From my undergraduate studies I know that a fundamental solution to a partial differential operator $P$ is a distribution $u$ such that $Pu= \delta$ (no reference to any boundary or initial condition). Now, while reading about the heat equation I see that the heat kernel is said to be a fundamental solution for this equation, defined as a distribution $u$ satisfying $Pu=0$ and $\mathbb{lim}_{t \rightarrow 0} u(t, \cdot) = \delta$ (with $P$ the heat operator). I'm confused: are there multiple concepts of "fundamental solution"?

  • $\begingroup$ The two definitions are the same: Pu = delta with u(0)= 0; is equivalent to Pu = 0 with u(0) = delta. $\endgroup$ – Juan Ospina Sep 5 '14 at 20:01
  • $\begingroup$ @JuanOspina: Agreed, but you need an initial condition ($u(0)=0$) in order to have them equivalent, whereas the first definition that I mention does not need it. $\endgroup$ – Alex M. Sep 6 '14 at 16:48
  • $\begingroup$ These definitions come from physics, not from mathematics. So don't be surprised if they are context dependent. $\endgroup$ – Han de Bruijn Jul 16 '17 at 12:17

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