I'm trying to model diffusion of a chemical species X inside a Y-shaped tube, whose diameter (thickness) is constant everywhere. The diffusion constant of X is $D$ ($\mu$m$^2$/s), so the concentration of X, denoted by $u$, follows the diffusion equation $$\frac{\partial u}{\partial t}=D\frac{\partial^2 u}{\partial x^2},$$ except for the branch point in the center. (Here $x$ is a one-dimensional spatial coordinate.)

My question is how to numerically calculate the temporal concentration profile, correctly taking the Y-shaped geometry (especially the branch point) into account. Is there any good (and preferably simple) method for this? I'm using MATLAB for numerical simulation and I know how to numerically calculate diffusion in a one-dimensional tube. Thank you.

Note: The actual equation I'm trying to solve is $$\frac{\partial u}{\partial t}=D\frac{\partial^2 u}{\partial x^2}-k(u-u_0),$$ where $k$ and $u_0$ are positive constants. i.e. the chemical species X can "leak" out of the tube when $u$ is higher than the basal value $u_0$.

Note 2: The diameter of the tube is approximately 0.1 $\mu$m (or less), whereas the length of each branching tube is approximately 10 $\mu$m. The concentration of X at the far ends of the branching tubes are kept at $u_0$ (Dirichlet boundary condition). Because the tube is thin, I expect that diffusion is mostly one-dimensional except for the branch point.


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