Can a differential equation be applied to aperiodic data?

Question

I need to know if it’s appropriate to simply apply differential operators to variables from some dataset to form a differential equation and then solving. Or does it require additional steps or a different methodology?

For example, let’s say I have a weather dataset with the following variables:

1. Longitude: x
2. Latitude: y
3. Geopotential Height: z
4. Divergence: $d = d(x,y,z)$

The data when plotted looks something like this with divergence contoured in yellow. Just ignore the wind barbs and streamlines.

The differential equation for velocity potential, $\chi = \chi(x,y,z)$ is $\frac{\partial^2{\chi}}{\partial{x}^2} + \frac{\partial^2{\chi}}{\partial{y}^2}=d(x,y,z) $

Can I simply plug in the divergence for the right-hand-side and solve for the the value $\chi$?

Answer 1

I think you are asking whether you can reproduce the velocity data as gradient of $\chi$. The answer is probably no, for two reasons.

1. Without some boundary conditions $\chi$ is not unique; you can add any harmonic function to it, such as $x^2-y^2$.
2. Even if you deal with item 1, you get a gradient velocity field, while the real field is not really a gradient. Could be a reasonable approximation, or not.