Answering on some posts on MSE about Laplace transform and Fourier transform I stumbled upon a question to which I cannot answer myself (not having a good ground in pure mathematics).

The question is the following:

Is there some mathematical constraint that doesn't let us use both Fourier and Laplace transform on the same equation?

I'm not saying that it would be useful in any case, I was just wondering if it's feasible! Just as an example I could use both transforms to solve the one dimensional (or three, doesn't change much) wave equation with some external force $$\begin{cases}\partial^2_{t}u(x,t) - c^2\partial^2_{x}u(x,t) = f(x,t)\\u(x,0) = \partial_tu(x,t)|_{t=0}=0\\-\infty\lt x\lt\infty\;\;\;\;t\gt0\end{cases}$$

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    $\begingroup$ You can use both as any times as makes sense. The two transforms are very similar in nature and the multidimensional versions are just iterated one dimensional transforms. A long winded way of saying yes. $\endgroup$ – copper.hat Aug 5 '18 at 21:35
  • $\begingroup$ Thanks! If you want, write it as an answer $\endgroup$ – Davide Morgante Aug 5 '18 at 21:45

The Laplace transform was really invented by Oliver Heaviside, who was studying time evolution systems, especially for the equations of Electromagnetics, and for electronic circuits. He reasoned that time evolution systems for unchanging systems must have an exponential property. That is, if you start with the initial state of the system $x_0$ and evolve it through $t$ seconds to obtain $E(t)x_0$, and then evolve that state through $t'$ more seconds, you should get the same thing if you were to evolve the initial state through $t+t'$ seconds. That led to the abstract solution operator with an exponential property: $$ E(t')E(t)x_0 = E(t'+t)x_0. $$ And that grew into the Laplace transform. The Laplace transform is normally used for the time variable on $[0,\infty)$. The Laplace transform is ideally suited for studying the time variable because of how it was designed, and it allows for instable systems with exponential blow-up in time, which happens with instable circuits, for example.

The Fourier transform was designed to work on a full spatial variable on $(-\infty,\infty)$ though the Fourier sine and cosine transforms work equally as well on $(0,\infty)$. These transforms are better suited for situations where the functions in the original variable vanish at the infinite endpoint(s).

There is nothing that would keep you from using the Laplace transform in time with a Fourier transform in space. But usually application of one or the other is enough to replace the derivatives in the transformed coordinate to something algebraic, and that may be the only reduction you need.

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