I have these two equation $$ \frac{\partial q(t,x)}{\partial t} = \frac{\partial^2 q(t,x)}{\partial x^2} - \frac{L_1 a(t, x) q(t, x)}{1 + \frac{L_2}{L} (1 - q(t,x))}\\ \frac{\partial a(t,x)}{\partial t} = \frac{\partial^2 a(t,x)}{\partial x^2} - \frac{L_1 a(t, x) q(t, x)}{1 + \frac{L_2}{L} (1 - q(t,x))}\\ $$ with boundary condition $$ q\big|_{t=0} = 0, \quad a\big|_{t=0} = y, \quad q\big|_{t>=0, x-> infinity} = 0, \quad a\big|_{t>=0, x-> infinity} = y, \quad da/dx\big|_{x=0} = 0, \quad q\big|_{x=0} = \frac{1}{1 + \exp(f - g t)}, $$ and $f,g,L_1,L_2,L$ are all constants.
I need to know what is $\frac{\partial q(t,x)}{\partial x}$ at $x=0$.
I think I need to use Crank-Nickolson but don't know how. The info on the website wasn't enough. :(
I'd really appreciate if you help me.
Thanks a lot.