Following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential equation for FEM simulation.

$(\epsilon_{\parallel} - \epsilon_{\perp}) E^2 sin(u) cos(u) + (k_{3} - k_{1})sin(u)cos(u) (\frac{d u}{d z})^2 + ( k_{1} cos^2(u) + k_{3} sin^2(u) ) \frac{d^2 u}{dz^2} = 0$

where $\epsilon_{\parallel}$, $\epsilon_{\perp}$, $k_{1}$ and $k_{2}$ are constants and E is the input that is to be given in.

Can anyone tell what would be the correct weak form of the equation?

The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential equation for FEM simulation. $$ \begin{split} (\epsilon_{\parallel} - \epsilon_{\perp}) E^2 \sin(u) \cos(u) &+ (k_{3} - k_{1})\sin(u)\cos(u) \Big(\frac{d u}{d z}\Big)^2\\ &+ \big( k_{1} \cos^2(u) + k_{3} \sin^2(u) \big) \frac{d^2 u}{dz^2} = 0 \end{split} $$

where $\epsilon_{\parallel}$, $\epsilon_{\perp}$, $k_{1}$ and $k_{2}$ are constants and $E$ is the input parameter to be set.

Can anyone tell what would be a correct weak form of this equation?