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 Differential Equations Ordinary and Partial Differential Equations Math Forum

 December 1st, 2016, 11:11 AM #1 Newbie   Joined: Dec 2016 From: California Posts: 9 Thanks: 0 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 December 1st, 2016, 11:16 AM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,549 Thanks: 1399 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. Thanks from hydronate December 1st, 2016, 12:34 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 Math Focus: Mainly analysis and algebra 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. Thanks from hydronate December 1st, 2016, 01:44 PM #4 Newbie   Joined: Dec 2016 From: California Posts: 9 Thanks: 0 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 01:56 PM. December 1st, 2016, 01:59 PM #5 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2216 What do you solve to obtain $N(r)$? December 1st, 2016, 02:08 PM #6 Newbie   Joined: Dec 2016 From: California Posts: 9 Thanks: 0 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 02:11 PM. December 1st, 2016, 02:16 PM #7 Global Moderator   Joined: Dec 2006 Posts: 20,968 Thanks: 2216 Why the "dr" on both sides? Also, is that equation dimensionally correct? December 1st, 2016, 03:39 PM #8 Newbie   Joined: Dec 2016 From: California Posts: 9 Thanks: 0 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 $\displaystyle N(r)dr = N 4 \pi r^2 \rho \exp \Bigg( \frac{-4 \pi r^3}{3} \rho \Bigg) dr$ December 1st, 2016, 04:45 PM #9 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 Math Focus: Mainly analysis and algebra You might be interested in the profile of $N$ for different radii changes over time. In that case, the equation becomes a PDE. Thanks from hydronate December 2nd, 2016, 07:47 AM #10 Newbie   Joined: Dec 2016 From: California Posts: 9 Thanks: 0 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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