I've been trying to figure out this problem for a while and was wondering what you thought. I have the PDE:

\begin{equation*} \frac{\partial c}{\partial t}+\frac{K}{r^2}\frac{\partial c}{\partial r}+\frac{Da~c}{c+n}=0 \end{equation*}

The initial condition is $c(r,0) = c_{in}$ and boundary condition is $\partial c(R_{in},t) \over \partial r $ $= 0 $, where $R_{in}$ is the radius of an inner sphere (and $R_{out}$ the radius of an outer sphere). $K$ and $Da$ are both constants. $r$ is the radius measured from the centre of the sphere outwards, and $t$ time.

I have attempted to solve this using the method of characteristics but the solution keeps reducing to only depend on time. And even then I don't end up using the second Neumann boundary condition.

Does anyone have any ideas as to where I'm going wrong. Is it just the boundary conditions that are causing the problem?

The following pde is (approximately) the leading order homogenized form of the local mass transport equation with a non-linear metabolism (in symmetrical spherical co-ordinates):

\begin{equation*} \frac{\partial c}{\partial t}+\frac{K}{r^2}\frac{\partial c}{\partial r}+\frac{Da~c}{c+n}=0 \end{equation*}

My question is, if this is applied over some 3D sphere (with an outer and an inner sphere) then what would be the appropriate boundary/initial conditions? A no flux BC on the inner sphere seems to imply that concentration should be zero throughout the domain. (K,Da,n are constants).

I've been trying to figure out this problem for a while and was wondering what you thought. I have the PDE:

$\partial c \over \partial t$ + $K \over r^2$ $\partial c \over \partial r$ + $Da \ c \over c+n$ $=0$

The initial condition is $c(r,0) = c_{in}$ and boundary condition is $\partial c(R_{in},t) \over \partial r $ $= 0 $, where $R_{in}$ is the radius of an inner sphere (and $R_{out}$ the radius of an outer sphere). $K$ and $Da$ are both constants. $r$ is the radius measured from the centre of the sphere outwards, and $t$ time.

I have attempted to solve this using the method of characteristics but the solution keeps reducing to only depend on time. And even then I don't end up using the second Neumann boundary condition.

Does anyone have any ideas as to where I'm going wrong. Is it just the boundary conditions that are causing the problem?

I've been trying to figure out this problem for a while and was wondering what you thought. I have the PDE:

\begin{equation*} \frac{\partial c}{\partial t}+\frac{K}{r^2}\frac{\partial c}{\partial r}+\frac{Da~c}{c+n}=0 \end{equation*}

The initial condition is $c(r,0) = c_{in}$ and boundary condition is $\partial c(R_{in},t) \over \partial r $ $= 0 $, where $R_{in}$ is the radius of an inner sphere (and $R_{out}$ the radius of an outer sphere). $K$ and $Da$ are both constants. $r$ is the radius measured from the centre of the sphere outwards, and $t$ time.

I have attempted to solve this using the method of characteristics but the solution keeps reducing to only depend on time. And even then I don't end up using the second Neumann boundary condition.

Does anyone have any ideas as to where I'm going wrong. Is it just the boundary conditions that are causing the problem?