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December 1st, 2016, 12:11 PM   #1
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Whether to define system as PDE or ODE

Hey everyone.

I'm a grad student revising a journal article. Within the article, I layout an equation which describes of population "$\displaystyle n(r)$" through time. The parameters of this population vary with some variable "$\displaystyle r$"; however, $\displaystyle r$ does not change with time.

When I solve this system (written out below), I first solve for the "$\displaystyle N$-value" at every value of r. Afterwards, I use an ODE solver to evaluate the time evolution for "$\displaystyle n(r)$" at every $\displaystyle r$-value.

$\displaystyle \frac{dn(r)}{dt} = \frac{\dot{D}}{D_{0}} \bigg( N(r) - n(r) \bigg) - n(r) \exp \bigg( -\Delta E/k_B T \bigg) \frac{P(r) s}{P(r) + s}$

My question is this: is it strictly correct to describe this as an ordinary differential equation? Specifically, when I write the derivative $\displaystyle dn(r)/dt$, is this a mistake? A reviewer suggested that it is a mistake, writing "since you talk about a function of two variables, $\displaystyle n(r,t)$, the derivative should be written as $\displaystyle \partial n(r,t) / \partial t$." My reasoning in treating it as an ODE is that the solution to this equation requires differentiation with respect to $\displaystyle t$ only.

Thanks in advance!
Nathan
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December 1st, 2016, 12:16 PM   #2
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if the function is

$n(r, t)$ where $n()$ truly depends on time as well as on $r$ then your reviewer is correct.

On the other hand if the function is rather

$n(r(t))$ where $n()$ happens to vary with time only because it's single parameter $r$ varies with time then I'd say your notation is fine.
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December 1st, 2016, 01:34 PM   #3
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There is also the consideration as to whether $r$ is a variable or a parameter.

That is to say that if, for any given evolution of $n(t)$, the parameter $r$ remains fixed, then we have an equation in a single free variable which is therefore an ODE.

However, if $r$ is free to change within an evolution of $n$, we have a function $n(r,t)$ of two free variables and thus we have a PDE.
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December 1st, 2016, 02:44 PM   #4
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Yes, v8archie, this was my initial reasoning for describing the equation as an ODE. For every evolution of $\displaystyle n(t)$, the $\displaystyle r$-value is constant at all $\displaystyle t$-values.

In other words, the final goal is to describe how $\displaystyle n(t)$ evolves for every $\displaystyle r$-value, but the $\displaystyle r$-values are considered one-by-one during differentiation.

Last edited by hydronate; December 1st, 2016 at 02:56 PM.
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December 1st, 2016, 02:59 PM   #5
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What do you solve to obtain $N(r)$?
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December 1st, 2016, 03:08 PM   #6
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This is the expression which is evaluated first to determine the $\displaystyle N$-value associated with every $\displaystyle r$-value (the N and rho values on the right-hand side of the equation are constants):

$\displaystyle N(r)dr = N 4 \pi r^2 \rho \exp \Bigg( \frac{-4 \pi r^3}{3} \rho \Bigg) dr$

Last edited by hydronate; December 1st, 2016 at 03:11 PM.
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December 1st, 2016, 03:16 PM   #7
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Why the "dr" on both sides? Also, is that equation dimensionally correct?
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December 1st, 2016, 04:39 PM   #8
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That equation has been in the literature for a while now. It is certainly dimensionally correct on the r.h.s. of the equation. The l.h.s. is reproduced correctly from the literature, but is difficult to justify! (Seems to be m^-2)

The physical meaning is this. There are different concentrations of $\displaystyle N$ ($\displaystyle m^{-3}$) electrons and $\displaystyle \rho$ ($\displaystyle m^{-3}$) holes. As you move radially outwards from an electron site, the probability that you will find a hole within a thin shell between distance $\displaystyle r$ and $\displaystyle r+dr$ is $\displaystyle 4 \pi r^2 \rho \exp \Bigg( \frac{-4 \pi r^3}{3} \rho \Bigg) dr$. Assuming that the number of electrons is much less than the number of holes, the number of electrons with a nearest neighbor between $\displaystyle r$ and $\displaystyle r+dr$ is therefore $\displaystyle N(r)dr = N 4 \pi r^2 \rho \exp \Bigg( \frac{-4 \pi r^3}{3} \rho \Bigg) dr$
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December 1st, 2016, 05:45 PM   #9
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You might be interested in the profile of $N$ for different radii changes over time. In that case, the equation becomes a PDE.
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December 2nd, 2016, 08:47 AM   #10
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I am interested in how the profile of $\displaystyle N(r)$ changes through time, that is true. However, I only ever evaluate over the same $\displaystyle r$-values for all of my time evolutions.
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