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Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?

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Off hand, I don't know if this is generally true (I would expect not), but if the system involves N ODEs that are each of a single distinct variable, such that our original equation was

$\mathbb{O}_{i}\phi_{i} = 0$

where $\mathbb{O}_{i}$ is the relevant differential operator of a single variable, then it is always the case that you can replace the system with the PDE:

$$\left(\sum_{i} \mathbb{O}_{i}\right)\left(\Pi\phi_{i}\right) = 0$$

where $\Pi$ denotes repeated multiplication of all of the $\phi_{i}$. This is really just the inverse of the seperation of variables used in solving partial differential equations--i.e., when solving the hydrogen atom, you'll convert the PDE into three ODEs by assuming a form for the solution, and doing this in reverse.

Edit: example:

Concretely, you just promote the derivatives to partial derivatives. If f(x) is a solution to $\frac{d^{2}f}{dx^{2}} + V(x)f = 0$ and g(y) is a solution to $\frac{d^{2}g}{dy^{2}} + a(y)\frac{dg}{dy} = 0$, then $\phi(x,y) = f(x)g(y)$ is a solution to $\left(\frac{\partial^{2}}{\partial y^{2}} + a(y) \frac{\partial}{\partial y} + \frac{\partial^{2}}{\partial x^{2}} + V(x)\right)\phi = 0$, which should be obviously true.

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But the differential operators in the PDE case are different right? You use the same notation. Any chance you could show a minimal example? –  Sergio Parreiras Dec 17 '13 at 5:22
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@SergioParreiras: see edit. –  Jerry Schirmer Feb 8 at 17:47

In general, the answer is "no" (see, e.g., Commun. Math. Phys. 269, 545–556 (2007)). Sometimes it is possible though. For example, I proved that in a general case the Dirac equation is equivalent to a fourth-order partial differential equation for just one component (Journal of Mathematical Physics 52 (8): 082303 or http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf )

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