# Applying the fourier transform to a PDE

1. Nov 19, 2013

### _Stew_

I have a tutorial question for maths involving the heat equation and fourier transform.

${\frac{∂u}{∂t}} = {\frac{∂^2u}{∂x^2}}$

you are given the initial condition:

$u(x,0) = 70e^{-{\frac{1}{2}}{x^2}}$

$u(x,t) = {\frac{70}{\sqrt{1+2t}}}{e^{-{\frac{x^2}{2+4t}}}}$

In this course the definition of a fourier transform is:

$U(k) = F${$u(x)$}$= {\frac{1}{\sqrt{2π}}}{\int_{-∞}^{∞}{{u(x)}{e^{-ikx}}}\,dx}$

$u(x) = F^{-1}${$U(k)$}$= {\frac{1}{\sqrt{2π}}}{\int_{-∞}^{∞}{{U(k)}{e^{+ikx}}}\,dk}$

MY SOLUTION:

So far I can use the transform with the pde to obtain
$U(k,t) = A(k).e^{-k^{2}t}$
$U(k,0) = A(k)$

From the initial condition and a transform table:
$U(k,0) = 70.e^{-{\frac{1}{2}}k^2}$
$U(k,t) = 70.e^{-{\frac{1}{2}}k^2}.e^{-k^{2}t}$

from here it looks like I would have to apply the inverse fourier transform to get u(x,t). This is a difficult integration an I don't remember it being this complicated the first time. So basically:

Is there an easier way to do this question?
Any tips for the inverse fourier integration below?

$u(x,t) = {\frac{1}{\sqrt{2π}}}{\int_{-∞}^{∞}{{70.e^{-{\frac{1}{2}}k^2}.e^{-k^{2}t}}{e^{+ikx}}}\,dk}$

2. Nov 21, 2013

### vela

Staff Emeritus
You have a Fourier transform of a Gaussian
$$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty e^{-\alpha k^2} e^{ikx}\,dk$$ where $\alpha = t+\frac{1}{2}$, so you could just look up the answer in a table of Fourier transform pairs. If you want to derive the result, combine the exponentials and then complete the square.