Problem with variational form

Hi, I have a PDE (L_{loc} = 1)

\frac{\partial P}{\partial L}(a,L) = \frac{1}{L_{loc}}\bigg((a^2 - 1)\frac{\partial^2}{\partial a^2} - a\frac{\partial}{\partial a} +1 \bigg)P(L,a)

I approximate via \frac{\partial P}{\partial L} \approx = \frac{P^{n+1} - P^n}{\Delta L} to obtain

P^{n+1} - \frac{\Delta L}{L_{loc}}\bigg((a^2 - 1)\frac{\partial^2 P^{n+1}}{\partial a^2} - a\frac{\partial P^{n+1}}{\partial a} \bigg) - P^{n} = 0

then I have the bilinear form

a(P,v) = \int_{\Omega}Pv(a) - \frac{\Delta L}{L_{loc}}\bigg((a^2 - 1)\frac{\partial^2P}{\partial a^2} - a\frac{\partial P}{\partial a} + 1 \bigg)v(a)\,da,
L_{n+1}(v) = \int_{\Omega}P^nv(a)\,da

My problem is trying to reduce the second derivatives, I am always left with extra terms which don’t allow me to get the weak form…

Any help on how to get around this much appreciated.

To better understand your issue, what terms do you end up with when doing integrations by parts?
Please write out your derviation, and set L_{loc} to 1.
Couldn’t his problem be written as
\frac{\partial P}{\partial t}=(x^2-1) \frac{\partial^2P}{\partial x^2} -x\frac{\partial P}{\partial x} + P ?