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I am seeking the Green's function of a diffusion-reaction type differential equation in cylindrical coordinates. For the sake of the discussion, assume that I am trying to find the Green's function associated with

$$ \frac{\partial u(r,t)}{\partial t} = \frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial u(r,t)}{\partial r} - k H(1-x) $$

with $ 0 \leq x < \infty$ and a reflective inner boundary condition

$$ \frac{\partial u(0,t)}{\partial r} = 0 $$

Here, $k$ is a constant and $H$ is the Heaviside function. Is it possible to explicitly express the solution of the differential equation in terms of the Green's functions without sink term, $G_0(r,r_0,t)$, and the Green's function with a homogeneous sink term (i.e. with $H$ substituted by 1), $G_0 \exp(-k t)$? Furthermore, can the solution for the problem subject to the initial condition $u(r,0) = 1$ and $\lim_{r \to\infty} u(r,t) = 1$ be obtained analytically?

Sorry if these are obvious questions. It has been a while since I have been concerned with this type of problems.

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$\frac{1}{r}\frac{\partial}{\partial t}r\frac{\partial u(r,t)}{\partial r}$ should be $\frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial u(r,t)}{\partial r}$ –  mike Jun 29 at 19:49
    
@mike Good spot! Thank you. –  daniel Jun 30 at 7:48

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