Could someone please explain this notation for a partial differential equation?

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The example is from encyclopedia of mathematics.

I'm familiar with the notation $u_x=\frac{\delta{u}}{\delta{x}}$.

But I do not understand what the numeral 4 represents in $u_{x^4}$.

Is it the fourth derivate perhaps?

This type of notation is not explained on the wikipedia page for PDE notation.

  • 2
    My guess would be that this is formatted as (for example) u_{x^4}, which is rendered here as $u_{x^4}$. This would be a slightly nonstandard way of writing $u_{xxxx}$, i.e. the fourth derivative in the direction of $x$. Personally, I would prefer $$\frac{\partial^4}{\partial x^4} u, $$ or possibly, using multiindex notation, $\partial^{(4,0,0)} u(x,y,z)$. – Xander Henderson May 22 at 0:35
up vote 2 down vote accepted

It means the fourth derivative with respect to the same variable. It is written like that just to avoid long indices. For example :

$$u_{x^4} = u_{xxxx}, \; u_{y^4} = u_{yyyy}, \; u_{z^4} = u_{zzzz}$$

  • Excellent thanks. I will add that explanation to wikipedia. – Bill May 22 at 0:27
  • No problem ! Sometimes notations are not widespread used but one can interpret them from the given problem (for example elliptic equations are with respect to bigger differentiate factors, and the numeral index couldn't mean many more things). If you consider your question answered you can accept the answer ! – Rebellos May 22 at 0:29
  • I have to wait 8 mins before accepting. – Bill May 22 at 0:30
  • An editor at wikipedia removed my edit of the page saying "Extremely uncommon notation". I guess someone should inform encyclopedia of mathematics about that. – Bill May 23 at 19:52
  • It's indeed uncommon but makes sense. – Rebellos May 23 at 19:56

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