# Conjugated Diffusion Equations

I am new to mathematical-biology and I have to solve the following (diffusion-like) equation $$\begin{eqnarray} \frac{\partial a(x,t)}{\partial t}= D \frac{\partial^2 a(x,t)}{\partial x^2}\\ \frac{\partial b(x,t)}{\partial t}=-v \frac{\partial b(x,t)}{\partial x}-\mu a(x,t) b(x,t) \\ \frac{\partial c(x,t)}{\partial t}= \gamma \frac{\partial^2 c(x,t)}{\partial x^2}+k b(x,t) \end{eqnarray}$$

for the functions a(x,t), b(x,t) and c(x,t), subject to boundary conditions $$a(x,t\rightarrow \infty)=0$$, $$a(x\rightarrow \infty,t)=0$$, $$\frac{\partial c(x,t)}{\partial(x)}|_{x=0}=\frac{\partial a(x,t)}{\partial x}|_{x=0}$$, $$b(L,t)=b_L$$, $$c(0,t)=0$$, and $$\frac{\partial c(x,t)}{\partial x}|_{x=L}=0$$, where $$D, v, \mu,\gamma, k$$ and $$b_l$$ are constants.

I tried to attack this problem using, first, the separation of variables method. Although I do obtain a "simple" answer for $$a(x,t)$$ written in terms of exponentials, I am stuck at the second equation. More precisely, this term with $$-\mu a(x,t) b(x,t)$$ became a nightmare to me. I would NDSolveValue but, so far, only errors pop out from my Mathematica notebook.

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• Post the actual code that you used (Raw InputForm). You should not use D as a variable name since it already has a defined meaning in Mathematica. – Bob Hanlon 2 days ago
• Hi Bob, indeed, in my code I replaced "D" by another constant. pde = {D[aa[x, t],t]==k1*D[aa[x,t],x,x], D[bb[x, t], t]==-k2*D[bb[x,t],x]-k3*aa[x, t]*bb[x, t], D[cc[x, t], t]==k4*D[cc[x,t],x,x]+k5*bb[x, t]}; where k1,k2...k5 are just positive constants (used instead of the greek letters for sake of simplicity). I am then strugling to make the boundary conditions work. I can set a x=xout whete xout would be my "inifity" for the numerical solution – Gui Volp 14 hours ago