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Suppose I have a function of time $t$ and position $(x,y)$ such that \begin{equation} p_t \,dt = p \,dy - p_x (1-x) \,dx + p_y \,dy\end{equation} where the subscript denotes a differentiation. In this case, I am able to derive a (partial) differential equation from this form.

I'd love to have your help to address the case in which, for example, $dy$ appears also with higher orders. Something like: \begin{equation} p_t \,dt = p \,dy - p_x (1-x) \,dx + (dy)^2 (p-(1-y)p_y). \end{equation} or simpler (the key point is the presence of $(dy)^2$). I expect that in this case the pde will be second order...

Any idea?

P.S. I posted yesterday a similar question on the math.stackexchange but maybe it is more a physics-like question :)

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(i) I've never seen $dy^2$ before. I'm not sure if you can use differentials separately if they are not first order. (ii) You have a unit problem because all summands have the units of p, except the last one which has the units of p times the units of y. If this doesn't work out, probably everything else doesn't either. –  Rafael Reiter Jan 30 '13 at 9:39
    
Cross-posted from math.stackexchange.com/q/289777/11127 –  Qmechanic Jan 30 '13 at 9:42
    
Ok, maybe I am wrong at deriving the first equation. The original problem is discrete. Maybe I have done some mistake in the process. The problem is the following. Imagine to have the following relation describing the evolution of a system: $p_{i,j,k+1} = p_{i,j,k}(j/N^2+i/N)+p_{i,j-1,k} (N^2+1-j)/N^2 + p_{i-1,j,k} (1-i/N)$. Imagine that $k$ is the discrete time, while $i$ and $j$ are discrete coordinates. The initial condition is $p_{i,j,0}=\delta_i \delta_j$. When $N$ goes to infinity, this relation should be represented by a PDE where, for example, $i/N$ tends to $x$, etc –  Guido Jan 30 '13 at 11:06
    
One can see $p(x,y,t)$ for a fixed time as the superficial density of a pollutant and the PDE should be something like transport and diffusion... This is my idea, but I do not still know how to derive this PDE... –  Guido Jan 30 '13 at 11:09
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2 Answers 2

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The $(dy)^2$ term is totally negligible, it's as if it was not there. If you had two differentials everywhere, then yes, it would lead to a 2nd order diff. equation.

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I am taking the comments and equations in the OP edits as part of this question which therefore has both a mathematical part and a physics part.

If we take the equation

$$ p_{i,j,k+1} = p_{i,j,k}(j/N^{2} + i/N) + .... $$

as the initial discrete equation here, then if we take the limit as $N \rightarrow \infty$ such that $i/N \rightarrow x$ and $j/N \rightarrow y$ we are going to find that $i/N^2 \rightarrow 0$ and $j/N^2 \rightarrow 0$. Consequently those (inverse) square terms drop out of the "continuum" version and we are left with just the first order terms and not any second order terms. (Incidentally when I do this approximation to the equation above I dont quite get the first term in the Question, but a similar first order expression, but I am not sure which of these variants is really significant, for reasons I shall discuss next.)

However the reason you give for wanting a second order equation in the first place is to model the movement of pollutants. Now these are modelled by equations like the Convection-Diffusion Equation. Here is a simplification:

$$ \frac{\partial c}{\partial t} = D \nabla ^2 c - v \cdot \nabla c $$

This equation is first order in time and has two components on the RHS:

$D \nabla ^2 c$ deriving from the Diffusion current (D is diffusion constant) and is second order.

$ v \cdot \nabla c $ deriving from the Advection current and is first order.

So it is the diffusion process which introduces second order derivatives into this equation. This in turn is because of Ficks first Law, and you might want to study the derivation of that expression at the end of the WP article, since it is derived in one dimension from movement in a grid $\Delta x$ space units wide and in time units $\Delta t$.

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Thanks for the link in WP. Maybe in my case a continuous version of the discrete problem is not suited for the analysis of the discrete problem itself. My idea was that, since the recurrence equation under analysis is difficult to solve, the solution of a continuous version such as a PDE could have been useful... –  Guido Jan 30 '13 at 15:04
    
@Guido, Actually your discrete problem looks like a kind of "one way diffusion". I wonder whether modelling it on a Cellular Automata framework would be possible? –  Roy Simpson Jan 30 '13 at 16:37
    
Maybe, but I do not think that in that way it would be easier to find a closed form solution. And yes, it is just like a one way diffusion and the transport is in the positive $x$ positive $y$ direction. In the discrete model, as time goes to infinity, the solution tends to $(1,1)$ (when normalized with respect to $N$. If the vertical motion is not present in the continuous form, I do not know if this would still be the case. –  Guido Jan 30 '13 at 18:30
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