I want to solve a diffusion-convection problem numerically. I start from a PDE of the form $$\partial_t f(x,t) = \partial_x(K(x)\;f(x,t))+\partial_{xx}\;f(x,t)^{m}\;.$$ It is possible to calculate a stationary solution $f_{st}(x)$, which looks something like this $$ f_{st}(x) = (a \cosh(x-b))^c\;. $$ Now I try to simplify the original PDE be assuming that the solution is similar to the stationary one, the only difference being that the coefficients $a$ and $b$ are now time dependent, i.e. $a(t)$ and $b(t)$. This allows me to calculate the RHS of the PDE, reducing it formally to a first order ODE in time. My problem is that I'm not sure how to feed it to the computer, because due to the chain rule the spatial derivatives 'extract' additional instances of $a(t)$ and $b(t)$ out of $f$. In other words, I end up with a ODE like $$ \partial_t(g(t)\;h(u(t)) = f(g(t),u(t)) $$ Intuitively I'd expect there to be a problem because I have two unknowns and just one equation but I'm trying to retrace a paper that seems to be doing just that. Anyways, thanks for reading, hope you can understand what I mean.


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