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I am interested in if there is an analytic solution to the following set of coupled reaction-diffusion equations in polar coordinates (although there is no $\theta$ dependency) $$\frac{\partial f(r,t)}{\partial t}=D_f\nabla^2f(r,t)-kf(r,t)$$ $$\frac{\partial g(r,t)}{\partial t}=D_g\nabla^2g(r,t)+kf(r,t)$$ where the Laplacian in polar coordinates is given by $$\nabla^2=\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}$$

and where $f$ is initially given by a Gaussian, $g$ is initially $0$ over all space, and both functions go to $0$ as $r\to\infty$: $$f(r,0)=\frac{F}{2\pi\sigma^2}\exp\left(-\frac{r^2}{2\sigma^2}\right)$$ $$g(r,0)=0$$ $$f(\infty,t)=g(\infty,0)=0$$ $$\left.\frac{\partial f}{\partial r}\right|_{r=0}=\left.\frac{\partial g}{\partial r}\right|_{r=0}=0$$ The boundary condition at $r=0$ is because the system exhibits radial symmetry here. $D_f$, $D_g$, $k$, $F$, and $\sigma$ are all constants.

I know that the first equation can be solved on its own as $$f(r,t)=\frac{F}{4\pi D_f\,t+2\pi\sigma^2}\cdot\exp\left(-\frac{r^2}{4D_f\,t+2\sigma^2}-kt\right)$$ however, I cannot seem to get a solution for the second equation with this $f(r,t)$ substituted into it. I have tried using Fourier Transforms as well as Green's functions (which are probably trying to do the same thing anyway).

Is there an analytic solution to this set of equations? If not, is there a different $f(r,0)$ that starts at a maximum and drops to $0$ monotonically that would allow for an analytic solution? (I picked the Gaussian because those usually behave nicely with diffusion).

Also if there is not an analytic solution, would there still be a way to determine a sort of "effective diffusion constant" for $g$ that depends on the constant parameters?

Also also, this system is fairly easy to move between polar and rectangular coordinates for, so if solutions are easier to work out in that way then that is fine.

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Once you have solved the equation for $f$, the second one should be easy. I assume you found $f$ by computing the Green's $G_f$ function for the linear operator $D_f\nabla^2-k$, so you can also find the Green's function $G_g$ for the linear operator $D_g\nabla^2$. Then, the solution can be found using Duhamel's formula: \begin{align} g(t)=G_g(t)*g(0)+k\int_0^tG_g(t-s)*f(r,s)ds \end{align}

Of course, I cannot guarantee that you can explicitly calculate the second integral, but this is as good as it gets.

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  • $\begingroup$ Yeah, I was asking if there is a way to explicitly calculate the integral. From what I can tell I don't think so. Thanks though. $\endgroup$ Oct 9, 2019 at 11:55

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