Projection of two-dimensional diffusion in a narrow channel onto the longitudinal dimension

Diffusion in a narrow two-dimensional channel of width A(x), depending on the longitudinal coordinate x, is the object of our study. We show how the 2+1 dimensional diffusion equation can be projected onto a 1+1 dimensional one, governing corresponding one-dimensional density, in a steady-state approximation. Then we demonstrate the method on a nontrivial exactly solvable case for A(x)=x and discuss projection of the initial condition.