# Variational formulation and boundary condition

Hi,
We consider in 2D domain : [0,L] x[0,H], the system of equations:
dV/dt = D* Delta V - dV/dx - V
dU/dt= V - alpha* U
with boundary conditions
x=0,L: dV/dx = 0, y=0: dV/dy=- U + V; y=H: dV/dy=0
and initial condition V(x,y,0)= V0

Question 1: this system is no linear. So to resolve it, we have to use arf and matrix? Or it is sufficient to use the variational formulation: problem ([U,V],[UH,VH])?

like this:

problem MainSys([V,VH],[U,UH],solver=LU)= int2d(Th)(V*VH/dt)-int2d(Th)(oldV*VH/dt)+int2d(Th)(U*UH/dt)-int2d(Th)(oldU*UH/dt)
+int2d(Th)(D*(dx(V)*dx(VH)+dy(V)*dy(VH)))
+int2d(Th)(dx(V)*Vh)+int2d(Th)(V*VH)
+int2d(Th)(alpha*U*UH)-int2d(Th)(V*UH);
-int1d(Th,1)(D*V*VH)+int1d(Th,1)(D*U*UH);


is it correct?
Question 2: for boundary condition, we write it using \int 1d(Th,… or we have to use on(…?

Regards

1. remove the kind of solver=LU by default the solver is better than LU

warning: V,VH are unknown and V,VH are test function but you want
V,U unknot and VH,UH test fonction …

the formulation can be : ( not tested )
problem MainSys([V,U],[VH,UH])=
int2d(Th)(VVH/dt)-int2d(Th)(oldVVH/dt)
+int2d(Th)(UUH/dt)-int2d(Th)(oldUUH/dt)
+int2d(Th)(D*(dx(V)dx(VH)+dy(V)dy(VH)))
+int2d(Th)(dx(V)Vh)+int2d(Th)(VVH)
+int2d(Th)(alpha
U
UH)-int2d(Th)(VUH)
-int1d(Th,1)(D
VVH)
+int1d(Th,1)(D
U*UH);