3
$\begingroup$

I am currently working on a diffusion problem involving partial differential equations and am a bit confused about how the Laplacian (Laplace?) operator works on different independent variables such as time and space. Here is a picture of the question: PDE Problem involving space and time.

Now what I initially tried to do was expand out the laplace term into a second derivative in x and a second derivative in time. However, in the solutions which are provided in the following picture, they do not expand out a second derivative in time and I am confused as to why they would only do it in x. Solution to PDE. Does the laplace operator only work on spacial variables then? Thank you for your help.

Note the method being used to solve the problem is a separation of variables.

$\endgroup$
1
  • 1
    $\begingroup$ The adjectival analogue of spacial is temporal, by the way. $\endgroup$ Commented Feb 12, 2018 at 2:20

1 Answer 1

3
$\begingroup$

Usually when people write a Laplacian when there is a clear "time" variable and "space" variable(s), the Laplacian only hits the space variables. Sometimes people will explicitly write $\Delta_x$ or $\nabla^2_x$ to denote this, but usually it's clear from context. Here for instance they are talking about diffusion which definitely does not have a second time derivative involved.

$\endgroup$
4
  • $\begingroup$ Would there ever be a case where the second derivative in time was involved in a problem? Is that possible or even physically meaningful to talk about the second derivative in time? $\endgroup$ Commented Feb 12, 2018 at 1:12
  • 3
    $\begingroup$ You may be interested in studying $$\Delta u = u_{tt}$$, yes. This is the wave equation. $\endgroup$ Commented Feb 12, 2018 at 1:23
  • 1
    $\begingroup$ @AndrewSchroeder sure it's physically meaningful, this is acceleration! $\endgroup$ Commented Feb 12, 2018 at 3:59
  • $\begingroup$ Oh rip me lol, didn't even come to mind aha $\endgroup$ Commented Feb 13, 2018 at 1:17

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .