Problem with variational form

variational formulation
1

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.

2

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 ?