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Consider the following partial differential equation:

$\frac{\partial y(x,t)}{\partial t} = \theta \nabla^2y(x,t)$

where $x \in R^2$ is the spatial index and $t \in Z^+$ is the temporal index. Can you please confirm if the Laplacian operation is obtained from the Hessian matrix which is $3 \times 3$ (spatial and temporal factors) or $2 \times 2$ (only spatial factors) ?

I do not know much about PDEs and my apologizes if this question is too elementary.

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    $\begingroup$ Conventionally the Laplacian is only the spacial variables. $\endgroup$ – Chappers Aug 25 '17 at 21:56
  • $\begingroup$ unless otherwise noted, Chappers is right. Sometimes we'll denote $\Delta_t$ to say if we're talking about a laplacian w.r.t time or another variable. $\endgroup$ – DaveNine Aug 26 '17 at 5:50

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