question

please what is the method to find the Green function associated to the following fourth order ode $$ \begin{cases} u^{(4)}(t)+\beta\, u^{(2)}(t)-\alpha\,u(t) =0,\,\, t\in[0,1]\\ u(0)=u(1)=u''(0)=u''(1)=0 \end{cases} $$

Answer 1

The Green function $G(t,s)$ solves for fixed $s$ the system for $u(t)=G(t,s)$ \begin{cases} u^{(4)}(t)+\beta\, u^{(2)}(t)-\alpha\,u(t) =\delta(t-s),\,\, t\in[0,1]\\ u(0)=u(1)=u''(0)=u''(1)=0 \end{cases} so that $u(t)=\int_0^1G(t,s)f(s)ds$ is the general solution if the right side is $f(t)$. This means that $$ G(t,s)=\begin{cases} u_L(t),&t<s,\\u_R(t),&t\ge s, \end{cases} $$ where

• $u_L, u_R$ are solutions of the homogeneous equation,
• $u_L$ satisfies the left boundary conditions,
• $u_R$ the right boundary conditions,
• $u_L(s)=u_R(s)$, $u_L'(s)=u_R'(s)$, $u_L''(s)=u_R''(s)$ and
• $u_L'''(s)+1=u_R'''(s)$ so that the 4th derivative produces the delta term.