Find adjoint operator $L^*$ for the 3rd order operator $Lu(x,t)=u_t(x,t)-u_x(x,t)+\gamma^2u(x,t)$, with $wLu-uL^*w$ (1) is divergence expression.

Question

I'm working out of the Zauderer PDEs book and am having some trouble on the adjoint operators section (3.6). Specifically this problem:

"Obtain the adjoint operator $L^*$ for the third order operator $L$ given as $Lu(x,t)=u_t(x,t)-u_x(x,t)+\gamma^2u_{xxx}(x,t)$, such that $wLu-uL^*w$ (1) is a divergence expression."

The book and my notes shows how to do first and second order, but I have no idea how to go about third order. I was thinking that since (1) has to be equal to the divergence expression and I have $L$, I could just work backwards, but that's nasty. There has to be a more elegant way.

Answer 1

Hint:

$$ \langle u_{xxx}, v \rangle = (-1)^3 \langle u, v_{xxx} \rangle $$