I'm currently studying the heat equation, and I find myself confused by a discrepancy in the sources I've referred to about using the fundamental solution to find general solutions of the homogeneous case.
According to the Wikipedia article, a fundamental solution $G$ for a linear differential operator $L$ is the solution to $Lu = \delta(x)$. You can find a solution to the inhomogeneous equation $Lu = f$ by taking the convolution of $G$ with $f$.
Now according to the Wikipedia article on the Heat equation, you can find a solution to the initial value problem
$$[\partial_t-\partial^2_x]u = 0, u(x, 0) = f(x)$$ by taking the convolution of $f$ with the fundamental solution of the heat equation,
$$S(x, t) = \frac{1}{\sqrt{4\pi t}} e^{\frac{-x^2}{4t}}.$$
This second proposition seems contradictory to the first one, but I have seen this framework for both in multiple other sources too.
Is this just an unspoken change in terminology/definitions, or is there something going on that I'm missing?
Best Answer
The thing you missed here is the difference between the two problems. One of the key concepts in studying PDEs is understanding the difference between elliptic and parabolic PDEs, and understanding which problems are relevant in which setting. The first problem, $Lu=f$, is an elliptic problem. There is no natural time coordinate here. Hence, there is no initial condition, but a function $f$ that holds everywhere in space. However, in the second problem, a parabolic problem, there is a very clear separation between the variable $t$ and $x$. In this case, having an initial condition $f$ feels natural, but having a function $f$ that would hold everywhere in space for all time would feel unnatural. Hence, in order to resolve your confusion, you must understand (and appreciate) the different roles played by $f$ in the two problems