Why is the Fourier Transform of symbolic Laplacian function (2nd partial derivative) not being found?
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I am needing to find the Fourier Transform of the following symbolic expression:
syms U(x,y,z) beta k
LHS = laplacian(U) + beta.^2*U
LHS_FT = fourier(LHS)
That is, ![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/709412/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/709412/image.png)
I'm needing to take the spatial 3D Fourier Transform of LHS.
This is the output I get:
LHS(x, y, z) =
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/709417/image.png)
LHS_FT(y, z) =
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/709422/image.png)
I am stuck here, any help would be much appreciated! Thank you in advance!
7 Comments
David Goodmanson
on 18 Aug 2021
Edited: David Goodmanson
on 18 Aug 2021
Hi Paul,
the answer was
LHS_FT =
beta^2*fourier(fourier(fourier(U(x, y, z), x, kx), y, ky), z, kz)
- kx^2*fourier(fourier(fourier(U(x, y, z), x, kx), y, ky), z, kz)
+ fourier(fourier(fourier(diff(U(x, y, z), y, y), x, kx), y, ky), z, kz)
+ fourier(fourier(fourier(diff(U(x, y, z), z, z), x, kx), y, ky), z, kz)
so it could do the conversion d^2/dx^2 --> -kx^2, but it couldn't convert d^2/dy^2 or d^2/dz^2, which would have made the nice symmetric expression
beta^2*fourier(fourier(fourier(U(x, y, z), x, kx), y, ky), z, kz)
- kx^2*fourier(fourier(fourier(U(x, y, z), x, kx), y, ky), z, kz)
- ky^2*fourier(fourier(fourier(U(x, y, z), x, kx), y, ky), z, kz)
- kz^2*fourier(fourier(fourier(U(x, y, z), x, kx), y, ky), z, kz)
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