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.

why is it a 3rd order operator? – mookid Mar 30 '14 at 22:43 — mookid, Mar 30 '14 at 22:43
Because I misquote it :P – Desperate Fluffy Mar 30 '14 at 23:03 — Desperate Fluffy, Mar 30 '14 at 23:03

score 0 · Answer 1 · answered Mar 30 '14 at 23:05

0

Hint:

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

answered Mar 30 '14 at 23:05

mookid

28,236

Why? Where does that come from? – Desperate Fluffy Mar 30 '14 at 23:07
this is the definition of the weak derivative (similar to an integration by parts.) – mookid Mar 30 '14 at 23:08
Ok, then 2nd question: Zauderer describes the divergence form as (1)=$\sum_{i=1}^n \frac{\partial P_i}{\partial x_i}$ where $P_i=\sum_{j=1}^n [a_{ij}w\frac{\partial u}{\partial x_j}-u\frac{\partial (a_{ij}w}{\partial x_j}]+b_iuw$. How does the weak derivative fit into that? – Desperate Fluffy Mar 30 '14 at 23:14
I have no idea... – mookid Mar 30 '14 at 23:18

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.

1 Answers1

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.