Hi, I am new to fenics and I need to solve a challenging non-linear system of PDEs. The system has functions A,B and u which are all functions of x and t. It looks like:
\begin{equation}
-(A^{-1} \sqrt{AB}x^2 u_t)_t + (B^{-1}\sqrt{AB}x^2 u_x)_x + \frac{dV(u)}{du} \sqrt{AB}x^2 = 0
\end{equation}
\begin{equation}
A_x = A\frac{B - 1}{x} + x(A u_x^2 - A B u^2 + B u_t^2)
\end{equation}
\begin{equation}
B_x = -B\frac{B - 1}{x} + x B (\frac{B}{A} u_t^2 + u_x^2 + B V(u))
\end{equation}
where $V(u) = (1-exp(-u))^2$.
In the end I would like to solve the system on a square mesh and have the following boundary conditions: $u(t,x=0) = 0.1$, $u(t,x=xmax) = 0$, $u(t=0,x) = u(t=tmax)$, $B(t,x=0) = 1$, $A(t,x=0) = 0$.
I would greatly appreciate if someone would help me getting started with this problem.
