# Can I take the Fourier transform of a PDE in both sides? [duplicate]

I want to know if it is possible to use a PDE in the function FourierTransform instead of a function.

Consider for example the simple case for the heat equation

Clear["Global*"]
uf=
FourierTransform[
Derivative[2, 0][u[x, t]] == Derivative[0, 1][u[x, t]], x, k]


I tried to run this but it doesn't work, is there any way I could make this work?

Als

## 2 Answers

You can do it in two steps. Let $$U(\omega, t)$$ be the Fourier transform of $$u(x,t)$$, and using the relation that $$F \left(\frac{\partial u}{\partial t}\right) = \frac{\partial}{\partial t} U(\omega,t)$$ then we have (ps. I am using $$\omega$$ for your $$k$$ as I find it more clear)

Clear["Global*"]
uf = FourierTransform[D[u[x,t],{x,2}],x,w]==FourierTransform[D[u[x,t],t],x,w]
uf = uf/.FourierTransform[u[x,t],x,w]->U[w,t]
uf/.FourierTransform[(u^(0,1))[x,t],x,w]->D[U[w,t],t]


Using ApplySides:

uf = ApplySides[FourierTransform[#, x, w] &, D[u[x, t], {x, 2}] == D[u[x, t], t]]


And you do the following that @Nasser points out.