Fourier Transform of y'+y''

Questions
1

Hello, Can someone help me?
Mathematica is able to compute fourier transform of y’ and y’’ in term of unknown function y -individually- but appears it can’t evaluate Fourier transform of y’+y’’ , See picture:

In[1]:= FourierTransform[ \!\(
\*SubscriptBox[\(\[PartialD]\), \(r\)]\(y[r]\)\), r, \[Xi]]

Out[1]= -I \[Xi] FourierTransform[y[r], r, \[Xi]]

In[2]:= FourierTransform[ \!\(
\*SubscriptBox[\(\[PartialD]\), \({r, 2}\)]\(y[r]\)\), r, \[Xi]]

Out[2]= -\[Xi]^2 FourierTransform[y[r], r, \[Xi]]

In[3]:= FourierTransform[ \!\(
\*SubscriptBox[\(\[PartialD]\), \(r\)]\(y[r]\)\) + \!\(
\*SubscriptBox[\(\[PartialD]\), \({r, 2}\)]\(y[r]\)\), r, \[Xi]]

Out[3]= FourierTransform[
 Derivative[1][y][r] + (y^\[Prime]\[Prime])[r], r, \[Xi]]
2

Also Integrate does not apply linearity by default:

Integrate[y'[x] + y''[x], x]
Integrate[y'[x] + y''[x], {x, 0, 1}]

Perhaps it is because the two terms may have singularities that cancel out when summed.

You can force FourierTransform to apply linearity, for example with a replacement rule:

fourierLinearity = 
 HoldPattern@FourierTransform[a_ + b_, c__] :> 
  FourierTransform[a, c] + FourierTransform[b, c]; 
FourierTransform[y'[x] + y''[x], x, \[Xi]] //. fourierLinearity
3

I agree, I think it’s a bug in FourierTransform[], it should apply linearity!

I don’t see the problem with Integrate:

In[38]:= Integrate[y’[x] + y’'[x], x]

Out[38]= y[x] + Derivative[1][y][x]
4

It looks like the other integral transforms apply linearity as well:
In[1]:= LaplaceTransform[y’[x] + y’'[x], x, s]

Out[1]= s LaplaceTransform[y[x], x, s] + 
 s^2 LaplaceTransform[y[x], x, s] - y[0] - s y[0] - Derivative[1][y][0]

In[2]:= FourierSinTransform[y'[t] + y''[t], t, \[Omega]]

Out[2]= -\[Omega] (FourierCosTransform[y[t], 
    t, \[Omega]] + \[Omega] FourierSinTransform[y[t], t, \[Omega]] - 
   Sqrt[2/\[Pi]] y[0])

In[3]:= FourierTransform[y'[t] + y''[t], t, \[Omega]]

Out[3]= FourierTransform[
 Derivative[1][y][t] + (y^\[Prime]\[Prime])[t], t, \[Omega]]

In[4]:= FourierCosTransform[y'[t] + y''[t], t, \[Omega]]

Out[4]= -\[Omega]^2 FourierCosTransform[y[t], 
   t, \[Omega]] + \[Omega] FourierSinTransform[y[t], t, \[Omega]] - 
 Sqrt[2/\[Pi]] (y[0] + Derivative[1][y][0])

I didn’t test other generalities:

5

If Integrate applied linearity by default, it would get the following integral wrong:

Integrate[1/x - 1/Sin[x], {x, 0, 1}]
6

Thank you very much.
I enjoyed from your helpful Comments.
I learned very important points … Good luck.

7

Unfortunately version 10.4 still has not tackled the Fourier transform.

8

Unfortunately 10.4.1 still hasn’t fixed the FourierTransform linearity issue :frowning: maybe someone can pass this on to the right person?

In[6]:= FourierTransform[y'[t], t, \[Omega]]

Out[6]= -I \[Omega] FourierTransform[y[t], t, \[Omega]]

In[7]:= FourierTransform[y''[t], t, \[Omega]]

Out[7]= -\[Omega]^2 FourierTransform[y[t], t, \[Omega]]

In[8]:= FourierTransform[y'[t] + y''[t], t, \[Omega]]

Out[8]= FourierTransform[
 Derivative[1][y][t] + (y^\[Prime]\[Prime])[t], t, \[Omega]]