Inconsistent results for second derivative of inverse fourier transform

Questions
,
1

Hi,

I am trying to get the green’s function of a toy diffusion equation with Mathematica 9
1.PNG

Then solve by inverse fourier transform

In[3]:= u[x_, t_] := InverseFourierTransform[U[k, t], k, x]
D[u[x, t], {x, 2}]  
D[D[u[x, t], x], x]  

Out[4]= 0  
Out[5]= InverseFourierTransform[-k^2 U[k, t], k, x]

But can someone tell me why Out[4] and Out[5] are different? Thanks for your kindness.

2

Hi Guo,

Indeed this is strange, your second command in[4] should do the same I think as In[5].

Also:

D[u[x, t], {x, 2}]  
D[u[x, t], x,x]

give different answers, strange…

3

Hi Huisman,
Thanks for your reply. I’ve just contacted Wolfram technical support, and it should be a bug

Hello Guo Zhou,
Thank you for brining this issue to our attention.
I recreated the behavior you described and I found the issue can be seen when only examining the first derivative. Please see the attached notebook for details. I have filed a note with the appropriate members of our development team regardng this.
We are always interested in improving Mathematica and I want to thank you again for taking the time to contact us about this issue. If you have any further questions, or if you find further issues, please feel free to contact us again.

attached notebook

Simplifying issue.
Immediately below is what you found:
u[x_, t_] := InverseFourierTransform[U[k, t], k, x]
D[u[x, t], {x, 2}]
0
D[D[u[x, t], x], x]
InverseFourierTransform[-k^2 U[k, t], k, x]
However, the issue becomes more clear when we see that the first derivative is being handled differently:
D[u[x, t], x]
InverseFourierTransform[-I k U[k, t], k, x]
D[u[x, t], {x, 1}]
0
4

Hi Guo,

Sorry there is this bug in the system, unfortunately I don’t have the power/resources to fix it. For now, you have to find another way to solve your problem.