Largely following \href{https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0117949}{Exact Solutions of Linear Reaction-Diffusion Processes on a Uniformly Growing Domain: Criteria for Successful Colonization}, M J Simpson, PLoS One (2015).
\section{Kinematics}
We consider a linear reaction-diffusion process on a growing disk labelled by points $\vect{r}=(r, \varphi)$ with $0 < r < R(t)$ and $0\leq\varphi < 2\pi$ where $R(t)$ is the increasing radius of the domain. Domain growth is associated with a \emph{radial} velocity field $\vect{v}= v(r,t)\,\uvec{r}$ which causes points on a circle of radius $r$ to move to a circle of radius $r + v(r,t)\,\tau$ in a short time $\tau$. By considering the expansion of an element of initial size $\Delta r$, we can derive an expression relating $R(t)$ and $v(r,t)$ which can be written as
\eqn{
\frac{dR(t)}{dt} = \int_0^{R(t)}\frac{\partial v}{\partial r} dr
.
\label{eq:growing_radius}
}
We consider uniform growth, i.e., $\partial_r v$ is independent of $r$ but potentially depends on time $t$, so that we have $\partial_r v =\sigma(t)$. Combining this with \eqref{eq:growing_radius} gives
Assuming that the circular domain elongates with its center fixed, i.e., $v(0,t)=0$, integrating \eqref{eq:growth_rate} gives
\eqn{
v(r,t) = \frac{r}{R(t)}\frac{dR(t)}{dt}.
}
\section{Mass conservation}
We now consider the conservation of a mass density $C(\vect{r}, t)$, assuming that it evolves according to a linear reaction-diffusion mechanism. The associated conservation statement on the growing domain can be written as
\eqn{
\partial_t C = D \nabla^2 C - \nabla\cdot (\vect{v} C) + k \, C,
\label{eq:general_reaction_diffusion}
}
where $D>0$ and $k$ are the diffusion constant and reaction rate respectively.
We assume azimuthal symmetry $C(\vect{r}, t)= C(r, t)$, i.e., the dynamics is independent of $\varphi$. Then
\eqn{
\frac{\partial C}{\partial t} = D \,\frac{1}{r^2}\,\frac{\partial}{\partial r}\left(r^2 \frac{\partial C}{\partial r}\right) - \frac{1}{r^2}\,\frac{\partial (r^2 v C)}{\partial r} + k \, C,
\label{eq:azimuthal_symmetry}
}
and impose zero diffusive flux conditions $\partial_r C =0$ at $r=0$ and at $r=R(t)$. With some given initial conditions $C(r, 0)$ our aim is to find the exact solution of \eqref{eq:azimuthal_symmetry}.
\section{Exact solution}
We first transform the spatial variable to a fixed domain $\xi=\dfrac{r}{R(t)}$. Then
