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The solution to 1-D heat equation can be expressed via 1-D convolution. More formally, let $\frac{\partial u}{\partial t} = -k \frac{\partial^{2}}{\partial x^{2}}$, then we can find a solution $u = \int f(s)h_t(x-s)ds$.

However, when we consider the cartesian 3D coordinates ($\frac{\partial u}{\partial t} = -k \Delta u$), why cannot the solution be expressed as a convolution in 3 dimensions?

Thank you.

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    $\begingroup$ It can be, see en.m.wikipedia.org/wiki/Heat_kernel $\endgroup$ – lisyarus Aug 14 at 7:54
  • $\begingroup$ Thank you for your comment, I can't seem to understand the how link illustrates 3D convolution. Can you please elaborate with more details and put it as an answer, so I can clearly understand it? Thanks a lot. $\endgroup$ – SameeraR Aug 15 at 2:35

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