I'm analyzing a 1D problem in cylindrical geometry where the quantity of interest U(r,t) satisfies the convection-diffusion equation:
d/dt U(r,t) = divergence( F(r,t) ) + S(r,t) where F(r,t) = c(r)*grad U(r,t) - d(r)*U(r,t) is the radial flux of U(r,t) and S(r,t) is the source
all with a nice and smooth initial condition U0(r) and known c(r), d(r) and S(r,t)
On axis (r=0) the natural boundary condition is given by F=0 (i.e. Flux=0)
I can solve the equation numerically if I use either a prescribed value for U(r=1,t)=UBnd(t) or if I give the flux through the outer boundary F(r=1,t)=FluxBnd(t).
The problem is that at the moment I only know that in steady state the flux through the outer boundary must match the steady state integrated source (otherwise it would not be a steady state situation).
In principle I only need to solve the equation near the steady state as the source is of the form S(r,t)=SS(r)+epsilon*SN(r,t) i.e. it is perturbed to produce perturbed U. However, even is this case I do not know what should I use for FluxBnd(t). Physically the perturbation in S(r,t) would show up delayed in FluxBnd(t) but I have no idea what is this delay. I still only know that <FluxBnd(t)> should match the average of the volume integrated source.
Can anybody tell me useful references for solving this type of problems or first of all point me in the right direction in defining a problem that can be solved. Would Fourier treatment for time help me or not with the boundary issue?