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

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Emmanuel J Rodriguez 2021-8-11

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https://ww2.mathworks.cn/matlabcentral/answers/896597-why-is-the-fourier-transform-of-symbolic-laplacian-function-2nd-partial-derivative-not-being-found

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 2021-8-18

编辑：David Goodmanson 2021-8-18

在 MATLAB Online 中打开

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 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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