Why is the Fourier Transform of symbolic Laplacian function (2nd partial derivative) not being found?

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Emmanuel J Rodriguez
评论： Emmanuel J Rodriguez 2021-8-18
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,
I'm needing to take the spatial 3D Fourier Transform of LHS.
This is the output I get:
LHS(x, y, z) =
LHS_FT(y, z) =
I am stuck here, any help would be much appreciated! Thank you in advance!
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David Goodmanson
David Goodmanson 2021-8-18
编辑：David Goodmanson 2021-8-18
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)
Emmanuel J Rodriguez
Emmanuel J Rodriguez 2021-8-18
@David Goodmanson Thank you for that insight of having to include 3 k's, one corresponding to each spatial dimension. I may have to resort to solving this numerically, instead of symbolically to bypass the limitation.
@Paul The expected result should be:
** Expected result **
Applying 3D Fourier Transform
Here the represents the Fourier coefficient.
NOTE: I have put in a service request to MathWorks regarding this, and their response is "currently working with my colleagues to determine the nature of workflow and if this behavior is a potential enhancement/bug I will reach out to you at the earliest, once I have an update."
So it appears this is just not us :), and there is a potential limitation that MW will need to work out.

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