# 3D heat diffusion equation in terms of convolution

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.

• It can be, see en.m.wikipedia.org/wiki/Heat_kernel – lisyarus Aug 14 at 7:54
• 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. – SameeraR Aug 15 at 2:35